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User's Guide to Pari/GP - PARI/GP Development Headquarters

1. 008 257 3 10 Plotting functions sean a ak D A ee ae O a A 272 3 11 Programming in GP control statements a a 278 3 12 Programming in GP other specific functions ooa a e 287 3 13 Parallel programming ooa 302 BTA GP defaults mana ake ata AA A a ae A eee ee s 304 Appendix A Installation Guide for the UNIX Versions 315 TIE lt lt ae Sle eee la hee a A e Beh Genes a das 325 Chapter 1 Overview of the PARI system 1 1 Introduction PARI GP is a specialized computer algebra system primarily aimed at number theorists but has been put to good use in many other different fields from topology or numerical analysis to physics Although quite an amount of symbolic manipulation is possible PARI does badly compared to systems like Axiom Magma Maple Mathematica Maxima or Reduce on such tasks e g mul tivariate polynomials formal integration etc On the other hand the three main advantages of the system are its speed the possibility of using directly data types which are familiar to mathe maticians and its extensive algebraic number theory module from the above mentioned systems only Magma provides similar features Non mathematical strong points include the possibility to program either in high level scripting languages or with the PARI library a mature system development started in the mid eighties that was used to conduct and disseminate original mathematical research wh
2. 267 ONG sk ep oe a e Oe RKO 70 and yi deo He Bog Gee Bow ee ee we A eT anell aros of ew oe oes A ee 133 anellsmall 133 APPLY eea bc a ee e a 10 290 afea irc ee rA ee 130 OL de ooo ee a ek Ban dO 90 Artin L function 166 Artin root number 166 ASTD e e ul bn amp Only ax eo SS Oe 90 asin 6 a tee g oe hh eS SEES RSD 90 atan 25 2 406 oe ee GS bE See 90 atanh AI 91 automatic simplification 311 available commands 57 325 B AAA e eee ee ee 130 e eee o HAG OR Ga ee oe 130 DO Sass Be eek Ge ee ee ee 130 DE aaa s Goa RS Bois ace a 130 backslash character 16 basistoal Po ei a tae eee es 187 Berlekamp s sue cee anid ee ee ES 110 bermtrac 6 x x wees Bbw a ae Bw Ees 91 Bernoulli numbers 91 98 Bernoulli polynomial 91 Dermpol 2 25 6 bebe rastaga pri 91 bernreal a p e a po ew He ER dod ew a 91 DEYNVEC es 4 4 hb beh oO OE ek eS a a 91 besselbl of wo ee beds Be ee EOS 91 besselh2 o o 91 bessellc sisi al as es 91 DESSOL reos ede o s gi besseljh s s a sate hie wc on aH ae as 92 bess lk Visores REE EDS 92 besseln 0 0008 92 bestappr 78 100 101 bestapprPade 101 Bezout relation 114 DOZOUE esbirros Sa Se 101 113 DEZOUTTES 22 be eee ee ee 215 A ee ee A 44 154 DIA x acs dae ace aes A eke ee he ee 155 bigomega o
3. 42 TECULSION Fasa ba SA ee es ee 42 recursive P O 274 FOdimag escocesa 123 TOTS AL ee a ee ea e ea e 123 redrealnod 0 123 reduceddiscsmith 220 reduction sos e aos ague por AE miN 122 123 reference card o 56 TOS 3 ei ee we eS be ee ee 155 removeprimes 125 306 BEtuUrM a este e A doe a Se a Be 50 287 Thoreal 24 24 46 408 Se Soe cda 123 rhorealnod 2 24244 244 oe e bs 123 Riemann zeta function 40 98 TN oo oe eg a BPS ae ee ee ee ew G 153 rnfaletobasl8 dvi 202 o A vs ee ee oh hei ae A 202 rnfbasistoalg8 c0o o 6 be eee g 202 rnfcharpoly gt x es ta soaa ap o 202 mnfconductor 2 8 ee won a ea ee A a 209 rnfdedekind 203 204 a Gew ma eg Dei aa e A N oe h 204 PNTdIS G e dd a a E ada A 204 rnfdiscf 12 204 rnfeltabstorel 204 rnfeltdown 204 205 rnfeltnorm s e s o sate eee as 205 rnfeltreltoabs 205 rnfelttrace 205 206 rmnfeltup 2 26855 ee eH ras 206 rnfequation 206 207 rnfequationO 207 rnfequation2 c eces 268 822 h4 44 207 YNTHNTDASTS s x eere aoa a e a 207 rnfidealabstorel 207 rnfidealdown 207 rnfidealhnf 208 rnfidealmul 208 rnfidealnormabs 208 rnfidealnormrel 208 rnfidealreltoabs 208 rnfidealtwoelement 208 r
4. Then the Taylor series expansion of the function around X 0 where X is the main variable is computed to a number of terms depending on the number of terms of the argument and the function being computed Under gp this again is transparent to the user When programming in library mode however it is strongly advised to perform an explicit conversion to a power series first as in x gtoser x seriesprec where the number of significant terms seriesprec can be specified explicitly If you do not do this a global variable precdl is used instead to convert polynomials and rational functions to a power series with a reasonable number of terms tampering with the value of this global variable is deprecated and strongly discouraged e If the argument is a vector or a matrix the result is the componentwise evaluation of the function In particular transcendental functions on square matrices which are not implemented in the present version 2 7 0 will have a different name if they are implemented some day 3 3 1 If y is not of type integer x y has the same effect as exp y log x It can be applied to p adic numbers as well as to the more usual types The library syntax is GEN gpow GEN x GEN n long prec for xn 3 3 2 Catalan Catalan s constant G gt p gt 0 Ashe 0 91596 Note that Catalan is one of the few reserved names which cannot be used for user variables The library syntax is GEN mpcatalan long prec 3 3 3 Euler
5. derivnum x 0 sin exp x cos 1 1 1 262177448 E 29 A clumsier approach which would not work in library mode is f x sin exp x f 0 cos 1 1 1 262177448 E 29 When a is a power series compute derivnum t a f as f a f a a The library syntax is derivnum void E GEN eval void GEN GEN a long prec Also available is GEN derivfun void E GEN eval void GEN GEN a long prec which also allows power series for a 257 3 9 2 intcirc X a R expr taby Numerical integration of 2i1 7 expr with respect to X on the circle X a R In other words when expr is a meromorphic function sum of the residues in the corresponding disk tab is as in intnum except that if computed with intnuminit it should be with the endpoints 1 1 p105 intcirc s 1 0 5 zeta s 1 1 2 398082982 E 104 7 94487211 E 107xI The library syntax is intcirc void E GEN eval void GEN GEN a GEN R GEN tab long prec 3 9 3 intfouriercos X a b z expr taby Numerical integration of expr X cos 272X from a to b in other words Fourier cosine transform from a to b of the function represented by expr Endpoints a and b are coded as in intnum and are not necessarily at infinity but if they are oscillations i e 1 a1 are forbidden The library syntax is intfouriercos void E GEN eval void GEN GEN a GEN b GEN z GEN tab long prec 3 9 4 intfourierexp X
6. x OK where all three proposed continuations would be valid 48 Runtime errors from the evaluator are nicer because they answer a correctly worded query otherwise the bytecode compiler would have protested first here is a slightly pathological case if siN x lt eps do_something at top level if siN x lt eps do_someth Hook not a function in function call no arrow The code is syntactically correct and compiled correctly even though the siN function a typo for sin was not defined at this point When trying to evaluate the bytecode however it turned out that siN is still undefined so we cannot evaluate the function call siN x Library runtime errors are even nicer because they have more mathematical content which is easier to grasp than a parser s logic 1 Mod 2 4 at top level 1 Mod 2 4 40k Wee aa xxx _ _ impossible inverse in Fp_inv Mod 2 4 telling us that a runtime error occurred while evaluating the binary operator the surrounding the operator are placeholders more precisely the Fp_inv library function was fed the argument Mod 2 4 and could not invert it More context is provided if the error occurs deep in the call chain f x 1 x g N for i N N i 0 5 g 10 kk at top level g 10 xk dust KK in function g for i N N f i xk Pesos KK in function f 1 x CK ae xxx _ _ impossible inverse in ginv 0 b In this ex
7. 257 d l ese eee he OR aaa 230 COCO sea 6 teas ee ee amp we Meee ee aS 235 OG 2 a be ae ieee des he GR nw ae 235 detint siroa iie ess eo do 235 diagonal sico 3 eae ee Gs 236 Giff sd oa baw Sow ke A ee ee ee eS 155 difference 00004 64 dIELOD is See e eG eS 216 ditlopO sig cavas Bare e 216 digits sessa se eu bra eoe a s 80 dilog eame ee AA we 92 GQirdiy 44644 24284 Cie Ree es 105 Gireuler og eo wn GE KE ee A a ea 105 Dirichlet series 105 168 Girmul ua pa i ac 2h oS oS 105 dirzetak lt soo oeoa oe amope ee es 168 SC pd 6 644 gi 25 ei soe 2A 130 155 GIVUSOES osor ans p ghee E Ehe a ht 105 279 divremn aci a Bae 33 68 AVE asis i ea ae Eas ae BG wd 62 dynamic scoping 34 E OCHO cera teoa ao hey i eR es 57 305 ECM e cee e e e 2 a a a a a A 99 110 editing characters 16 Egyptian fraction 104 Gigon si a ek Le Ra Re ee 236 GING 2 5 eae eek ee Ses ee OG oe 92 elementary divisors 242 Cl nd RGB Re EAR RRS 44 130 141 ell bbe m aop ee ee ew 140 elladd cocos ede a ey ce eee 133 ellak pee we e eee ee Eek Ee 133 CUAD dis a ete eee de oe ee 133 ellanalyticrank 132 133 134 ellap e si ge as eee oS 134 135 ellbil weed we bs He ee ee Hwee 135 ellcard isis ia oR eee 8 135 ellchangecurve 135 ellchangepoint 135 ellchangepointinv 135 136 ellconvertname 136 146 elldata 1
8. a b b c 2 3 The PARI types We see here how to input values of the different data types known to PARI Recall that blanks are ignored in any expression which is not a string see below A note on efficiency The following types are provided for convenience not for speed t_INTMOD t_FRAC t_PADIC t_QUAD t_POLMOD t_RFRAC Indeed they always perform a reduction of some kind after each basic operation even though it is usually more efficient to perform a single reduction at the end of some complex computation For instance in a convolution product gt jan CY in Z NZ common when multiplying polynomials it is quite wasteful to perform n reductions modulo N In short basic individual operations on these types are fast but recursive objects with such components could be handled more efficiently programming with libpari will save large constant factors here compared to GP 2 3 1 Integers t_INT After an optional leading or type in the decimal digits of your integer No decimal point 1234567 1 1234567 3 42 3 1 oops not an integer 3 1 000000000000000000000000000 2 3 2 Real numbers t_REAL Real numbers are represented approximately in a floating point system internally in base 2 but converted to base 10 for input output purposes A t_REAL object has a given accuracy or precision gt 0 it comprises e asign s 1 1 or 0 e a mantissa m a multiprecision integer
9. c6 216 e disc discriminant of the curve This is only required to be non zero not necessarily a unit e j j invariant of the curve These are used as follows E ellinit 0 0 0 a4 a6 E b4 42 2 xa4 E disc 3 64 a473 432 a672 130 3 5 1 2 Curves over R This in particular includes curves defined over Q All member functions in this section return data as it is currently stored in the structure if present and otherwise compute it to the default accuracy that was fixed at the time of ellinit via a t_REAL D domain argument or realprecision by default The function ellperiods allows to recompute and cache the following data to current realprecision e area volume of the complex lattice defining E e roots is a vector whose three components contain the complex roots of the right hand side g x of the associated b model Y g x If the roots are all real they are ordered by decreasing value If only one is real it is the first component e omega w1 w2 periods forming a basis of the complex lattice defining E The first com ponent w is the positive real period in other words the integral of dx 2y a x az over the connected component of the identity component of E R The second component wa is a complex period such that 7 FF belongs to Poincar s half plane positive imaginary part not necessarily to the standard fundamental domain e eta is a row vector containing the quasi p
10. for i 1 1074 sum i 1 100 i 1 x7i time 3 985ms Polynomials are always printed as univariate polynomials with monomials sorted by decreasing degree x y 1 72 11 x72 Qey 2 x y72 Qty 1 Univariate polynomial in x whose coefficients are polynomials in y See Section 2 5 for valid variable names and a discussion of multivariate polynomial rings 2 3 11 Power series t_SER Typing 0 X k where k is an integer yields an X adic 0 of accuracy k representing any power series in X whose valuation is gt k Of course X can be replaced by any other variable name To input a general non 0 power series type in a polynomial or rational function in X say and add 0 X k to it The discussion in the t_POL section about variables remains valid a constructor Ser replaces Pol and Polrev Caveat Power series with inexact coefficients sometimes have a non intuitive behavior if k significant terms are requested an inexact zero is counted as significant even if it is the coefficient of lowest degree This means that useful higher order terms may be disregarded If a series with a zero leading coefficient must be inverted then as a desperation measure that coefficient is discarded and a warning is issued 7C 0 y 0 y72 1 C xxx _ _ Warning normalizing a series with O leading term 2 y 1 0 1 The last output could be construed as a bug since it is a priori impossible to deduce such a resu
11. forgfvec v 3 2 2 3 3 print v o 1 1 0 Et ls The library syntax is void forqfvecO GEN v GEN q NULL GEN b The following func tion is also available void forgfvec void E long fun void GEN double GEN q GEN b Evaluate fun E v m on all v such that q v lt b where v is a t_VECSMALL and m q v is a C double The function fun must return 0 unless forgfvec should stop in which case it should return 1 3 8 5 lindep v flag 0 finds a small non trivial integral linear combination between compo nents of v If none can be found return an empty vector If v is a vector with real complex entries we use a floating point variable precision LLL algorithm If flag 0 the accuracy is chosen internally using a crude heuristic If flag gt 0 the computation is done with an accuracy of flag decimal digits To get meaningful results in the latter case the parameter flag should be smaller than the number of correct decimal digits in the input lindep sqrt 2 sqrt 3 sqrt 2 sqrt 3 1 1 1 1 If v is p adic flag is ignored and the algorithm LLL reduces a suitable dual lattice lindep 1 2 3 3 2 3 3 374 0 375 231 12 1 2 If v is a matrix flag is ignored and the function returns a non trivial kernel vector combination of the columns lindep 1 2 3 4 5 6 7 8 9 13 1 2 1 If v contains polynomials or power series over some base field finds a l
12. 1 my E ellinit 1 3 V vector 12 i randomprime 27200 parapply p gt ellcard E p V computes the order of E F for 12 random primes of 200 bits The library syntax is GEN parapply GEN f GEN x 3 13 2 pareval x Parallel evaluation of the elements of x where x is a vector of closures The closures must be of arity 0 must not access global variables or variables declared with local and must be free of side effects The library syntax is GEN pareval GEN x 3 13 3 parfor i a b expr1 j expr2 Evaluates the sequence expr2 dependent on i and j for i between a and b in random order computed in parallel in this sequence expr2 substitute the variable j by the value of expr1 dependent on i If b is omitted the loop will not stop It is allowed for expr2 to exit the loop using break next return however in that case expr2 will still be evaluated for all remaining value of less than the current one unless a subsequent break next return happens 3 13 4 parforprime p a b expr1 j expr2 Evaluates the sequence expr2 dependent on p and j for p prime between a and b in random order computed in parallel Substitute for j the value of expr1 dependent on p If b is omitted the loop will not stop It is allowed fo expr2 to exit the loop using break next return however in that case expr2 will still be evaluated for all remaining value of p less than the current one unless a subsequent break next
13. The factors are normalized so that their leading coefficient is a power of p The method used is a modified version of the round 4 algorithm of Zassenhaus If pol has inexact t_PADIC coefficients this is not always well defined in this case the poly nomial is first made integral by dividing out the p adic content then lifted to Z using truncate coefficientwise Hence we actually factor exactly a polynomial which is only p adically close to the input To avoid pitfalls we advise to only factor polynomials with exact rational coefficients The library syntax is factorpadic GEN f GEN p long r The function factorpadicoO is deprecated provided for backward compatibility 3 7 7 intformal z v formal integration of x with respect to the variable v wrt the main variable if v is omitted Since PARI cannot represent logarithmic or arctangent terms any such term in the result will yield an error intformal x 2 1 1 3 x 3 intformal x 2 y 12 y x 2 intformal 1 x at top level intformal 1 x kkk xxx intformal domain error in intformal residue series pole 0 The argument x can be of any type When z is a rational function we assume that the base ring is an integral domain of characteristic zero By definition the main variable of a t_POLMOD is the main variable among the coefficients from its two polynomial components representative and modulus in other words assuming a polmod represents a
14. The library syntax is GEN gtopoly GEN t long v 1 where v is a variable number 3 2 7 Polrev t v x Transform the object t into a polynomial with main variable v If t is a scalar this gives a constant polynomial If t is a power series the effect is identical to truncate i e it chops off the O X The main use of this function is when t is a vector it creates the polynomial whose coefficients are given by t with t 1 being the constant term Pol can be used if one wants t 1 to be the leading coefficient Polrev 1 2 3 1 3 x72 Qex 1 Pol 1 2 3 2 2 2 24x 3 The reciprocal function of Pol resp Polrev is Vec resp Vecrev The library syntax is GEN gtopolyrev GEN t long v 1 where v is a variable number 3 2 8 Qfb a b c D 0 Creates the binary quadratic form ax bry cy If b 4ac gt 0 initialize Shanks distance function to D Negative definite forms are not implemented use their positive definite counterpart instead The library syntax is GEN QfbO GEN a GEN b GEN c GEN D NULL long prec Also available are GEN qfi GEN a GEN b GEN c assumes b 4ac lt 0 and GEN qfr GEN a GEN b GEN c GEN D assumes b 4ac gt 0 3 2 9 Ser s v x d seriesprecision Transforms the object s into a power series with main variable v x by default and precision number of significant terms equal to d the default seriesprecision by default
15. The library syntax is GEN sqrtnint GEN x long n 3 4 83 stirling n k flag 1 Stirling number of the first kind s n k flag 1 default or of the second kind S n k flag 2 where n k are non negative integers The former is 1 times the number of permutations of n symbols with exactly k cycles the latter is the number of ways of partitioning a set of n elements into k non empty subsets Note that if all s n k are needed it is much faster to compute Y s n k a x 1 z n 1 k Similarly if a large number of S n k are needed for the same k one should use ak ii 1 a 1 ka Should be implemented using a divide and conquer product Here are simple variants for n fixed list of s n k k 1 n vecstirling n Vec factorback vector n 1 i 1 i x list of S n k k 1 n 126 vecstirling2 n my Q x n 1 t vector n i t divrem Q x i Q t 1 t 2 The library syntax is GEN stirling long n long k long flag Also available are GEN stirling1 ulong n ulong k flag 1 and GEN stirling2 ulong n ulong k flag 2 3 4 84 sumdedekind h k Returns the Dedekind sum associated to the integers h and k corre sponding to a fast implementation of s h k sum n 1 k 1 n k frac h n k 1 2 The library syntax is GEN sumdedekind GEN h GEN k 3 4 85 sumdigits n Sum of decimal digits in the integer n sumdigits 123
16. sumnum n 1 2 3 21 1 n sqrt n tab 1 zeta 3 2 time 690 ms 1 1 19 E 305 fast and correct sumnum n 1 2 3 21 1 sqrt n 3 tab 1 zeta 3 2 time 730 ms 12 1 55 nonsense However sumnum n 1 2 3 2 1 n7 3 2 tab 1 zeta 3 2 time 8 990 ms 3 1 19 E 305 perfect as 1 n yn above but much slower For exponentially decreasing functions sumnum is given for completeness but one of suminf or sumpos should always be preferred If you experiment with such functions and sumnum anyway indicate the exact rate of decrease and increase m by 1 or 2 suminf n 1 27 n 1 time 10 ms 1 1 11 E 308 fast and perfect sumpos n 1 27 n 1 time 10 ms 12 2 78 E 308 also fast and perfect sumnum n 1 2 2 n 1 3 1 321115060 E320 0 E311 I nonsense sumnum n 1 2 log 2 27 n omitted 1 1 of real type time 5 860 ms 14 1 5 E 236 slow and lost 70 decimals m intnumstep 4b 9 sumnum n 1 2 log 2 2 n m 1 1 1 time 11 770 ms 16 1 9 E 305 now perfect but slow The library syntax is sumnum void E GEN eval void GEN GEN a GEN sig GEN tab long flag long prec 3 9 25 sumnumalt X a sig expr tab flag 0 Numerical summation of 1 expr X the variable X taking integer values from ceiling of a to 00 where expr is assumed to be a holomorphic function for R X
17. 2 1 000000000000000000000000000 check 0 3 0 9999999999999999999999999994 check 5 4 1 00000000000000007549266557 check 11 5 0 9999752641047824902660847745 check 1 2 14 134 1I very close to a non trivial zero 6 1 000000000000000000003747432 7 62329066 E 21x 1 check 1 10 I 7 1 000000000000000000000002511 2 989950968 E 24 I Now wait a minute not only are we summing a series which is certainly no longer alternating it has complex coefficients but we are also way outside of the region of convergence and still get decent results No programming mistake this time sumalt is a magic function providing very good convergence acceleration in effect we are computing the analytic continuation of our original function To convince ourselves that sumalt is a non trivial implementation let us try a simpler example sum n 1 1077 1 n n 0 log 2 approximates the well known formula time 7 417 ms 1 0 9999999278652515622893405457 sumalt n 1 1 n n log 2 accurate and fast time O ms 2 1 000000000000000000000000000 No we are not using a powerful simplification tool here only numerical computations Remember PARI is not a computer algebra system sumalt is heuristic but its use can be rigorously justified for a given function in particular our s formula Indeed Peter Borwein An efficient algorithm for the Riemann zeta function CMS Co
18. 3 6 5 Class field theory A modulus in the sense of class field theory is a divisor supported on the non complex places of K In PARI terms this means either an ordinary ideal I as above no Archimedean component or a pair 7 a where a is a vector with r 0 1 components corresponding to the infinite part of the divisor More precisely the i th component of a corresponds to the real embedding associated to the i th real root of K roots That ordering is not canonical but well defined once a defining polynomial for K is chosen For instance 1 1 1 is a modulus for a real quadratic field allowing ramification at any of the two places at infinity and nowhere else A bid or big ideal is a structure output by idealstar needed to compute in Zx 1 where I is a modulus in the above sense It is a finite abelian group as described above supplemented by technical data needed to solve discrete log problems Finally we explain how to input ray number fields or bnr using class field theory These are defined by a triple A B C where the defining set A B C can have any of the following forms bnr bnr subgroup bnf mod bnf mod subgroup The last two forms are kept for backward compatibility but no longer serve any real purpose see example below no newly written function will accept them e bnf is as output by bnfinit where units are mandatory unless the modulus is trivial bnr is as output by bnrinit Th
19. 3 8 8 listkill L Obsolete retained for backward compatibility Just use L List instead of listkill L In most cases you won t even need that e g local variables are automatically cleared when a user function returns The library syntax is void listkill GEN L 3 8 9 listpop list n Removes the n th element of the list list which must be of type t_LIST If n is omitted or greater than the list current length removes the last element If the list is already empty do nothing This runs in time O L n 1 The library syntax is void listpop GEN list long n 232 3 8 10 listput list x n Sets the n th element of the list list which must be of type t_LIST equal to x If n is omitted or greater than the list length appends x You may put an element into an occupied cell not changing the list length but it is easier to use the standard list n x construct This runs in time O L in the worst case when the list must be reallocated but in time O 1 on average any number of successive listputs run in time O L where HL denotes the list final length The library syntax is GEN listput GEN list GEN x long n 3 8 11 listsort L flag 0 Sorts the t_LIST list in place with respect to the somewhat arbitrary universal comparison function cmp In particular the ordering is the same as for sets and setsearch can be used on a sorted list L List 1 2 4 1 3 11 listsort L L 1 List 1 1
20. 332 MIVEDSE ads BASS Ze ea 67 inverseimage 240 isdiagonal s ciales GR em S k a 240 isfundamental 114 isideal cia Bes ea 195 isirreduci bles oca bos we ais 222 ispolygonal 114 ISPOWOT s m bee os 114 115 igpowerful mias sea 22h See aed 115 eprint a Re eos ee 115 116 isprimepower 116 isprincipalray 166 ispseudoprime 110 115 116 118 120 125 issquare ss oseon Gee ee es 116 117 issquareall 117 issquarefree 99 117 TSCOTIONG a a A a 117 J E oh BOK ae aS 130 Jacobi esac Fe he dee ee eae 248 jbessel sor s arsa ee eee ee ee es 91 JDESSSID dit ua tk Ue en ie a 92 POU de dh weirs Pond tp dees oe GR A 142 K kbessel 2 44 256 Gedo ew eS 92 Ker ui he Megs oe PE Be Be ae es la 241 Keri gt bee non eae oo oe eee A 241 KOTITE 2 e em saa de 4 SS Ge bw S 241 keyword a sosa ma sce ate ha Gea a a 45 RILI e ara a Ow wo 294 RILLO ear a ee a A 294 Kodalra goea ao de a ew 142 Kronecker symbol 117 kromeck6r soa oos p pok a KOE a 117 L lamberti p ici rr Sa ee eS 94 Laplace isos sesos nas 224 E A ee a a 117 Leech lattice 250 Legendre polynomial 222 Legendre symbol tiy Length 4 G6 aad OS a INE eee E 80 Lenstra hea be es Bee ee a 110 VOX cave Go ia eo Sek we ee eS 68 LOXCMP 224552480555 2 eee aa 69 lexical scoping 34 TIPA 6 4 w
21. K bnfinit x 2 23 L bnrdisclist K 10 s L 1 2 1 Mat 8 1 0 O 0 Mat 9 11 0 O 0 bnfdecodemodule K s 1 1 Lo 2 0 o 1 The library syntax is GEN decodemodule GEN nf GEN m 3 6 11 bnfinit P flag 0 tech Initializes a bnf structure Used in programs such as bnfisprincipal bnfisunit or bnfnarrow By default the results are conditional on the GRH see 3 6 7 The result is a 10 component vector bnf This implements Buchmann s sub exponential algorithm for computing the class group the regulator and a system of fundamental units of the general algebraic number field K defined by the irreducible polynomial P with integer coefficients Tf the precision becomes insufficient gp does not strive to compute the units by default flag 0 When flag 1 we insist on finding the fundamental units exactly Be warned that this can take a very long time when the coefficients of the fundamental units on the integral basis are very large If the fundamental units are simply too large to be represented in this form an error message is issued They could be obtained using the so called compact representation of algebraic numbers as a formal product of algebraic integers The latter is implemented internally but not publicly accessible yet tech is a technical vector empty by default see 3 6 7 Careful use of this parameter may speed up your computations but it is mostly obsolete an
22. Principal branch of tan x log 1 ix 1 ix 2i In particular the real part of atan x belongs to 7 2 7 2 The branch cut is in two pieces ico i continuous with quadrant IV and Ji i00 continuous with quadrant II The function satisfies iatan x iatanh ix for all x 4 i The library syntax is GEN gatan GEN x long prec 3 3 14 atanh x Principal branch of tanh log 1 x 1 x 2 In particular the imaginary part of atanh x belongs to 7 2 7 2 if x R and x gt 1 then atanh x is complex The library syntax is GEN gatanh GEN x long prec 3 3 15 bernfrac x Bernoulli number Bz where By 1 By 1 2 Ba 1 6 expressed as a rational number The argument x should be of type integer The library syntax is GEN bernfrac long x 3 3 16 bernpol n v x Bernoulli polynomial B in variable v bernpol 1 1 x 1 2 bernpol 3 2 x 3 3 2 x 2 1 2 x The library syntax is GEN bernpol long n long v 1 where v is a variable number 3 3 17 bernreal x Bernoulli number B as bernfrac but By is returned as a real number with the current precision The library syntax is GEN bernreal long x long prec 3 3 18 bernvec 1 Creates a vector containing as rational numbers the Bernoulli numbers Bo Ba Baz This routine is obsolete Use bernfrac instead each time you need a Bernoulli number in exact form Note This routine is implemen
23. attention to variable scope Here is an example used to retrieve the coefficient array of a multivari ate polynomial a non trivial task due to PARI s unsophisticated representation for those objects coeffs P nbvar if type P t_POL for i 1 nbvar P P return P vector poldegree P 1 i coeffs polcoeff P i 1 nbvar 1 If P is a polynomial in k variables show that after the assignment v coeffs P k the coefficient of x x in P is given by v n 1 ngt1 The operating system automatically limits the recursion depth dive n dive n 1 dive 0 42 Hook at dive n 1 4K as xxx in function dive dive n 1 OK pt last 2 lines repeated 19 times deep recursion There is no way to increase the recursion limit which may be different on your machine from within gp To increase it before launching gp you can use ulimit or limit depending on your shell and raise the process available stack space increase stacksize 2 7 5 Function which take functions as parameters This is done as follows calc f x f x calc sin Pi 42 5 04870979 E 29 g x x 2 calc g 3 714 9 If we do not need g elsewhere we should use an anonymous function here calc x gt x 2 3 Here is a variation funs cos sin tan x gt x 3 1 an array of functions call i x funs i x evaluates the appropriate function on argument x provided 1
24. equal entries with respect to the sorting criterion is not changed If cmpf is omitted we use the standard comparison function lex thereby restricting the possible types for the elements of x integers fractions or reals and vectors of those If cmpf is present it is understood as a comparison function and we sort according to it The following possibilities exist e an integer k sort according to the value of the k th subcomponents of the components of x e a vector sort lexicographically according to the components listed in the vector For example if cmpf 2 1 3 sort with respect to the second component and when these are equal with respect to the first and when these are equal with respect to the third e a comparison function t_CLOSURE with two arguments x and y and returning an integer which is lt 0 gt 0 or 0if x lt y x gt y or x y respectively The sign function is very useful in this context vecsort 3 0 2 1 0 2 sort columns according to lex order n 0 2 3 o 2 1 vecsort v x y gt sign y x reverse sort vecsort v x y gt sign abs x abs y sort by increasing absolute value cmpf x y my dx poldisc x dy poldisc y sign abs dx abs dy vecsort x 2 1 x73 2 x74 5 x 1 cmpf The last example used the named cmpf instead of an anonymous function and sorts polynomials with respect to the absolute value of their discriminant A more efficient ap
25. in particular this is not the same as as idealpow nf x k followed by reduction The library syntax is GEN idealpowO GEN nf GEN x GEN k long flag See also GEN idealpow GEN nf GEN x GEN k and GEN idealpows GEN nf GEN x long k flag 0 Corresponding to flag 1 is GEN idealpowred GEN nf GEN vp GEN k 3 6 65 idealprimedec nf p Computes the prime ideal decomposition of the positive prime number p in the number field K represented by nf If a non prime p is given the result is undefined The result is a vector of prid structures each representing one of the prime ideals above p in the number field nf The representation pr p a e f mb of a prime ideal means the following a and is an algebraic integer in the maximal order Zg and the prime ideal is equal to p pZk aZx e is the ramification index f is the residual index finally mb is the multiplication table associated to the algebraic integer b is such that p gt Zg 6 pZx which is used internally to compute valuations In other words if p is inert then mb is the integer 1 and otherwise it s a square t_MAT whose j th column is 6 nf zk 3 The algebraic number a is guaranteed to have a valuation equal to 1 at the prime ideal this is automatic if e gt 1 The components of pr should be accessed by member functions pr p pr e pr f and pr gen returns the vector p a K nfinit x 3 2 L idealprimedec K 5 7 HL 2 primes above 5 in
26. lt lt 110 factofint soe anse ses EOS 106 110 factormod miis o ek Re 107 110 factormodd 110 FactorD eis ab bw AS Bee eS 107 168 factorpadic 217 218 factor_add_primes 288 factor_proven 100 106 110 Jamat coronarias 45 2 54 wa 152 AA III 44 ffigen mom norma 19 110 111 140 LLIDIC gt ote ee Ry a 19 110 111 ELO 2 Seo fogs ee Re 111 112 ffnDirred 4 5 4 eRe c A 112 finbirredO lt c ee A A T12 Fford r sos a s irea ei A ie ffprimroot 111 112 113 ffrandom sa s ei e ai eiii OE a 85 ffsumnbirred 112 fiDO emos ra 113 TIDONACCL 4 0 6 4 66 a4 amp Pb ew Rd 113 field discriminant 187 filename ces a eee a 55 finite field element 4 8 19 finite field se acrea da a ets praa 20 fixed floating point format 306 POG sa aa E aR OG aa i ER a A 63 FELOOE ae n 6k A a 80 PLEKR ie ih ana ae n aE EAE GOR ae a Se i 27l TOL Ha Ga eh E re G 279 forcomposit se sco doe Shae ew ae 279 A 6 Stee a Ghee ee 185 LOEC V Laprida MO ee Ee ae Ee 279 forelli ares rr eae oe 182 280 formal integration 218 formal SUITE ss 2 2 Bas canarios es 226 Format ah aca a e Boe ae we oS 306 310 DOLPATD Bi asa a eo ele we ee 280 281 FOYPYiMe 24 2 ee ek ee ee a 281 Torgi vot de pasa hs aisha ats wee a 231 forgqivecO socios ieies ee dg 231 Tortop sasea ek Be AA ee ae A 281 TOTSUDETOUD ses ee ew Se So a a 282
27. o o 153 Prompt 44 0 4 4 8 sr eis A 309 PSdraW cna chard ee sates de ees 27 pseudo basis o o 153 pseudo matrit o 153 PS ile resis 272 310 A Rodel E wie a oa GSR i ae 96 PSPLOUN 6 i g va te he a oe oe ee A 277 psplothraw eco soad aa 278 Python east Sob ae EK Gr eee A 54 pto GEN 2 23 6 6 je a ee 111 Q Q AULO este rs A 245 Q AUtOO s o ie s eaa a e a oe ae 245 gfa toexport s s sacair a a 245 QD cecer ae ke Hos perei nrs 74 QDO 9 3 Gal rd e a x 74 qfbclassno 121 124 qfbclassnoO ooo o 122 EDCOMPLAW decir e da ey es 122 qfbhclassno 123 Q DiL posre a e ad e 246 250 qfbnucomp gt gt e heen eee ee a es 122 QfONUPOW i si eee a ees 123 qfbpowraw o 123 qfbprimeform 123 qfbred escocia Peed bes 123 q bredO cole co ala cd oA 123 QEDSOIVE so s sacs es rantes e 123 124 qigalissred 2 i144 gee eee de eee 246 qfgaussred_positive 246 L orcas ass be ee eh ee 74 QPISOM 684 Hie Hee ea ee HE AAS 247 axis OM coc sm ale Boe dS ee Be a 247 qfisomi it s so se sa Se 247 QLISOMINDITO 0 du Soke ai a A 247 G Jacobl s sod e suede Soa ee aS 236 247 qllll wea ean eee nd rs 228 248 QUITO ceca ee dt asad vd dasa 248 gfililtgram mosso a Ce eo 248 qflliteram tee ae ke bea ee He 249 QEMINIM o 4 ya ee a re wt 249 251 qfiminimO 2 326 2 eee ewe ee ee ss 250 GNO oh e e a a 246 250 251 qfperfection 251 QE secs A Grease 74 A TOD 2
28. 1 rnfeltreltoabs L Mod y K pol 15 Mod x 2 x 4 1 The library syntax is GEN rnfeltreltoabs GEN rnf GEN x 205 3 6 136 rnfelttrace rnf x rnf being a relative number field extension L K as output by rn finit and z being an element of L returns the relative trace Nz g x as an element of K K nfinit y 2 1 L rnfinit K x 2 y rnfelttrace L Mod x L pol 42 0 rnfelttrace L 2 43 4 rnfelttrace L Mod x x 2 y The library syntax is GEN rnfelttrace GEN rnf GEN x 3 6 137 rnfeltup rnf x rnf being a relative number field extension L K as output by rnfinit and x being an element of K computes x as an element of the absolute extension L Q as a polynomial modulo the absolute equation rnf pol K nfinit y 2 1 L rnfinit K x 2 y L pol 12 x 4 1 rnfeltup L Mod y K pol L4 Mod x 2 x74 1 rnfeltup L y 45 Mod x 2 x 4 1 rnfeltup L 1 2 in terms of K zk 46 Mod 2 x72 1 x74 1 The library syntax is GEN rnfeltup GEN rnf GEN x 3 6 138 rnfequation nf pol flag 0 Given a number field nf as output by nfinit or simply a polynomial and a polynomial pol with coefficients in nf defining a relative extension L of nf computes an absolute equation of L over Q The main variable of nf must be of lower priority than that of pol see Section 2 5 3 Note that for efficiency this does not check whether the relative equation is irreducible ov
29. 1 change it in all rectwindows This only works in the gnuplot interface 276 3 10 22 plotpointtype w type Change the type of points subsequently plotted in rectwindow w type 1 corresponds to a dot larger values may correspond to something else w 1 changes highlevel plotting This is only taken into account by the gnuplot interface 3 10 23 plotrbox w dx dy Draw in the rectwindow w the outline of the rectangle which is such that the points 1 y1 and x1 dx yl dy are opposite corners where x1 yl is the current position of the cursor Only the part of the rectangle which is in w is drawn The virtual cursor does not move 3 10 24 plotrecth w X a b expr flag 0 n 0 Writes to rectwindow w the curve output of ploth w X a b expr flag n Returns a vector for the bounding box 3 10 25 plotrecthraw w data flags 0 Plot graph s for data in rectwindow w flag has the same significance here as in ploth though recursive plot is no more significant data is a vector of vectors each corresponding to a list a coordinates If parametric plot is set there must be an even number of vectors each successive pair corresponding to a curve Otherwise the first one contains the x coordinates and the other ones contain the y coordinates of curves to plot 3 10 26 plotrline w dz dy Draw in the rectwindow w the part of the segment x1 y1 x1 dx yl dy which is inside w where x1 y1 is
30. 1 return only the polynomial P If flag 2 return P x F where P and x are as above and F is the factorization of gal pol over the field defined by P where variable v y by default stands for a root of P The priority of v must be less than the priority of the variable of gal pol see Section 2 5 3 Example G galoisinit x 4 1 galoisfixedfield G G group 2 2 12 x72 2 Mod x73 x x 4 1 x72 y x 1 x72 y x 1 computes the factorization 1 1 x y 2x 1 x y 2x 1 The library syntax is GEN galoisfixedfield GEN gal GEN perm long flag long v 1 where v is a variable number 3 6 36 galoisgetpol a b s Query the galpol package for a polynomial with Galois group isomorphic to GAP4 a b totally real if s 1 default and totally complex if s 2 The output is a vector pol den where e pol is the polynomial of degree a e den is the denominator of nfgaloisconj pol Pass it as an optional argument to ga loisinit or nfgaloisconj to speed them up pol den galoisgetpol 64 4 1 G galoisinit pol time 352ms galoisinit pol den passing den speeds up the computation time 264ms h h 74 1 same answer If b and s are omitted return the number of isomorphism classes of groups of order a The library syntax is GEN galoisgetpol long a long b long s Also available is GEN galoisnbpol long a when b and s are omitted 169 3
31. 3 3 f 10 1 59049 Mat 3 10 x 6 3 x 3 63 3 1 0 III 9 page 184 3 3 N FaN T V genus2red x 3 x 2 1 x72 x AM X_1 13 global reduction p 13 potential stable reduction 5 Mod 0 13 Mod 0 13 reduction at p I O II 0 page 159 f 2 N 3 169 FaN 4 Mat 13 2 in particular good reduction at 2 T 5 x 6 58 x75 1401 x 4 18038 x 3 130546 x 2 503516 x 808561 V 6 13 5 Mod 0 13 Mod 0 13 I O II O page 159 We now first describe the format of the vector V Vp in the case where p was specified local re duction at p it is a triple p stable red The component stable type vecj contains information about the stable reduction after a field extension depending on types the stable reduction is e 1 smooth i e the curve has potentially good reduction The Jacobian J C has potentially good reduction e 2 an elliptic curve E with an ordinary double point vecj contains j mod p the modular invariant of E The potential semi abelian reduction of J C is the extension of an elliptic curve with modular invariant j mod p by a torus e 3 a projective line with two ordinary double points The Jacobian J C has potentially multiplicative reduction e 4 the union of two projective lines crossing transversally at three points The Jacobian J C has potentially multiplicative reduction e 5 the union of two ell
32. 76 Strexpand gt s ca egeo eee Cee s 75 SLT TIMO oec ereng he RE RSs 55 309 SLT CLATES oa c omara eae woe 40 312 Strictmatch eL 4 ee wes 312 string contezt ooo 46 SITING one eee tolea e poa 7 26 45 StRPEinth gt s css ee ee ee dees 287 306 SETS ws ge ead baie ee we Oa SS 8 76 StrtoGEN ec ade we 75 SCUIM rta Be e OO a ea 223 Sturmpart gt o sne 455 68 eee ee 223 subfield meis sa e aota aa Ane 197 SUDGTOWD t so e ea eB eh Be i 152 Subgroup s j be ees as 282 subgrouplist 213 282 subgrouplist0 o rosa 60k Be ace 214 subresultant algorithm 113 220 222 SUDST a coe sos anune oee e 225 228 Substpol gt s sasse raanei ss 225 SUDSIVEC s s pas o He He ke a a 225 SUM e a eee ee Se ee E we 64 SUM escasea eh e ae es a 267 sumalt 264 267 268 271 339 SUuMalt2 mos aari de boa 4 Bea a 268 sumdedekind 127 SUMGIGITS sp ori a u ap a ee t27 SUMdIYV eka soo po eee ee ee a 126 268 SUMNGIVE ui Be Bek Bee se we a 126 SUNdIVMUIG coe eb ba ea ee ek 268 sumformal e ss r ea eus a eee Be ee 226 SUMINT aura 267 268 270 SUMMUM 2 26k ak we BRS o a 268 270 SUMMUMAIT sso reri sa we See 270 271 SUMNUM N t aaa a 271 sumpos 268 270 271 SUMPOS2 omo SED ee ee ae 271 SUPPL s em Gs a eH ee ee i E 244 sylvestermatrix 224 symmetric powers 224 SYSTEM cios 47 292 299 311 T O 155 Tamagawa number 137 142
33. GEN seq NULL 3 12 37 type x This is useful only under gp Returns the internal type name of the PARI object x as a string Check out existing type names with the metacommand t For example type 1 will return t_INT The library syntax is GEN typeO GEN x The macro typ is usually simpler to use since it returns a long that can easily be matched with the symbols t_ The name type was avoided since it is a reserved identifier for some compilers 300 3 12 38 uninline Experimental Exit the scope of all current inline variables 3 12 39 version Returns the current version number as a t_VEC with three integer compo nents major version number minor version number and patchlevel if your sources were obtained through our version control system this will be followed by further more precise arguments in cluding e g a git commit hash This function is present in all versions of PARI following releases 2 3 4 stable and 2 4 3 testing Unless you are working with multiple development versions you probably only care about the 3 first numeric components In any case the lex function offers a clever way to check against a particular version number since it will compare each successive vector entry numerically or as strings and will not mind if the vectors it compares have different lengths if lex version 2 3 5 gt O code to be executed if we are running 2 3 5 or more recent gt compatibility
34. If you really insist on using these in your program we warned you the source plot c should be readable enough for you to achieve something 3 10 5 plot X a b expr Yminy Ymazr Crude ASCII plot of the function represented by expression expr from a to b with Y ranging from Ymin to Ymaz If Ymin resp Ymar is not given the minimum resp the maximum of the computed values of the expression is used instead 3 10 6 plotbox w 12 y2 Let 11 y1 be the current position of the virtual cursor Draw in the rectwindow w the outline of the rectangle which is such that the points xl yl and 12 y2 are opposite corners Only the part of the rectangle which is in w is drawn The virtual cursor does not move 3 10 7 plotclip w clips the content of rectwindow w i e remove all parts of the drawing that would not be visible on the screen Together with plotcopy this function enables you to draw on a scratchpad before committing the part you re interested in to the final picture 3 10 8 plotcolor w c Set default color to c in rectwindow w This is only implemented for the X windows fltk and Qt graphing engines Possible values for c are given by the graphcolormap default factory setting are 1 black 2 blue 3 violetred 4 red 5 green 6 grey 7 gainsborough but this can be considerably extended 3 10 9 plotcopy sourcew destw dx dy flag 0 Copy the contents of rectwindow sourcew to rectwindow destw with offset
35. R s lt 2 but the right hand side provides an analytic continuation which may be evaluated at s 2 Multivariate integration Using successive univariate integration with respect to different formal parameters it is immediate to do naive multivariate integration But it is important to use a suitable intnuminit to precompute data for the internal integrations at least For example to compute the double integral on the unit disc x y lt 1 of the function x y we can write tab intnuminit 1 1 intnum x 1 1 intnum y sqrt 1 x 2 sqrt 1 x 2 x 2 y 2 tab tab The first tab is essential the second optional Compare tab intnuminit 1 1 time 30 ms intnum x 1 1 intnum y sqrt 1 x 2 sqrt 1 x 2 x 2 y72 time 54 410 ms slow intnum x 1 1 intnum y sqrt 1 x 2 sqrt 1 x 2 x 2 y72 tab tab time 7 210 ms AN faster However the intnuminit program is usually pessimistic when it comes to choosing the integration step 27 It is often possible to improve the speed by trial and error Continuing the above example test M tab intnuminit 1 1 M intnum x 1 1 intnum y sqrt 1 x 2 sqrt 1 x 2 x 2 y72 tab tab Pi 2 m intnumstep what value of m did it take 11 7 test m 1 264 time 1 790 ms 2 2 05 E 104 11 4 2 times faster and still OK test m 2 time 430 ms 3 1 11 E 104 11 16 24 times faster and sti
36. S 1 1 1 In degree 2 Sa 2 1 1 In degree 3 43 C3 3 1 1 S3 6 1 1 In degree 4 Cy 4 1 1 Va 4 1 1 Da 8 1 1 Ag 12 1 1 S4 24 1 1 In degree 5 Cs 5 1 1 Ds 10 1 1 Moo 20 1 1 As 60 1 1 Ss 120 1 1 In degree 6 Cg 6 1 1 3 6 1 2 De 12 1 1 A4 12 1 1 Gig 18 1 1 S 24 1 1 As x Co 24 1 2 Sf 24 1 1 G3 36 1 1 Gd 36 1 1 Sa x 198 C 48 1 1 As PSLa 5 60 1 1 Gro 72 1 1 Ss PGLa 5 120 1 1 Ag 360 1 1 S 720 1 1 In degree 7 C7 7 1 1 Dz 14 1 1 Mo 21 1 1 Mas 42 1 1 PSLa 7 PSL3 2 168 1 1 A7 2520 1 1 S7 5040 1 1 This is deprecated and obsolete but for reasons of backward compatibility we cannot change this behavior yet So you can use the default new_galois_format to switch to a consistent naming scheme namely k is always the standard numbering of the group among all transitive subgroups of Sn If this default is in effect the above groups will be coded as In degree 1 S 1 1 1 In degree 2 S2 2 1 1 In degree 3 43 C3 3 1 1 S3 6 1 2 In degree 4 Cy 4 1 1 Va 4 1 2 Da 8 1 3 Ag 12 1 4 S4 24 1 5 In degree 5 Cs 5 1 1 Ds 10 1 2 Moo 20 1 3 As 60 1 4 Ss 120 1 5 In
37. The library syntax is GEN nfnewprec GEN nf long prec See also GEN bnfnewprec GEN bnf long prec and GEN bnrnewprec GEN bnr long prec 195 3 6 112 nfroots nf x Roots of the polynomial x in the number field nf given by nfinit without multiplicity in Q if nf is omitted x has coefficients in the number field scalar polmod polynomial column vector The main variable of nf must be of lower priority than that of x see Section 2 5 3 However if the coefficients of the number field occur explicitly as polmods as coefficients of x the variable of these polmods must be the same as the main variable of t see nffactor It is possible to input a defining polynomial for nf instead but this is in general less efficient since parts of an nf structure will be computed internally This is useful in two situations when you don t need the nf or when you can t compute its discriminant due to integer factorization difficulties In the latter case addprimes is a possibility but a dangerous one roots will probably be missed if the true field discriminant and an addprimes entry are strictly divisible by some prime If you have such an unsafe nf it is safer to input nf pol The library syntax is GEN nfroots GEN nf NULL GEN x See also GEN nfrootsQ GEN x corresponding to nf NULL 3 6 113 nfrootsof1 nf Returns a two component vector w 2 where w is the number of roots of unity in the number field nf and z is a primitive w t
38. The library syntax is GEN partitions long k GEN a NULL GEN n NULL 3 4 56 polrootsff x p a Returns the vector of distinct roots of the polynomial x in the field F defined by the irreducible polynomial a over Fp The coefficients of x must be operation compatible with Z pZ Either a or p can omitted in which case both are ignored if x has t_FFELT coefficients polrootsff x 2 1 5 y 2 3 over F_5ly 1 y72 3 F_25 1 Mod Mod 3 5 Mod 1 5 y 2 Mod 3 5 Mod Mod 2 5 Mod 1 5 y 2 Mod 3 5 t ffgen y 2 Mod 3 5 t A a generator for F_25 as a t_FFELT polrootsff x 2 1 not enough information to determine the base field xkk at top level polrootsff x 2 1 oK x polrootsff incorrect type in factorff polrootsff x 2 t70 make sure one coeff is a t_FFELT 43 3 2 polrootsff x 2 t 1 4 2xt 1 3 t 4 Notice that the second syntax is easier to use and much more readable The library syntax is GEN polrootsff GEN x GEN p NULL GEN a NULL 3 4 57 precprime z Finds the largest pseudoprime see ispseudoprime less than or equal to x x can be of any real type Returns 0 if x lt 1 Note that if x is a prime this function returns x and not the largest prime strictly smaller than x To rigorously prove that the result is prime use isprime The library syntax is GEN precprime GEN x 3 4 58 prime n The nt prime number prime 1079
39. a b z expr tab Numerical integration of expr X exp 2i72X from a to b in other words Fourier transform from a to b of the function represented by expr Note the minus sign Endpoints a and b are coded as in intnum and are not necessarily at infinity but if they are oscillations i e 1 a1 are forbidden The library syntax is intfourierexp void E GEN eval void GEN GEN a GEN b GEN z GEN tab long prec 3 9 5 intfouriersin X a b z expr tab Numerical integration of expr X sin 272X from a to b in other words Fourier sine transform from a to b of the function represented by expr Endpoints a and b are coded as in intnum and are not necessarily at infinity but if they are oscillations i e 1 a1 are forbidden The library syntax is intfouriersin void E GEN eval void GEN GEN a GEN b GEN z GEN tab long prec 3 9 6 intfuncinit X a b expr flag 0 m 05 Initialize tables for use with integral transforms such as intmellininv etc where a and b are coded as in intnum expr is the function s X to which the integral transform is to be applied which will multiply the weights of integration and m is as in intnuminit If flag is nonzero assumes that s X s X which makes the computation twice as fast See intmellininvshort for examples of the use of this function which is particularly useful when the function s X is lengthy to compute such as a gamma product The l
40. bnf bnfinit x 2 x 100000 bnf fu EK at top level bnf fu OK Fez _ fu missing units in fu u 119836165644250789990462835950022871665178127611316131167 379554884019013781006303254896369154068336082609238336 bnfisunit bnf u 13 1 Mod 0 2 The given u is the inverse of the fundamental unit implicitly stored in bnf In this case the fundamental unit was not computed and stored in algebraic form since the default accuracy was too low Re run the command at gl or higher to see such diagnostics The library syntax is GEN bnfisunit GEN bnf GEN x 161 3 6 17 bnfnarrow bnf bnf being as output by bnfinit computes the narrow class group of bnf The output is a 3 component row vector v analogous to the corresponding class group component bnf clgp bnf 8 1 the first component is the narrow class number v no the second component is a vector containing the SNF cyclic components v cyc of the narrow class group and the third is a vector giving the generators of the corresponding v gen cyclic groups Note that this function is a special case of bnrinit The library syntax is GEN buchnarrow GEN bnf 3 6 18 bnfsignunit bnf bnf being as output by bnfinit this computes an r x r r2 1 matrix having 1 components giving the signs of the real embeddings of the fundamental units The following functions compute generators for the totally positive units exponents of totally positive unit
41. dx dy If flag s bit 1 is set dx and dy express fractions of the size of the current output device otherwise dx and dy are in pixels dx and dy are relative positions of northwest corners if other bits of flag vanish otherwise of 2 southwest 4 southeast 6 northeast corners 3 10 10 plotcursor w Give as a 2 component vector the current scaled position of the virtual cursor corresponding to the rectwindow w 273 3 10 11 plotdraw list flag 0 Physically draw the rectwindows given in list which must be a vector whose number of components is divisible by 3 If list wl x1 yl w2 2x2 y2 the windows wl w2 etc are physically placed with their upper left corner at physical position a1 yl 12 y2 respectively and are then drawn together Overlapping regions will thus be drawn twice and the windows are considered transparent Then display the whole drawing in a special window on your screen If flag 4 0 x1 yl etc express fractions of the size of the current output device 3 10 12 ploth X a b expr flags 0 n 0 High precision plot of the function y f z represented by the expression expr x going from a to b This opens a specific window which is killed whenever you click on it and returns a four component vector giving the coordinates of the bounding box in the form zmin zmaz ymin ymaz Important note ploth may evaluate expr thousands of times given the relatively low resolution of p
42. factor abs nf disc determines ramified primes 7 2 2 3 5 idealfactor nf 2 18 2 0 0 1 3 1 0 1 0 31 p3 Huge discriminants helping nfdisc In case pol has a huge discriminant which is difficult to factor it is hard to compute from scratch the maximal order The special input format pol B is also accepted where pol is a polynomial as above and B has one of the following forms e an integer basis as would be computed by nfbasis a vector of polynomials with first element 1 This is useful if the maximal order is known in advance e an argument listP which specifies a list of primes see nfbasis Instead of the maximal order nfinit then computes an order which is maximal at these particular primes as well as the primes contained in the private prime table see addprimes The result is unconditionnaly correct when the discriminant nf disc factors completely over this set of primes The function nfcertify automates this pol polcompositum x 5 101 polcyclo 7 1 nf nfinit pol 1073 nfcertify nf 3 A priori nf zk defines an order which is only known to be maximal at all primes lt 10 no prime lt 10 divides nf index The certification step proves the correctness of the computation If flag 2 polis changed into another polynomial P defining the same number field which is as simple as can easily be found using the polredbest algorithm and all the s
43. for instance both rational numbers If y is omitted and zx is a vector returns the gcd of all components of zx i e this is equivalent to content x When z and y are both given and one of them is a vector matrix type the GCD is again taken recursively on each component but in a different way If y is a vector resp matrix then the result has the same type as y and components equal to gcd x y i resp gcd x y i1 Else if x is a vector matrix the result has the same type as x and an analogous definition Note that for these types gcd is not commutative The algorithm used is a naive Euclid except for the following inputs e integers use modified right shift binary plus minus variant e univariate polynomials with coefficients in the same number field in particular rational use modular gcd algorithm e general polynomials use the subresultant algorithm if coefficient explosion is likely non modular coefficients If u and v are polynomials in the same variable with inexact coefficients their gcd is defined to be scalar so that a x 0 0 gcd a a Zi 1 b y x 0 y gcd b b 42 y c 4 x 0 273 gcd c c 12 4 A good quantitative check to decide whether such a gcd should be non trivial is to use polre sultant a value close to 0 means that a small deformation of the inputs has non trivial gcd You 113 may also use bezout which does try to compute an approximate gcd d and provi
44. from cell m to cell n last cell If a function name is given instead of a number or range outputs info on the internal structure of the hash cell this function occupies a struct entree in C If the range is reduced to a dash outputs statistics about hash cell usage 2 13 12 1 logfile Switches log mode on and off If a logfile argument is given change the default logfile name to logfile and switch log mode on 2 13 13 m As a but using prettymatrix format 2 13 14 o n Sets output mode to n 0 raw 1 prettymatrix 3 external prettyprint 2 13 15 p n Sets realprecision to n decimal digits Prints its current value if n is omitted 57 2 13 16 ps nj Sets seriesprecision to n significant terms Prints its current value if n is omitted 2 13 17 g Quits the gp session and returns to the system Shortcut for quit see Sec tion 3 12 29 2 13 18 r filename Reads into gp all the commands contained in the named file as if they had been typed from the keyboard one line after the other Can be used in combination with the w command see below Related but not equivalent to the function read see Section 3 12 30 in particular if the file contains more than one line of input there will be one history entry for each of them whereas read would only record the last one If filename is omitted re read the previously used input file fails if no file has ever been successfully read in the current ses
45. i e the product of the Dedekind zeta function by its gamma and exponential factors CAVEAT This implementation is not satisfactory and must be rewritten In particular e The accuracy of the result depends in an essential way on the accuracy of both the zetakinit program and the current accuracy Be wary in particular that x of large imaginary part or on the contrary very close to an ordinary integer will suffer from precision loss yielding fewer significant digits than expected Computing with 28 digits of relative accuracy we have zeta 3 1 1 202056903159594285399738161 zeta 3 1e 20 2 1 202056903159594285401719424 zetak zetakinit x 3 1e 20 13 1 2020569031595952919 MAN 5 digits are wrong zetak zetakinit x 3 1e 28 4 25 33411749 junk e As the precision increases results become unexpectedly completely wrong p100 zetak zetakinit x 2 5 1 1 30 1 7 26691813 E 108 perfect p150 zetak zetakinit x 2 5 1 1 30 2 2 486113578 E 156 MN perfect p200 zetak zetakinit x 2 5 1 1 30 13 4 47 E 75 more than half of the digits are wrong p250 214 zetak zetakinit x 2 5 1 1 30 14 1 6 E43 junk The library syntax is GEN gzetakall GEN nfz GEN x long flag long prec See also GEN glambdak GEN znf GEN x long prec or GEN gzetak GEN znf GEN x long prec 3 6 164 zetakinit bnf Computes a number of initialization data concerning the num
46. if you want to specify a given value of nrpid you must give some values as well for c and c2 and provide a vector c1 cz nrpid Note also that you can use an nf instead of P which avoids recomputing the integral basis and analogous quantities 3 6 8 bnfcertify bnf flag 0 bnf being as output by bnfinit checks whether the result is correct i e whether it is possible to remove the assumption of the Generalized Riemann Hypothesis It is correct if and only if the answer is 1 If it is incorrect the program may output some error message or loop indefinitely You can check its progress by increasing the debug level The bnf structure must contain the fundamental units K bnfinit x 3 27273 1 bnfcertify K kk at top level K bnfinit x 3 2 2 3 1 bnfcertify K RRR nnn nnn bnfcertify missing units in bnf K bnfinit x 3 27273 1 1 include units bnfcertify K 43 1 If flag is present only certify that the class group is a quotient of the one computed in bnf much simpler in general likewise the computed units may form a subgroup of the full unit group In this variant the units are no longer needed K bnfinit x73 27273 1 bnfcertify K 1 44 1 The library syntax is long bnfcertify0 GEN bnf long flag Also available is GEN bn fcertify GEN bnf flag 0 157 3 6 9 bnfcompress bnf Computes a compressed version of bnf from bnfinit a small Buch mann s number field or sb
47. n k Returns the vector of partitions of the integer k as a sum of positive integers parts for k lt 0 it returns the empty set and for k 0 the trivial partition no parts A partition is given by a t_VECSMALL where parts are sorted in nondecreasing order partitions 3 1 Vecsmal1 3 Vecsmall 1 2 Vecsmal1 1 1 1 correspond to 3 1 2 and 1 1 1 The number of unrestricted partitions of k is given by numbpart partitions 50 1 204226 numbpart 50 12 204226 Optional parameters n and a are as follows en nmaz resp n nmin nmaz restricts partitions to length less than nmaz resp length between nmin and nmax where the length is the number of nonzero entries 119 ea amaz resp a amin amaz restricts the parts to integers less than amaz resp between amin and amaz partitions 4 2 parts bounded by 2 1 Vecsmall 2 21 Vecsmall 1 1 21 Vecsmal1 1 1 1 1 partitions 4 2 at most 2 parts 2 Vecsmall 4 Vecsmal1 1 3 Vecsmal1 2 2 partitions 4 0 3 2 at most 2 parts 3 Vecsmal11 4 Vecsmal1 1 3 Vecsmal1 2 2 By default parts are positive and we remove zero entries unless amin lt 0 in which case nmin is ignored and X is of constant length nmaz partitions 4 0 3 AN parts between O and 3 1 Vecsmall 0 O 1 3 Vecsmall 0 0 2 2 Vecsmal1 0 1 1 2 Vecsmal1 1 1 1 1
48. normlp M 1 6 10 normlp T 1 7 2 4142135623730950488016887242096980786 The library syntax is GEN gnormlp GEN x GEN p NULL long prec 3 8 51 qfauto G fl G being a square and symmetric matrix with integer entries representing a positive definite quadratic form outputs the automorphism group of the associate lattice Since this requires computing the minimal vectors the computations can become very lengthy as the dimension grows G can also be given by an qfisominit structure See qfisominit for the meaning of fl The output is a two components vector o g where o is the group order and g is the list of generators as a vector For each generators H the equality G HGH holds The interface of this function is experimental and will likely change in the future This function implements an algorithm of Plesken and Souvignier following Souvignier s im plementation The library syntax is GEN qfauto0 GEN G GEN fl NULL Also available is GEN qfauto GEN G GEN f1 where G is a vector of zm 245 3 8 52 qfautoexport qfa flag qfa being an automorphism group as output by qfauto export the underlying matrix group as a string suitable for no flags or flag 0 GAP or flag 1 Magma The following example computes the size of the matrix group using GAP G qfauto 2 1 1 2 1 12 1 0 O 1 O 1 1 1 1 1 O 1 s qfautoexport G 2 Group 1 0 0 1 bios 11 1
49. polcompositum 197 polcompositumO 198 polcyclo concisa 219 220 poleyclofactore e essa sae id Ea ad 220 polcyclo_eval 220 poldegree o ooo ooo ooo 220 POLGISC essere pee De oo Bok A 220 poldiscO 2 222442 five ee psa 220 poldiscreduced 220 Polini sessa e eaoaai ada 168 polgalois 198 199 308 polgraecffe 26 ooo ee 220 polhensellift 221 polhermite o 221 polhermite_eval 221 Polint mirra e HO Sales 221 polinterpolate 221 poliscyclO reee aaneren epi raa dal poliscycloprod 221 222 polisirreducible 222 Pollard Rios e ee vids er ae g 99 110 pollead sess ek siai aaa 222 pollegendre 222 pollegendre_eval 222 PONOA ss ew s a sa ene we 7 POlMOd sise seeda eau sd 8 20 polr cip Lite sia ars 222 Polred aes a rc ea 199 200 polr ed2 ea sas emm a 200 polredabS coe beeen o 200 201 polredabsO se se G6 24 4 4 4 0 daa 200 polredbest 194 199 200 201 202 polredord 2266 ease eee ea Hess 202 polresultant e ses eee een 113 222 polresultantO 222 polresultantext 222 223 336 polresultantextO 215 223 PolreV fesse ee eee eee es 22 73 74 POLPOOCS ea era Me A ve cae 223 228 polrootst coco ba Pe eee ra 120 polrootsmod 127 223 polrootspadic 127 223 POlSturm osmosis se
50. polmods whose modulus is y are simply algebraic numbers in the finite extension defined by the polynomial y This allows us to work easily in number fields finite extensions of the p adic field Q or finite fields Note that this type is available for convenience not for speed each elementary operation involves a reduction modulo y If p is a t_POLMOD the following member functions are defined p pol return a representative of the polynomial class of minimal degree p mod return the modulus 20 Important remark Mathematically the variables occurring in a polmod are not free variables But internally a congruence class in R t y is represented by its representative of lowest degree which is a t_POL in R t and computations occur with polynomials in the variable t PARI will not recognize that Mod y y 2 1 is the same as Mod x x72 1 since x and y are different variables To avoid inconsistencies polmods must use the same variable in internal operations i e be tween polmods and variables of lower priority for external operations typically between a poly nomial and a polmod See Section 2 5 3 for a definition of priority and a discussion of PARI s idea of multivariate polynomial arithmetic For instance Mod x x72 1 Mod x x72 1 1 Mod 2 x x72 1 2i or 2i with i 1 x Mod y y 2 1 2 x Mod y y 2 1 Ain Q x y Mod x x72 1 43 Mod x y x72
51. sumnum can be slower than sumpos when the latter is applicable It is in general faster for slowly decreasing functions Finally if flag is nonzero we assume that the function f to be summed is of real type i e satisfies f z f Z which speeds up the computation p 308 a sumpos n 1 1 n 3 tn 1 time 1 410 ms tab sumnuminit 2 time 1 620 ms slower but done once and for all b sumnum n 1 2 1 n 3 n 1 tab time 460 ms 3 times as fast as sumpos a b 44 1 0 E 306 0 E 320x1 perfect sumnum n 1 2 1 n 3 n 1 tab 1 a function of real type time 240 ms 12 1 0 E 306 twice as fast no imaginary part c sumnum n 1 2 1 n72 1 tab 1 time 170 ms fast d sumpos n 1 1 n 2 1 time 2 700 ms slow d c time O ms 5 1 97 E 306 perfect For slowly decreasing function we must indicate singularities Xp 308 a sumnum n 1 2 n 4 3 time 9 930 ms slow because of the computation of n a zeta 4 3 time 110 ms 4 3 269 1 2 42 E 107 lost 200 decimals because of singularity at oo b sumnum n 1 2 4 3 n 4 3 omitted 1 of real type time 12 210 ms b zeta 4 3 3 1 05 E 300 better Since the complex values of the function are used beware of determination problems For instance p 308 tab sumnuminit 2 3 2 time 1 870 ms
52. tan acabas A be ee ee 97 tanh 2 eso kh a d bial ae Hoses 97 Taniyama Shimura Weil conjecture 133 Tate s we Anie Sa i we oe os ae ee tol Gay i eon doe ahs ond ie Ate e Bos He 226 Taylor series 2 000 65 taylor coo sm coco 226 TO ICA 2 845 be FeO Se eee Oa we 98 teichmuller 98 texMall ce Ge a bog eup oe ee a 308 309 TOXStYLO s on hak ech ake NE eek 304 308 Cheta 2 eip dence Se he Bae Bs A 98 thetamullk os s soe mece acae Re a 98 threadsSiZe s 4 4 4 meon dE ea a 312 TRUSE A oe ae amp sh eS A 226 227 thueinit sx se rrasa a 220 227 225 time expansion 55 timer e soaa aro aa a m ee wD 312 TACO cenar ee ee e a a 253 MABE sor gue ee aa GY GG be EE ed 168 Chaps Saka eee Ge a ee oe A7 299 trapO 2 655 sae eee me Popi pa 300 LEUCCCA seo wots Ga e eS ee 93 ETUDCO weed wee BRO eee ER Ow es 86 truncate sima 1 82 3 86 218 223 tSchirnhaus s i ssas m g W ior eRe we ws 202 CW E De E E 155 tULW 2 i ccnp rosado it 155 VECT V 4 444544 6 5444 racia 23 76 tutorial semea eedi a4 ace o 56 vecsearch 254 255 EYDE saed eee eee ewe Hate eee em 300 yeosmall foc ach dey ta SEA eee i 7 YPC kane e ke es o ee ee 300 Vecsmall mossos a or 76 t_CLOSURE Gea eee eee ae 20 VECSO De oe he Be ped A 254 255 GOL a ee pra da See 2 7 23 VOCSOLtOS das ms dc a e A Ore 256 ELCOMPLEX coso ac ea e aio ae eg 7 19 YECHIM etitaciesbean as bs 256 ERROR ssa oh SS See dadi Se
53. where vis a variable num ber 3 7 37 polzagier n m Creates Zagier s polynomial pm used in the functions sumalt and sumpos with flag 1 One must have m lt n The exact definition can be found in Convergence acceleration of alternating series Cohen et al Experiment Math vol 9 2000 pp 3 12 The library syntax is GEN polzag long n long m 3 7 38 serconvol x y Convolution or Hadamard product of the two power series x and y in other words if x Y az X and y bp X then serconvol z y Y ap by X The library syntax is GEN convol GEN x GEN y 3 7 39 serlaplace x x must be a power series with non negative exponents If x az k X then the result is Y az X The library syntax is GEN laplace GEN x 224 3 7 40 serreverse s Reverse power series of s i e the series t such that t s x s must be a power series whose valuation is exactly equal to one ps 8 t serreverse tan x 12 x 1 3 x73 1 5x x 5 1 7 x 7 0 x78 tan t 13 x 0 x78 The library syntax is GEN serreverse GEN s 3 7 41 subst z y z Replace the simple variable y by the argument z in the polynomial ex pression x Every type is allowed for x but if it is not a genuine polynomial or power series or rational function the substitution will be done as if the scalar components were polynomials of degree zero In particular beware that subst 1 x 1 2 3 4 1 1 0
54. x11 m11 x12 m12 x13 m13 x21 m21 x22 m22 x23 m23 Two last examples hist 10 20 returns all history entries from 10 to 420 neatly packed a single vector histlast 10 returns the last 10 history entries hist a b vector b a 1 i eval Str a 1 i histlast n vector n i eval Str i 1 2 10 Errors and error recovery 47 2 10 1 Errors Your input program is first compiled to a more efficient bytecode then the latter is evaluated calling appropriate functions from the PARI library Accordingly there are two kind of errors syntax errors produced by the compiler and runtime errors produced by the PARI library either by the evaluator itself or in a mathematical function Both kinds are fatal to your computation gp will report the error perform some cleanup restore variables modified while evaluating the erroneous command close open files reclaim unused memory etc and output its usual prompt When reporting a syntax error gp gives meaningful context by copying part of the expression it was trying to compile indicating where the error occurred with a caret as in factor too few arguments factor kkk 1 eK syntax error unexpected end 1 kkk possibly enlarged to a full arrow given enough trailing context if isprime 1 do_something syntax error unexpected if isprime 1 do_something 4K These error messages may be m
55. 0 lt m lt 10 e an exponent e a small integer in E E where E 2 log 2 and B 32 on a 32 bit machine and 64 otherwise This data may represent any real number x such that z sm10 lt 10 We consider that a t_REAL with sign s 0 has accuracy 0 so that its mantissa is useless but it still has an exponent e and acts like a machine epsilon for all accuracies lt e 17 After an optional leading or type a number with a decimal point Leading zeroes may be omitted up to the decimal point but trailing zeroes are important your t_REAL is assigned an internal precision which is the supremum of the input precision one more than the number of decimal digits input and the default precision For example if the default precision is 28 digits typing 2 yields a precision of 28 digits but 2 0 0 with 45 zeros gives a number with internal precision at least 45 although less may be printed You can also use scientific notation with the letter E or e As usual en is interpreted as x10 for all integers n Since the result is converted to a t_REAL you may often omit the decimal point in this case 6 02 E 23 or 1e 5 are fine but e10 is not By definition 0 E n returns a real 0 of exponent n whereas 0 returns a real 0 of default precision of exponent realprecision see Section 1 3 7 behaving like the machine epsilon for the current default accuracy any float of smaller absolute value is ind
56. 0 if e 1 and f 1 and 1 if f gt 1 and e 1 The following function computes the cardinality of a subgroup of G as given by the components of g card H my o H 2 prod i 1 0 o i nf nfinit x 6 3 gal galoisinit nf pr idealprimedec nf 3 1 g idealramgroups nf gal pr apply card g 4 6 6 3 3 3 cardinalities of the G_i nf nfinit x 6 108 gal galoisinit nf pr idealprimedec nf 2 1 iso idealramgroups nf gal pr 2 4 Vecsmal1 2 3 1 5 6 4 Vecsmal1 3 nfdisc galoisfixedfield gal iso 1 45 3 The field fixed by the inertia group of 2 is not ramified at 2 The library syntax is GEN idealramgroups GEN nf GEN gal GEN pr 181 3 6 68 idealred nf J v 0 LLL reduction of the ideal J in the number field nf along the direction v The v parameter is best left omitted but if it is present it must be an nf r1 nf r2 component vector of non negative integers What counts is the relative magnitude of the entries if all entries are equal the effect is the same as if the vector had been omitted This function finds a small a in T see the end for technical details The result is the Hermite normal form of the reduced ideal J rI a where r is the unique rational number such that J is integral and primitive This is usually not a reduced ideal in the sense of Buchmann K nfinit y 2 1 P idealprimedec K 5 1 idealred K P 1
57. 1 2 In particular Re acosh x gt 0 and In acosh x 7 7 0 if x R and z lt 1 then acosh x is complex The library syntax is GEN gacosh GEN x long prec 3 3 9 agm z y Arithmetic geometric mean of z and y In the case of complex or negative numbers the optimal AGM is returned the largest in absolute value over all choices of the signs of the square roots p adic or power series arguments are also allowed Note that a p adic agm exists only if x y is congruent to 1 modulo p modulo 16 for p 2 x and y cannot both be vectors or matrices The library syntax is GEN agm GEN x GEN y long prec 3 3 10 arg x Argument of the complex number x such that r lt arg x lt r The library syntax is GEN garg GEN x long prec 3 3 11 asin x Principal branch of sin t x ilog ix V1 22 In particular Re asin 2 7 2 7 2 and if x R and x gt 1 then asin x is complex The branch cut is in two pieces co 1 continuous with quadrant II and 1 oo continuous with quadrant IV The function satisfies iasin x asinh ix The library syntax is GEN gasin GEN x long prec 3 3 12 asinh x Principal branch of sinh x log a V1 22 In particular Im asinh x 7 2 7 2 The branch cut is in two pieces i oo i continuous with quadrant III and i i oo continuous with quadrant I The library syntax is GEN gasinh GEN x long prec 90 3 3 13 atan
58. 101 bDilhell s ce sear eG wa Boe wee ee we G 135 bihaife AS TE binary Te es a ss rr ae oe ee 302 binary file eiii eno ed 57 297 binary flag saaa ea us ea senses 63 binary quadratic form 22 74 DIMany oe eg ica be ee mS eek e a 76 binomial coefficient 101 DIROMIAL e aces lel ee Bl ce 101 102 binomialuu 102 Birch and Swinnerton Dyer conjecture 137 bitand a supa b ae ste aa eee ky be E 77 A eek ee Se Bee 2 Gi bitnegimply escitas e e Gob bee e ae etd bDitOr essa eee bbe ea ee ES oo 77 Bittest spea 4 e bree ea PE RG A 18 bitwise and s sorore m 4 64 hob dee es iii bitwise exclusive or 78 bitwise inclusive or 77 bitwise negation 0 77 Bitxor asis ee eG ened eet awa 78 DIG amp ve cae eo a a Aap ae A ne 44 151 157 DHE ceca bias he ee RA we a 155 bnficertify eo ccoo 157 bnfcertifyO cas gba eae cs 157 bnfcompress 157 158 bnfdecodemodule 158 165 bnfinit ss a miaa 124 151 157 158 182 PHENO ces ra e 159 bnfisintnorm 159 160 bnfisintnormabs 159 bnfisnorm ecc oros 159 160 bnfisprincipal 123 159 160 182 bnfisprincipal0 161 POTTSSUDIE se ate iia a A 161 bnfisunit s saie eee ee Re ee a 161 bntnarrow vse 2 dba 2a wd 124 161 DO NeWpTeC o ooo oo 195 DOTSIgDUNDIE sse iaa ee ee A 162 PO SUDA E soe asias 4 66 We ee bet 162 DNR cok wid ea
59. 11 E 1 0 1 1 extern echo Order s gap q 3 12 The library syntax is GEN qfautoexport GEN qfa long flag 3 8 53 qfbil x y q Evaluate the bilinear form q symmetric matrix at the vectors x y if q omitted use the standard Euclidean scalar product corresponding to the identity matrix Roughly equivalent to x q y but a little faster and more convenient does not distinguish between column and row vectors x 1 2 3 y 1 0 1 qfbil x y 1 2 q 1 2 3 2 2 1 3 1 0 qfbil x y q 2 13 for i 1 10 6 qfbil x y q 3 568ms for i 1 10 6 x q y 4 717ms The associated quadratic form is also available as qfnorn slightly faster for i 1 10 6 qfnorm x q time 444ms for i 1 10 6 qfnorm x time 176 ms for i 1 10 6 qfbil x y time 208 ms The library syntax is GEN qfbil GEN x GEN y GEN q NULL 3 8 54 qfgaussred q decomposition into squares of the quadratic form represented by the sym metric matrix q The result is a matrix whose diagonal entries are the coefficients of the squares and the off diagonal entries on each line represent the bilinear forms More precisely if a denotes the output one has a t Y ases aja i j i qfgaussred 0 1 1 0 Z1 1 2 1 1 1 2 This means that 2xy 1 2 x y 1 2 x y 246 The library syntax is GEN qfgaussred GEN q GEN qfgaussred_positive G
60. 2 H bnrclassnolist bnf L H 98 14 1 3 1 1 L 1 98 ids vector 1 i 1 i mod 1 5 98 88 0 1 14 O O 7 98 10 O 1 The weird 1 i mod 1 is the first component of 1 i mod i e the finite part of the con ductor This is cosmetic since by construction the Archimedean part is trivial I do not want to see it This tells us that the ray class groups modulo the ideals of norm 98 printed as 5 have respectively order 1 3 and 1 Indeed we may check directly bnrclassno bnf ids 2 16 3 The library syntax is GEN bnrclassnolist GEN bnf GEN list 3 6 23 bnrconductor A B C flag 0 Conductor f of the subfield of a ray class field as defined by A B C of type bnr bnr subgroup Conf modulus or bnf modulus subgroup Section 3 6 5 If flag 0 returns f If flag 1 returns f Cl H where Cl is the ray class group modulo f as a finite abelian group finally H is the subgroup of Cl defining the extension If flag 2 returns f bnr f H as above except Cl is replaced by a bnr structure as output by bnrinit f 1 The library syntax is GEN bnrconductorO0 GEN A GEN B NULL GEN C NULL long flag Also available is GEN bnrconductor GEN bnr GEN H long flag 3 6 24 bnrconductorofchar bnr chi bnr being a big ray number field as output by bnrinit and chi being a row vector representing a character as expressed on the generators of the
61. 2 Important warning For D lt 0 this function may give incorrect results when the class group has many cyclic factors because implementing Shanks s method in full generality slows it down immensely It is therefore strongly recommended to double check results using either the version with flag 1 or the function quadclassunit 121 Warning Contrary to what its name implies this routine does not compute the number of classes of binary primitive forms of discriminant D which is equal to the narrow class number The two notions are the same when D lt 0 or the fundamental unit e has negative norm when D gt 0 and Ne gt 0 the number of classes of forms is twice the ordinary class number This is a problem which we cannot fix for backward compatibility reasons Use the following routine if you are only interested in the number of classes of forms QFBclassno D qfbclassno D if D lt O norm quadunit D lt 0 1 2 Here are a few examples qfbclassno 400000028 time 3 140 ms 1 1 quadclassunit 400000028 no time 20 ms much faster 12 1 gfbclassno 400000028 time O ms 13 7253 correct and fast enough quadclassunit 400000028 no time O ms 14 7253 See also qfbhclassno The library syntax is GEN gfbclassnoO GEN D long flag The following functions are also available GEN classno GEN D flag 0 GEN classno2 GEN D flag 1 Finally GEN hclassno GE
62. 242 Matsnf wees vee hime a deus 237 242 natsit Or e a ak a8 ee 4 A 8 243 matsolve o cata ien oe e ennea 243 matsolvemod aoaaa a 243 matsolvemodO 243 matsupplement 243 mattranspose 244 MAX ok eth ky eee ee eee 69 member functions 44 130 155 Min bh a HS aR ed Sa Shes Ss 69 Minim sone 8 ba HOR ae a Be we we 250 MIDIMZ oye ee Hy ew Hae ee a A 250 minimal model 137 142 minimal polynomial 244 minimal vector 250 minin Taw s e ieem ee EOE a a 250 MINPOLY o a s eaii Ge a a a a we a a 244 Nod 2 86 85 448 BR Awe eR ee ae Dod oe ee ile ae hb hee eb SK ar ee A 155 MOdPF vip gS e ee Se a 195 MOGpPrinit 2 6 bee ee ee 190 modreverse 183 184 201 MOGULS esoe ob ee Swe HR Ee Eo 153 Moebius 08 99 118 MOCDIUS irc Yeas Se A AS 99 118 Mordell Weil group 137 139 140 146 Mpcatalam s ss ea Gtk e Oe a aS 89 Mpeuler oo coo oo ocn ss 89 Upiact eoria aa a By cece aia er hee A 110 MpPlLaCt roc cts db lA os 110 MPPl e e neasi ipat 89 MPOS asia ac o o ee 99 110 multivariate polynomial 42 MV a ere Gee ee ee ee a es 34 38 N Nbessel sb ee ee RS Se Ee BM 92 nbthreadS soa esea maant ew ee 308 newtonpoly 0 184 new_galois_format 199 305 NOK Ge ce og ws WE ag Bk A he Se 50 286 MEXtprime 2 22 bees be ee RS TLS 119 Nf Aaa e we a 44 151 DE a a a E its a 155 DEA ica a
63. 3 12 17 gettime Returns the time in milliseconds elapsed since either the last call to gettime or to the beginning of the containing GP instruction if inside gp whichever came last For a reentrant version see getabstime The library syntax is long gettime 3 12 18 global list of variables Obsolete Scheduled for deletion 3 12 19 inline z z Experimental declare x z as inline variables Such variables behave like lexically scoped variable see my but with unlimited scope It is however possible to exit the scope by using uninline When used in a GP script it is recommended to call uninline before the script s end to avoid inline variables leaking outside the script 3 12 20 input Reads a string interpreted as a GP expression from the input file usually standard input i e the keyboard If a sequence of expressions is given the result is the result of the last expression of the sequence When using this instruction it is useful to prompt for the string by using the print1 function Note that in the present version 2 19 of pari el when using gp under GNU Emacs see Section 2 16 one must prompt for the string with a string which ends with the same prompt as any of the previous ones a will do for instance 292 3 12 21 install name code gpname lib Loads from dynamic library lib the function name Assigns to it the name gpname in this gp session with prototype code see below Tf gpn
64. 3 6 102 nfhilbert nf a b pr If pr is omitted compute the global quadratic Hilbert symbol a b in nf that is 1 if 2 ay bz has a non trivial solution x y z in nf and 1 otherwise Otherwise compute the local symbol modulo the prime ideal pr as output by idealprimedec The library syntax is long nfhilbertO GEN nf GEN a GEN b GEN pr NULL Also available is long nfhilbert GEN bnf GEN a GEN b global quadratic Hilbert symbol 3 6 103 nfhnf nf x Given a pseudo matrix A I finds a pseudo basis in Hermite normal form of the module it generates The library syntax is GEN nfhnf GEN nf GEN x Also available GEN rnfsimplifybasis GEN bnf GEN x simplifies the pseudo basis given by x 4 1 The ideals in the list J are integral primitive and either trivial equal to the full ring of integer or non principal 3 6 104 nfhnfmod nf x detz Given a pseudo matrix A J and an ideal detz which is contained in read integral multiple of the determinant of A J finds a pseudo basis in Hermite normal form of the module generated by 4 1 This avoids coefficient explosion detz can be computed using the function nfdetint The library syntax is GEN nfhnfmod GEN nf GEN x GEN detx 192 3 6 105 nfinit pol flag 0 pol being a non constant preferably monic irreducible polynomial in Z X initializes a number field structure nf associated to the field K defined by pol As such it s a technical object
65. 446 26 be HS Gb Fo Se aS GE eo 251 337 GirepO 2 22428 48 5464 rasa 251 GESTION e is ad See Se eS 251 UPOP fas ee ee RS ee Oe ee 93 Qp gamma s re ea ee eee poe 94 UP L0S amp a eae a Eee Be 95 Qpssqrt 2 id eh oP aS a a ae i 96 97 QR decomposition 241 QE he ee aa ao 271 quadclassunit s s itie sn dass 124 quadclassunitO 124 Quaddisc 2 2445 4 59452 64 105 124 quadgen 2 2 he ees ea 124 125 quadhilbert sss seese rnaar 125 quadpoly se ea ss e poed ei ee gt 125 quadpolyO s ttn ke ed a SES 125 quadratic number 7 8 20 quadray so cra seen ba es ae ee gas 125 quadregula 124 quadregulator 125 QUAGUNEG s sa a ee ES 125 QUEb i en dos as doe A Ae Ae ke an S Df 297 QUOTE lt robes a obs bs 294 quotient oo se 6 bee s m 65 R Dl aba Bees ee St ee Ye a 155 T2 Se Se ao Bae eee ho Ge Be ee E 155 ramification group 181 random so esc s HoH 4a ee eysa 84 292 randomprime 125 TANK pos 6 605 e ron bg By we SS le eS 242 rational function 32 rational number 7 8 18 TOW fOTMGL o ce mc we a 308 read at Bie base Eon he 47 55 297 302 Head Line s i ica ee hoe a eee we 310 TOAGSEM k op ae ete AS eek ae ws 298 H6adVeG 253 AG ba Re RES 47 298 real number 8 LF Teal peat e aang a ew S 85 realprecision 18 57 88 310 TOCOMED so eor Ge ace Gee Be e 311 recursion depth
66. 6 37 galoisidentify gal gal being be a Galois group as output by galoisinit output the isomorphism class of the underlying abstract group as a two components vector o i where o is the group order and 7 is the group index in the GAP4 Small Group library by Hans Ulrich Besche Bettina Eick and Eamonn O Brien This command also accepts subgroups returned by galoissubgroups The current implementation is limited to degree less or equal to 127 Some larger easy orders are also supported The output is similar to the output of the function IdGroup in GAP4 Note that GAP4 IdGroup handles all groups of order less than 2000 except 1024 so you can use galoisexport and GAP4 to identify large Galois groups The library syntax is GEN galoisidentify GEN gal 3 6 38 galoisinit pol den Computes the Galois group and all necessary information for com puting the fixed fields of the Galois extension K Q where K is the number field defined by pol monic irreducible polynomial in Z X or a number field as output by nfinit The extension K Q must be Galois with Galois group weakly super solvable see below returns 0 otherwise Hence this permits to quickly check whether a polynomial of order strictly less than 36 is Galois or not The algorithm used is an improved version of the paper An efficient algorithm for the com putation of Galois automorphisms Bill Allombert Math Comp vol 73 245 2001 pp 359 375 A group G is s
67. 7 E ellinit ellfromj ffgen 5710 ellcard E 5 9762580 P random E ellorder E P 76 4881290 18 ti p 27160 7 E ellinit 1 2 p N ellcard E 1461501637330902918203686560289225285992592471152 o N factor N for i 1 100 ellorder E random E me 260 ms The parameter o is now mostly useless and kept for backward compatibility If present it represents a non zero multiple of the order of z see Section 3 4 2 the preferred format for this parameter is Lord factor ord where ord is the cardinality of the curve It is no longer needed 144 since PARI is now able to compute it over large finite fields was restricted to small prime fields at the time this feature was introduced and caches the result in E so that it is computed and factored only once Modifying the last example we see that including this extra parameter provides no improvement o N factor N for i 1 100 ellorder E random E o time 260 ms The library syntax is GEN ellorder GEN E GEN z GEN o NULL The obsolete form GEN orderell GEN e GEN z should no longer be used 3 5 36 ellordinate E x Gives a 0 1 or 2 component vector containing the y coordinates of the points of the curve E having x as x coordinate The library syntax is GEN ellordinate GEN E GEN x long prec 3 5 37 ellperiods w flag 0 Let w describe a complex period lattice w w 1 w2 or an ellinit structure
68. 93 1 2 3 4 5 6 25 M 2 3 44 2 3 5 6 8 9 All this is recursive so if M is a matrix of matrices of an expression such as M 1 1 3 4 1 is perfectly valid and actually identical toM 1 1 4 3 1 assuming that all matrices along the way have compatible dimensions Technical note design flaw Matrices are internally represented as a vector of columns All matrices with 0 columns are thus represented by the same object internally an empty vector and there is no way to distinguish between them Thus it is not possible to create or represent matrices with zero columns and an actual nonzero number of rows The empty matrix is handled as though it had an arbitrary number of rows exactly as many as needed for the current computation to make sense 1 2 3 4 5 6 LE 1 The empty matrix on the first line is understood as a 3 x 0 matrix and the result as a 2 x 0 matrix On the other hand it is possible to create matrices with a given positive number of columns each of which has zero rows e g using Mat as above or using the matrix function Note that although the internal representation is essentially the same a row vector of column vectors is not a matrix for example multiplication will not work in the same way It is easy to go from one representation to the other using Vec Mat though 1 2 3 4 5 6 1 1 2 3 4 5 6 Vech 12 1 4 2 5 3 6 M
69. Euler s constant y 0 57721 Note that Euler is one of the few reserved names which cannot be used for user variables The library syntax is GEN mpeuler long prec 3 3 4 I The complex number y 1 The library syntax is GEN gen_I 3 3 5 Pi The constant 7 3 14159 Note that Pi is one of the few reserved names which cannot be used for user variables The library syntax is GEN mppi long prec 89 3 3 6 abs xz Absolute value of x modulus if x is complex Rational functions are not allowed Contrary to most transcendental functions an exact argument is not converted to a real number before applying abs and an exact result is returned if possible abs 1 fi 1 abs 3 7 4 7 I 42 5 7 abs 1 I 3 1 414213562373095048801688724 If x is a polynomial returns x if the leading coefficient is real and negative else returns x For a power series the constant coefficient is considered instead The library syntax is GEN gabs GEN x long prec 3 3 7 acos x Principal branch of cos x ilog x iV1 22 In particular Re acos x 0 7 and if x R and x gt 1 then acos x is complex The branch cut is in two pieces co 1 continuous with quadrant II and 1 o00 continuous with quadrant IV We have acos x 7 2 asin x for all z The library syntax is GEN gacos GEN x long prec 3 3 8 acosh x Principal branch of cosh x 2log y 2 1 2
70. F 27277 1 1009 100003 factor F 1075 fast incomplete time O ms 4 1009 1 34029257539194609161727850866999116450334371 1 factor F 1079 very slow time 6 892 ms 6 1009 1 100003 1 340282366920938463463374607431768211457 1 factorint F 1 8 much faster all small primes were found time 12 ms 47 1009 1 100003 1 340282366920938463463374607431768211457 1 factor F complete factorisation time 112 ms 18 1009 1 100003 1 59649589127497217 1 5704689200685129054721 1 Over Q the prime factors are sorted by increasing size 107 Rational functions The polynomials or rational functions to be factored must have scalar coefficients In particular PARI does not know how to factor multivariate polynomials The following domains are currently supported Q R C Q finite fields and number fields See factormod and factorff for the algorithms used over finite fields factornf for the algorithms over number fields Over Q van Hoeij s method is used which is able to cope with hundreds of modular factors The routine guesses a sensible ring over which to factor the smallest ring containing all coefficients taking into account quotient structures induced by t_INTMODs and t_POLMODs e g if a coefficient in Z nZ is known all rational numbers encountered are first mapped to Z nZ different moduli will produce an error Factoring modulo a non prime n
71. GEN pol GEN a NULL GEN b NULL Also avail able is long sturm GEN pol total number of real roots 3 7 33 polsubcyclo n d vu xj Gives polynomials in variable v defining the sub Abelian extensions of degree d of the cyclotomic field Q where d n If there is exactly one such extension the output is a polynomial else it is a vector of polyno mials possibly empty To get a vector in all cases use concat polsubcyclo n d The function galoissubcyclo allows to specify exactly which sub Abelian extension should be computed The library syntax is GEN polsubcyclo long n long d long v 1 where v is a vari able number 3 7 34 polsylvestermatrix z y Forms the Sylvester matrix corresponding to the two polyno mials x and y where the coefficients of the polynomials are put in the columns of the matrix which is the natural direction for solving equations afterwards The use of this matrix can be essential when dealing with polynomials with inexact entries since polynomial Euclidean division doesn t make much sense in this case The library syntax is GEN sylvestermatrix GEN x GEN y 3 7 35 polsym x n Creates the column vector of the symmetric powers of the roots of the polynomial x up to power n using Newton s formula The library syntax is GEN polsym GEN x long n 3 7 36 poltchebi n v x Deprecated alias for polchebyshev The library syntax is GEN polchebyshevi long n long v 1
72. GEN y Also available is GEN ibitor GEN x GEN y which returns the bitwise ir of x and y two integers 77 3 2 23 bittest x n Outputs the n bit of x starting from the right i e the coefficient of 2 in the binary expansion of x The result is 0 or 1 bittest 7 3 1 1 the 3rd bit is 1 bittest 7 4 2 0 the 4th bit is 0 See Section 3 2 19 for the behavior at negative arguments The library syntax is GEN gbittest GEN x long n For a t_INT x the variant long bittest GEN x long n is generally easier to use and if furthermore n gt 0 the low level function ulong int_bit GEN x long n returns bittest abs x n 3 2 24 bitxor z y Bitwise exclusive or of two integers x and y that is the integer Y mixer yi See Section 3 2 19 for the behavior for negative arguments The library syntax is GEN gbitxor GEN x GEN y Also available is GEN ibitxor GEN x GEN y which returns the bitwise zor of x and y two integers 3 2 25 ceil x Ceiling of x When z is in R the result is the smallest integer greater than or equal to x Applied to a rational function ceil x returns the Euclidean quotient of the numerator by the denominator The library syntax is GEN gceil GEN x 3 2 26 centerlift x v Same as lift except that t_INTMOD and t_PADIC components are lifted using centered residues e for a t_INTMOD x Z nZ the lift y is such that n 2 lt y lt n 2 e a t_PADIC z is lifted
73. GP command as if directly input from the keyboard and returns its output x1 one x2 two n 1 eval Str x n 72 one Tof exp v 1 eval Str f v 4 2 7182818284590452353602874713526624978 Note that the first construct could be implemented in a simpler way by using a vector x one two x n and the second by using a closure f exp f v The final example is more interesting genmat u v matrix u v i j eval Str x i j genmat 2 3 MAN generic 2 x 3 matrix 12 x11 x12 x13 x21 x22 x23 A syntax error in the evaluation expression raises an e_SYNTAX exception which can be trapped as usual la EK unused characters la 44 Pi Elexpr iferr eval expr e print syntax error errname e e_SYNTAX E 1 1 11 2 E 1ta syntax error The library syntax is geval GEN x 217 3 7 6 factorpadic pol p r p adic factorization of the polynomial pol to precision r the re sult being a two column matrix as in factor Note that this is not the same as a factorization over Z p Z polynomials over that ring do not form a unique factorization domain anyway but approximations in Q p Z of the true factorization in Q X factorpadic x 2 9 3 5 1 1 0 375 xx 2 0 375 x 372 0 375 1 factorpadic x 2 1 5 3 2 1 0 573 x 2 5 2 572 0 573 1 1 0 0573 x 3 3 5 2 5 2 0 573 1
74. Instead of hard coded numerical flags one should rather use GEN isprincipalray GEN bnr GEN x for flag 0 and if you want generators bnrisprincipal bnr x nf_GEN 166 3 6 30 bnrrootnumber bnr chi flag 0 If x chi is a character over bnr not necessarily primitive let L s xX gt y x id N id be the associated Artin L function Returns the so called Artin root number i e the complex number W x of modulus 1 such that A 8 x W x A s X where A s x A x 7 s L s x is the enlarged L function associated to L The generators of the ray class group are needed and you can set flag 1 if the character is known to be primitive Example bnf bnfinit x 2 x 57 bnr bnrinit bnf 7 1 1 1 bnrrootnumber bnr 2 1 returns the root number of the character x of Cl7w o Q w229 defined by x 9193 e Here 91 92 are the generators of the ray class group given by bnr gen and e27 M a e27 N2 where N1 N2 are the orders of g and ga respectively N 6 and N 3 as bnr cyc readily tells us The library syntax is GEN bnrrootnumber GEN bnr GEN chi long flag long prec 3 6 31 bnrstark bnr subgroup bnr being as output by bnrinit 1 finds a relative equation for the class field corresponding to the modulus in bnr and the given congruence subgroup as usual omit subgroup if you want the whole ray class group The main variable of bnr must not be x and the ground field a
75. K zk is not an Ox module To add arbitrary Z modules generated by the columns of matrices A and B use mathnf concat A B The library syntax is GEN idealadd GEN nf GEN x GEN y 173 3 6 46 idealaddtoone nf x y x and y being two co prime integral ideals given in any form this gives a two component row vector a b such that a x b E yanda b 1 The alternative syntax idealaddtoone nf v is supported where v is a k component vector of ideals given in any form which sum to Zg This outputs a k component vector e such that e Exp for 1 lt i lt k and gt lt lt z eli 1 The library syntax is GEN idealaddtoone0 GEN nf GEN x GEN y NULL 3 6 47 idealappr nf x flag 0 If x is a fractional ideal given in any form gives an element o in nf such that for all prime ideals p such that the valuation of x at p is non zero we have Up Up x and vp a gt 0 for all other p If flag is non zero x must be given as a prime ideal factorization as output by idealfactor but possibly with zero or negative exponents This yields an element such that for all prime ideals p occurring in z vp a is equal to the exponent of p in x and for all other prime ideals vpla gt O This generalizes idealappr nf x 0 since zero exponents are allowed Note that the algorithm used is slightly different so that idealappr nf idealfactor nf x may not be the same as idealappr nf x 1 The library syntax is GEN i
76. LONG_MAX 23 1 for 32 bit machines or 263 1 for 64 bit machines if x is an exact object The library syntax is long padicprec GEN x GEN p 3 2 46 permtonum z Given a permutation x on n elements gives the number k such that x numtoperm n k i e inverse function of numtoperm The numbering used is the standard lexicographic ordering starting at 0 The library syntax is GEN permtonum GEN x 3 2 47 precision z n The function has two different behaviors according to whether n is present or not If n is missing the function returns the precision in decimal digits of the PARI object x If x is an exact object the largest single precision integer is returned precision exp 1e 100 1 134 134 significant decimal digits precision 2 x 12 2147483647 exact object precision 0 5 O x 13 28 floating point accuracy NOT series precision precision exp 1le 100 0 5 14 28 minimal accuracy among components The return value for exact objects is meaningless since it is not even the same on 32 and 64 bit machines The proper way to test whether an object is exact is isexact x precision x precision 0 If n is present the function creates a new object equal to x with a new precision n This never changes the type of the result In particular it is not possible to use it to obtain a polynomial from a power series for that see truncate Now the meaning of precision is
77. O 0 0 O 0 1 O O O O 4 O O 1 1 1 1 0 O O 1 O O 1 0 005 0 03005 0 01 0123 00 45 250 0 L 0 0 04 1 0 15 0 0 05 05 0 0 0 0 0 0 0 O O 0 2 4 1 1 O 0 1 1 O 1 1 1 1 0 dy Ona dis Le 0 Obs Ls Ls 0 L 4 003 05 15 05 L 1 0 dd 0 15 1L 249 1 1 Wynd 05 03 0 5 tg she 1 07 Lo 7 005 05 Lo de Os hy 00 15 0 0 0 0 0 O O O 0 1 O O O 1 4 O O O 1 O O O 0 O 0 0 O 0 0 O O 0 1 1 O O 1 1 O 4 O O O O 1 1 O O 0 0 1 0 2 0 0 0 0 O 0 1 0 0 0 O 4 1 1 1 0 O 1 1 1 0 0 1 0 0 1 O 1 0 1 1 1 0 0 0 1 4 O 1 1 O 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 1 1 0 1 0 4 0 1 1 O 1 1 1 1 0 1 1 0 1 O 1 1 1 0 1 0 O 1 1 0 4 0 O 1 1 0 0 1 1 1 O 0 1 0 O O 1 1 O O 1 O 1 1 O 4 1 O 1 0 Ly 1 do ds Tee Oi Os Or Og Ll 04 45 05 1 05 05 Ty O Lo 4 03 1 0 15 05 1 0 LL 46 05 Ty 05 40 1605 Os 0 Lo d Os ds 5050 4 13 2d 07 10 Oy Li O 0 50 50 0 03 000500315 0 45 y do dor de 45 4 gfminim x 0 the Leech lattice has 196560 minimal vectors of norm 4 time 648 ms 74 196560 4 gfminim x 0 2 safe algorithm Slower and unnecessary here time 18 161 ms 5 196560 4 000061035156250000 In the last example we store 0 vectors to limit memory use All minimal vectors are nevertheless enumerated Provided parisize is
78. Pw eS doe 94 LLAMA sos cs a ne A eee se a 94 e ae tbe ede eR ew ae ee es 113 gecd0 bk bebe ee bee ee Es 113 ESTIMO sk s wd ds west ea ww Aw 215 shell wees eae hee ce ee ees 139 imag co bee ee pren a es 80 Bilsanypower gt se toled E ope oa 115 gispriMe 2 si oe ee eee es 116 gispseudoprime 116 BUSSQUATE ssie g gime we ee a a 117 gissquareall 117 glambdak 0 c00 2 2 ee 215 glambertW soss seda saani eoe ae a 94 ELCO 2 22s bee ewe rr a 118 glength 2 26 be eee be be es 81 GINPAMMS e o o are eo e E a a 95 global sasaaa ee ea ee Rw a 292 Elg ecca ee Eee ee ee 95 CMAN ewan ee e Ge By Mia Ge 69 DM bee ea Sk ee a 69 pod sii ca ae ee ae 66 BMOGULO de sie i wien oS eee eh E 73 PWL woo AR EG ee eee 65 poulin os ew ae RS ee He as 69 330 POR aioe a bbe PEGA DEG SaaS 64 GOOT yb me a SS 82 a A ey ek eR ee ew a 244 PNOMMIP be ver ee EERE eee ee 245 BD repere del e ena e a Eh 5 GP 2 aie ee he eee Se ae G AA 5 EP mosca sabreis coa 13 EPDZLO 6 334 gee Soe ba o EE 5 EPDSTD See ah he Se ee ea ee 56 epinstall seer epay wae ee es 293 BPOlVals hoa de eee ee se ae we 87 EPOLYLOE sea ds A ke wea 96 A 67 89 EPEC a oeeie Re 13 32 54 58 GPRG en a e aria ete ate a Ors R 60 EDIC wk od Gow ER EG eee as 308 EPISC che SEY eR ee ed ee le dx 84 OPEL ac feeb ye kee dhe Ged Ae de he AS 96 gpwritebin sare esros ee eS 302 EPLLTACTOLO e on Sw aa ee oe 108 Bp BetenvV sesser es ee 2 291 gp_readvec_fil
79. Q 27 1 3 43 2 pi L 1 p2 L 2 pi e p1 f the first is unramified of degree 1 4 1 1 p2 e p2 f the second is unramified of degree 2 5 1 2 pl gen 6 za 5 2 1 0 nfbasistoalg K 21 a uniformizer for pl 47 Mod x 2 x 3 2 The library syntax is GEN idealprimedec GEN nf GEN p 180 3 6 66 idealprincipalunits nf pr k Given a prime ideal in idealprimedec format returns the multiplicative group 1 pr 1 pr as an abelian group This function is much faster than idealstar when the norm of pr is large since it avoids useless work in the multiplicative group of the residue field K nfinit y 2 1 P idealprimedec K 2 1 G idealprincipalunits K P 20 G cyc 512 256 4 2 512 x Z 256 x Z 4 G gen 15 1 2 1021 O 1 minimal generators of given order The library syntax is GEN idealprincipalunits GEN nf GEN pr long k 3 6 67 idealramgroups nf gal pr Let K be the number field defined by nf and assume that K Q is Galois with Galois group G given by gal galoisinit nf Let pr be the prime ideal Y in prid format This function returns a vector g of subgroups of gal as follow e g 1 is the decomposition group of YB e g 2 is Go B the inertia group of Y and for i gt 2 e gli is G _2 B the i 2 th ramification group of PY The length of g is the number of non trivial groups in the sequence thus is
80. This routine computes an LLL reduced basis for the ring of integers of Q X T then exam ines small linear combinations of the basis vectors computing their characteristic polynomials It returns the separable P polynomial of smallest discriminant the one with lexicographically smallest abs Vec P in case of ties This is a good candidate for subsequent number field computations since it guarantees that the denominators of algebraic integers when expressed in the power basis are reasonably small With no claim of minimality though It can happen that iterating this functions yields better and better polynomials until it sta bilizes p5 P X712 8 X 8 50 X 6 16 X 4 3069 X 2 625 poldisc P 1 12 1 2622 E55 P polredbest P poldisc P 1 14 2 9012 E51 P polredbest P poldisc P 1 6 8 8704 E44 In this example the initial polynomial P is the one returned by polredabs and the last one is stable If flag 1 outputs a two component row vector P a where P is the default output and Mod a P is a root of the original T P a polredbest x 4 8 1 1 x74 2 Mod x73 x74 2 charpoly a 201 12 x 4 8 In particular the map Q z T gt Ql z P x gt Mod a P defines an isomorphism of number fields which can be computed as subst lift Q x a if Q is a t_POLMOD modulo T b modreverse a returns a t_POLMOD giving the inverse of the above map wh
81. a strictly lower priority than any variable in use at this point To see the ordering used by gp at any given time type variable Given such an ordering multivariate polynomials are stored so that the variable with the highest priority is the main variable And so on recursively until all variables are exhausted A different storage pattern which could only be obtained via libpari programming and low level constructors would produce an invalid object and eventually a disaster In any case if you are working with expressions involving several variables and want to have them ordered in a specific manner in the internal representation just described the simplest is just to write down the variables one after the other under gp before starting any real computations This is not strictly true the variable x is predefined and always has the highest possible priority 32 You could also define variables from your gprc to have a consistent ordering of common variable names in all your gp sessions e g read in a file variables gp containing x y z t a b c d Important note PARI allows Euclidean division of multivariate polynomials but assumes that the computation takes place in the fraction field of the coefficient ring if it is not an integral domain the result will a priori not make sense This can become tricky for instance assume zx has highest priority which is always the case then y 7xhy wi 0 Pyhx 12 y th
82. a t_PADIC x the function GEN Qp_log GEN x is also available 3 3 43 polylog m x flag 0 One of the different polylogarithms depending on flag If flag 0 or is omitted m polylogarithm of z i e analytic continuation of the power series Li 2 gt gt 1 1 x lt 1 Uses the functional equation linking the values at z and 1 x to restrict to the case x lt 1 then the power series when x lt 1 2 and the power series expansion in log x otherwise Using flag computes a modified m polylogarithm of x We use Zagier s notations let Rm denote R or S depending on whether m is odd or even If flag 1 compute D a defined for x lt 1 by m 1 al k Mi m 1 Re E E i ae rosie log 1 a k k 0 95 If flag 2 compute D 1 defined for x lt 1 by am E Cay par k 2 m k 0 If flag 3 compute P 1 defined for x lt 1 by m 1 2F B l Or Bm oA Rm log z Lim 2 log z i k k 0 These three functions satisfy the functional equation fm 1 x 1 f x The library syntax is GEN polylogO long m GEN x long flag long prec Also avail able is GEN gpolylog long m GEN x long prec flag 0 3 3 44 psi x The 4 function of x i e the logarithmic derivative I x T 2 The library syntax is GEN gpsi GEN x long prec 3 3 45 sin x Sine of x The library syntax is GEN gsin GEN x long prec 3 3 46 sinh x Hyperbolic
83. about 50MB qfminim x succeeds in 2 5 seconds The library syntax is GEN gfminimO GEN x GEN b NULL GEN m NULL long flag long prec Also available are GEN minim GEN x GEN b NULL GEN m NULL flag 0 GEN minim2 GEN x GEN b NULL GEN m NULL flag 1 GEN minim_raw GEN x GEN b NULL GEN m NULL do not perform LLL reduction on x 3 8 61 qfnorm z q Evaluate the binary quadratic form q symmetric matrix at the vector z If q omitted use the standard Euclidean form corresponding to the identity matrix Equivalent to x q x but about twice faster and more convenient does not distinguish between column and row vectors x 1 2 3 qfnorm x 1 q 1 2 3 2 2 1 3 1 0 qfnorm x q A2 for i 1 10 6 qfnorm x q time 384ms for i 1 10 6 x q x time 729ms We also allow t_MATs of compatible dimensions for x and return x q x in this case as well 7 M 1 2 3 4 5 6 7 8 91 qfnorm M Gram matrix 15 66 78 90 78 93 108 90 108 126 for i 1 10 6 qfnorm M q time 2 144 ms for i 1 10 6 M q M 250 time 2 793 ms The polar form is also available as qfbil The library syntax is GEN qfnorm GEN x GEN q NULL 3 8 62 qfperfection G G being a square and symmetric matrix with integer entries representing a positive definite quadratic form outputs the perfection rank of the form That is gives the rank of the family of the s symmetric
84. alarm s 0 code If code is omitted trigger an e ALARM exception after s seconds cancelling any previously set alarm stop a pending alarm if s 0 or is omitted Otherwise if s is positive the function evaluates code aborting after s seconds The return value is the value of code if it ran to completion before the alarm timeout and a t_ERROR object otherwise p nextprime 10 25 q nextprime 10 26 N pxq E alarm 1 factor N type E 13 t_ERROR print E 4 error alarm interrupt after 964 ms alarm 10 factor N enough time 5 10000000000000000000000013 1 100000000000000000000000067 1 Here is a more involved example the function timefact N sec below tries to factor N and gives up after sec seconds returning a partial factorisation AN Time bounded partial factorization default factor_add_primes 1 timefact N sec 1 F alarm sec factor N if type F t_ERROR factor N 2 24 F We either return the factorization directly or replace the t_ERROR result by a simple bounded factorization factor N 2724 Note the factor_add_primes trick any prime larger than 274 discovered while attempting the initial factorization is stored and remembered When the alarm rings the subsequent bounded factorization finds it right away Caveat It is not possible to set a new alarm within another alarm code the new timer erases the parent one 3 12 4 alias newsym
85. and does nothing otherwise if a seq evaluates seq if a is equal to zero and does nothing otherwise You could get the same result using the not operator if a seq The value of an if statement is the value of the branch that gets evaluated for instance x if n 4 1 y z sets x to y if n is 1 modulo 4 and to z otherwise Successive else blocks can be abbreviated in a single compound if as follows if testl seq1 test2 seq2 iets seqn seqdefault is equivalent to if testl seql if test2 seq2 if testn seqn seqdefault For instance this allows to write traditional switch case constructions if x 0 do0 x 1 do1 x 2 do2 dodefault Remark The boolean operators amp amp and are evaluated according to operator precedence as explained in Section 2 4 but contrary to other operators the evaluation of the arguments is stopped as soon as the final truth value has been determined For instance if x 0 amp amp 1 x is a perfectly safe statement 283 Remark Functions such as break and next operate on loops such as forrxx while until The if statement is not a loop Obviously 3 11 17 iferr seq1 E seq2 pred Evaluates the expression sequence seg1 If an error occurs set the formal parameter E set to the error data If pred is not present or evaluates to true catch the error and evaluate seg2 Both pred and seq2 can reference E The er
86. assumed resulting in catastrophic loss of accuracy The library syntax is GEN intmellininvshort GEN sig GEN z GEN tab long prec 260 3 9 10 intnum X a b expr tab Numerical integration of expr on Ja b with respect to X The integrand may have values belonging to a vector space over the real numbers in particular it can be complex valued or vector valued But it is assumed that the function is regular on Ja b If the endpoints a and b are finite and the function is regular there the situation is simple intnum x 0 1 x 2 1 0 3333333333333333333333333333 intnum x 0 Pi 2 cos x sin x 72 1 000000000000000000000000000 1 000000000000000000000000000 An endpoint equal to 00 is coded as the single component vector 1 You are welcome to set e g oo 1 or INFINITY 1 then using 00 00 INFINITY etc will have the expected behavior oo 1 for clarity intnum x 1 00 1 x72 2 1 000000000000000000000000000 In basic usage it is assumed that the function does not decrease exponentially fast at infinity intnum x 0 00 exp x kk at top level intnum x 0 00 exp xk kk exp exponent expo overflow We shall see in a moment how to avoid the last problem after describing the last argument tab which is both optional and technical The routine uses weights which are mostly independent of the function being integrated evaluated at many sampl
87. batch of rect objects corresponding to the output of the related ploth function Finally the actual physical drawing is done using plotdraw The rectwindows are preserved so that further drawings using the same windows at different positions or different windows can be done without extra work To erase a window use plotkill It is not possible to partially erase a window erase it completely initialize it again then fill it with the graphic objects that you want to keep In addition to initializing the window you may use a scaled window to avoid unnecessary conversions For this use plotscale As long as this function is not called the scaling is simply the number of pixels the origin being at the upper left and the y coordinates going downwards Plotting functions are platform independent but a number of graphical drivers are available for screen output X11 windows hence also for GUI s based on X11 such as Openwindows and Motif and the Qt and FLTK graphical libraries The physical window opened by plotdraw or any of the ploth functions is completely separated from gp technically a fork is done and the non graphical memory is immediately freed in the child process which means you can go on working in the current gp session without having to kill the window first This window can be closed enlarged or reduced using the standard window manager functions No zooming procedure is implemented though yet Among these special
88. be integral given by a minimal model this function computes the value of the canonical bilinear form on z1 z2 ME 21422 h E 21 h E 22 2 where denotes of course addition on E In addition z1 or 22 but not both can be vectors or matrices The library syntax is GEN bilhel1 GEN E GEN z1 GEN z2 long prec 3 5 9 ellcard E p Let E be an ell structure as output by ellinit defined over Q or a finite field F The argument p is best left omitted if the curve is defined over a finite field and must be a prime number otherwise This function computes the order of the group E F as would be computed by ellgroup If the curve is defined over Q p must be explicitly given and the function computes the cardinal of the reduction over F the equation need not be minimal at p but a minimal model will be more efficient The reduction is allowed to be singular and we return the order of the group of non singular points in this case The library syntax is GEN ellcard GEN E GEN p NULL Also available is GEN ell card GEN E GEN p where p is not NULL 3 5 10 ellchangecurve F v Changes the data for the elliptic curve E by changing the coordi nates using the vector v u r s t ie if z and y are the new coordinates then x u x r y uy su x t E must be an ell structure as output by ellinit The special case v 1 is also used instead of 1 0 0 0 to denote the trivial coordinate change The librar
89. but beware that x 1 1 0 is not because the parentheses force the complete evaluation of x 1 and the result is not modifiable e Priority 10 unary postfix derivative with respect to the main variable If f is a function t_CLOSURE 28 f is allowed and defines a new function which will perform numerical derivation when evaluated at a scalar x this is defined as f a f a e 2 lt for a suitably small epsilon depending on current precision x72 y x y72 derive with respect to main variable x 1 2 x y SIN cos 2 cos SIN Pi 6 numerical derivation 3 0 5000000000000000000000000000 cos Pi 6 works directly no need for intermediate SIN 4 0 5000000000000000000000000000 unary postfix vector matrix transpose unary postfix factorial 2 x x 1 1 unary prefix logical not x returns 1 if x is equal to 0 specifically if gequal0 x 1 and 0 otherwise e Priority 9 unary prefix cardinality tx returns length z e Priority 8 powering This operator is right associative 2 7374 is understood as 2 374 e Priority 7 unary prefix toggles the sign of its argument has no effect whatsoever e Priority 6 multiplication exact division 3 2 yields 3 2 not 1 5 4 Euclidean quotient and remainder i e if qy r then x y q x y r If x and y are scalars then q is an integer and r s
90. code On a number of different machines version could return either of 1 2 3 4 released version stable branch 1 2 4 3 released version testing branch 1 2 6 1 15174 505ab9b development In particular if you are only working with released versions the first line of the gp introductory message can be emulated by M m p version printf GP PARI CALCULATOR Version s s s M m p If you are working with many development versions of PARI GP the 4th and or 5th components can be profitably included in the name of your logfiles for instance Technical note For development versions obtained via git the 4th and 5th components are liable to change eventually but we document their current meaning for completeness The 4th component counts the number of reachable commits in the branch analogous to svn s revision number and the 5th is the git commit hash In particular lex comparison still orders correctly development versions with respect to each others or to released versions provided we stay within a given branch e g master The library syntax is GEN pari_version 3 12 40 warning str Outputs the message user warning and the argument list each of them interpreted as a string If colors are enabled this warning will be in a different color making it easy to distinguish warning n is very large this might take a while 3 12 41 whatnow key If keyword key is t
91. columns of u If no solution exists returns zero The library syntax is GEN matsolvemodO GEN M GEN D GEN B long flag Also available are GEN gaussmodulo GEN M GEN D GEN B flag 0 and GEN gaussmodulo2 GEN M GEN D GEN B flag 1 243 3 8 46 matsupplement z Assuming that the columns of the matrix x are linearly independent if they are not an error message is issued finds a square invertible matrix whose first columns are the columns of x i e supplement the columns of x to a basis of the whole space matsupplement 1 2 1 1 0 2 1 Raises an error if x has 0 columns since due to a long standing design bug the dimension of the ambient space the number of rows is unknown in this case matsupplement matrix 2 0 eK at top level matsupplement matrix kkk xxx matsupplement sorry suppl empty matrix is not yet implemented The library syntax is GEN suppl GEN x 3 8 47 mattranspose z Transpose of x also x This has an effect only on vectors and matrices The library syntax is GEN gtrans GEN x 3 8 48 minpoly A v x minimal polynomial of A with respect to the variable v i e the monic polynomial P of minimal degree in the variable v such that P A 0 The library syntax is GEN minpoly GEN A long v 1 where v is a variable number 3 8 49 norml2 zx Square of the L norm of x More precisely if x is a scalar norm12 x is defined to be the
92. compositum GEN P GEN Q flag 0 and GEN compositum2 GEN P GEN Q flag 1 3 6 118 polgalois T Galois group of the non constant polynomial T Q X In the present version 2 7 0 T must be irreducible and the degree d of T must be less than or equal to 7 If the galdata package has been installed degrees 8 9 10 and 11 are also implemented By definition if K Q x T this computes the action of the Galois group of the Galois closure of K on the d distinct roots of T up to conjugacy corresponding to different root orderings The output is a 4 component vector n s k name with the following meaning n is the cardi nality of the group s is its signature s 1 if the group is a subgroup of the alternating group Ag s 1 otherwise and name is a character string containing name of the transitive group according to the GAP 4 transitive groups library by Alexander Hulpke k is more arbitrary and the choice made up to version 2 2 3 of PARI is rather unfortunate for d gt 7 k is the numbering of the group among all transitive subgroups of Sq as given in The transitive groups of degree up to eleven G Butler and J McKay Communications in Algebra vol 11 1983 pp 863 911 group k is denoted Ty there And for d lt 7 it was ad hoc so as to ensure that a given triple would denote a unique group Specifically for polynomials of degree d lt 7 the groups are coded as follows using standard notations In degree 1
93. course In the latter case the result may still be unconditionally correct see thue For instance in most cases where P is reducible not a pure power of an irreducible or conditional computed class groups are trivial or the right hand side is 1 then results are always unconditional The library syntax is GEN thueinit GEN P long flag long prec 3 8 Vectors matrices linear algebra and sets Note that most linear algebra functions operating on subspaces defined by generating sets such as mathnf qf111 etc take matrices as arguments As usual the generating vectors are taken to be the columns of the given matrix Since PARI does not have a strong typing system scalars live in unspecified commutative base rings It is very difficult to write robust linear algebra routines in such a general setting We thus assume that the base ring is a domain and work over its field of fractions If the base ring is not a domain one gets an error as soon as a non zero pivot turns out to be non invertible Some functions e g mathnf or mathnfmod specifically assume that the base ring is Z 3 8 1 algdep x k flag 0 x being real complex or p adic finds a polynomial of degree at most k with integer coefficients having x as approximate root Note that the polynomial which is obtained is not necessarily the correct one In fact it is not even guaranteed to be irreducible One can check the closeness either by a polynomial evaluation use
94. course the results above are obvious adding t places at infinity will add t copies of Z 2Z to the ray class group The following application is more typical L ideallist bnf 100 2 units are required now La ideallistarch bnf L 1 1 H bnrclassnolist bnf La H 98 6 2 12 2 The library syntax is GEN ideallistarch GEN nf GEN list GEN arch 178 3 6 59 ideallog nf x bid nf is a number field bid is as output by idealstar nf D and x a non necessarily integral element of nf which must have valuation equal to 0 at all prime ideals in the support of D This function computes the discrete logarithm of x on the generators given in bid gen In other words if g are these generators of orders d respectively the result is a column vector of integers x such that 0 lt x lt d and z o7 mod D Note that when the support of D contains places at infinity this congruence implies also sign conditions on the associated real embeddings See znlog for the limitations of the underlying discrete log algorithms The library syntax is GEN ideallog GEN nf GEN x GEN bid 3 6 60 idealmin nf iz vdir This function is useless and kept for backward compatibility only use idealred Computes a pseudo minimum of the ideal x in the direction vdir in the number field nf The library syntax is GEN idealmin GEN nf GEN ix GEN vdir NULL 3 6 61 idealmul nf x y flag 0 Ideal multiplic
95. creates the associated 2 x 2 matrix Otherwise this creates a 1 x 1 matrix containing zx Mat x 1 1 x 1 Vec matid 3 2 1 0 0 0 1 O O O 1 Mat 3 1 0 0 o 1 0 0 O 1 Coll 1 2 3 4 14 1 2 8 4 Mat 5 1 2 3 4 Mat Q b 1 2 3 6 1 1 1 3 The library syntax is GEN gtomat GEN x NULL 3 2 5 Mod a b In its basic form creates an intmod or a polmod a mod b b must be an integer or a polynomial We then obtain a t_INTMOD and a t_POLMOD respectively t Mod 2 17 t78 1 Mod 1 17 t Mod x x 2 1 t72 2 Mod 1 x72 1 If a b makes sense and yields a result of the appropriate type t_INT or scalar t_POL the operation succeeds as well Mod 1 2 5 13 Mod 3 5 Mod 7 0 376 3 44 Mod 1 3 Mod Mod 1 12 9 95 Mod 1 3 Mod 1 x x72 1 46 Mod 1 x 2 1 Mod exp x x74 AT Mod 1 6xx 3 1 2 x 2 x 1 x74 72 If a is a complex object base change it to Z bZ or K x b which is equivalent to but faster than multiplying it by Mod 1 b Mod 1 2 3 4 2 18 Mod 1 2 Mod 0 2 Mod 1 2 Mod 0 2 Mod 3 x 5 2 19 Mod 1 2 x Mod 1 2 Mod x 2 y x y73 y72 1 110 Mod 1 y7 2 1 x 2 Mod y y72 1 x Mod y y 2 1 This function is not the same as x y the result of which has no knowledge of the indended modulu
96. degree 6 Cg 6 1 1 S3 6 1 2 Dg 12 1 3 Ay 12 1 4 Gig 18 1 5 Ay x Co 24 1 6 Sf 24 1 7 Sp 24 1 8 G3 36 1 9 Gi 36 1 10 S4 x Cy 48 1 11 As PSL2 5 60 1 12 G72 72 1 13 Ss PGLa 5 120 1 14 Ag 360 1 15 Sg 720 1 16 In degree 7 C7 7 1 1 Dz 14 1 2 Mo gt 21 1 31 Mas 42 1 4 PSLa 7 PSL3 2 168 1 5 A7 2520 1 6 S7 5040 1 7 Warning The method used is that of resolvent polynomials and is sensitive to the current preci sion The precision is updated internally but in very rare cases a wrong result may be returned if the initial precision was not sufficient The library syntax is GEN polgalois GEN T long prec To enable the new format in library mode set the global variable new_galois_format to 1 3 6 119 polred T flag 0 This function is deprecated use polredbest instead Finds polynomials with reasonably small coefficients defining subfields of the number field defined by T One of the polynomials always defines Q hence is equal to x 1 and another always defines the same number field as T if T is irreducible All T accepted by nfinit are also allowed here in particular the format T listP is recommended e g with listP 10 or a vector containing all ramified primes Otherwise the maximal order of Q z Z must be computed The following binary digits
97. editing under gp an automatic context dependent completion and an editable history of commands e GNU emacs and the PariEmacs package The gp calculator can be run in an Emacs buffer with all the obvious advantages if you are familiar with this editor Note that readline is still useful in this case since it provides a better automatic completion than is provided by Emacs s GP mode e GNU gzip gunzip gzcat package enables gp to read compressed data e perl provides extended online help full text from the manual about functions and concepts The script handling this online help can be used under gp or independently 315 2 Compiling the library and the gp calculator 2 1 Basic configuration Type Configure in the toplevel directory This attempts to configure PARI GP without outside help Note that if you want to install the end product in some nonstandard place you can use the prefix option as in Configure prefix an exotic directory the default prefix is usr local For example to build a package for a Linux distribution you may want to use Configure prefix usr This phase extracts some files and creates a build directory names Oosname arch where the object files and executables will be built The osname and arch components depends on your architecture and operating system thus you can build PARI GP for several different machines from the same source tree the builds are independent and can be done si
98. entries are appended at the end of the vector if n gt 0 and prepended at the beginning if n lt 0 The dimension of the resulting vector is n Variant GEN gtovec GEN x is also available The library syntax is GEN gtovecO GEN x long n 3 2 16 Vecrev x nj As Vec x n then reverse the result In particular In this case Vecrev is the reciprocal function of Polrev the coefficients of the vector start with the constant coefficient of the polynomial and the others follow by increasing degree The library syntax is GEN gtovecrevO0 GEN x long n GEN gtovecrev GEN x is also avail able 3 2 17 Vecsmall z n Transforms the object x into a row vector of type t_VECSMALL The dimension of the resulting vector can be optionally specified via the extra parameter n This acts as Vec x n but only on a limited set of objects the result must be representable as a vector of small integers If x is a character string a vector of individual characters in ASCII encoding is returned Strchr yields back the character string The library syntax is GEN gtovecsmall0 GEN x long n GEN gtovecsmall GEN x is also available 76 3 2 18 binary xz Outputs the vector of the binary digits of x Here x can be an integer a real number in which case the result has two components one for the integer part one for the fractional part or a vector matrix The library syntax is GEN binaire GEN x 3 2 19 bitand z y Bitwise and of two integers
99. for the torsion units zk bnr bnf nf integral basis i e a Z basis of the maximal order zkst bnr structure of Zx m 155 Deprecated The following member functions are still available but deprecated and should not be used in new scripts futu bnr bnf luz uy w uz is a vector of fundamental units w generates the torsion units tufu bnr bnf w ui Ur us is a vector of fundamental units w generates the torsion units For instance assume that bnf bnfinit pol for some polynomial Then bnf clgp retrieves the class group and bnf clgp no the class number If we had set bnf nfinit pol both would have output an error message All these functions are completely recursive thus for instance bnr bnf nf zk will yield the maximal order of bnr which you could get directly with a simple bnr zk 3 6 7 Class group units and the GRH Some of the functions starting with bnf are implementations of the sub exponential algorithms for finding class and unit groups under GRH due to Hafner McCurley Buchmann and Cohen Diaz Olivier The general call to the functions concerning class groups of general number fields i e excluding quadclassunit involves a polynomial P and a technical vector tech c1 C2 nrpid where the parameters are to be understood as follows P is the defining polynomial for the number field which must be in Z X irreducible and monic In fact if you supply a non
100. function behaves like cos kx g x e a k x I assumes that the function behaves like sin kx g x Here it is critical to give the exact value of k If the oscillating part is not a pure sine or cosine one must expand it into a Fourier series use the above codings and sum the resulting contributions Otherwise you will get nonsense Note that cos kx and similarly sin kx means that very function and not a translated version such as cos ka a Note If f x cos kx g x where g x tends to zero exponentially fast as exp az it is up to the user to choose between 1 a and 1 kx1 but a good rule of thumb is that if the oscillations are much weaker than the exponential decrease choose 1 a otherwise choose 1 k 1 although the latter can reasonably be used in all cases while the former cannot To take a specific example in the inverse Mellin transform the integrand is almost always a product of an exponentially decreasing and an oscillating factor If we choose the oscillating type of integral we perhaps obtain the best results at the expense of having to recompute our functions for a different value of the variable z giving the transform preventing us to use a function such as intmellininvshort On the other hand using the exponential type of integral we obtain less accurate results but we skip expensive recomputations See intmellininvshort and intfuncinit for more explanations We shall now see
101. gt 2 See Section 3 6 4 1 for the definition of integral pseudo matrix briefly it is input as a 3 component row vector A I J where J b1 b and J a1 a are two ideal lists and A is a square n x n matrix with columns 41 An seen as elements in K with canonical basis e1 n This data defines the Zg module x given by bye D bnen a1 Az Ppp An An y The integrality condition is aj biaz for all i j If it is not satisfied then the d will not be integral Note that every finitely generated torsion module is isomorphic to a module of this form and even with b Zx for all i 196 The library syntax is GEN nfsnf GEN nf GEN x 3 6 115 nfsolvemodpr nf a b P Let P be a prime ideal in modpr format see nfmodprinit let a be a matrix invertible over the residue field and let b be a column vector or matrix This function returns a solution of a x b the coefficients of x are lifted to nf elements K nfinit y 2 1 P idealprimedec K 3 1 P nfmodprinit Kk E a y 1 y y 0 b 1 yl nfsolvemodpr K a b ey 15 1 2 The library syntax is GEN nfsolvemodpr GEN nf GEN a GEN b GEN P This function is normally useless in library mode Project your inputs to the residue field using nfM_to_FqM then work there 3 6 116 nfsubfields pol d 0 Finds all subfields of degree d of the number field defined by the monic integral polynomial pol all su
102. help If you now type n with 1 lt n lt 11 you get the list of commands corresponding to category n and simultaneously to Section 3 n of this manual If you type functionname where functionname is the name of a PARI function you will get a short explanation of this function If extended help see Section 2 13 1 is available on your system you can double or triple the sign to get much more respectively the complete description of the function e g sqrt or a list of gp functions relevant to your query e g elliptic curve or quadratic field If gp was properly installed see Appendix A a line editor is available to correct the command line get automatic completions and so on See Section 2 15 or readline for a short summary of the line editor s commands If you type you will get a short description of the metacommands keyboard shortcuts Finally typing will return the list of available pre defined member functions These are functions attached to specific kind of objects used to retrieve easily some information from complicated structures you can define your own but they won t be shown here We will soon describe these commands in more detail More generally commands starting with the symbols or are not computing commands but are metacommands which allow you to exchange information with gp The available metacommands can be divided into default setting commands explained below and simple co
103. in the same way as above modulo pPedicerec s if its valuation v is non negative if not returns the fraction p centerlift xp in particular rational reconstruction is not attempted Use bestappr for this For backward compatibility centerlift x v is allowed as an alias for lift x v The library syntax is centerlift GEN x 3 2 27 characteristic z Returns the characteristic of the base ring over which z is defined as defined by t_INTMOD and t_FFELT components The function raises an exception if incompatible primes arise from t_FFELT and t_PADIC components characteristic Mod 1 24 x Mod 1 18 y 1 6 The library syntax is GEN characteristic GEN x 78 3 2 28 component x n Extracts the n component of x This is to be understood as follows every PARI type has one or two initial code words The components are counted starting at 1 after these code words In particular if x is a vector this is indeed the n component of x if x is a matrix the nt column if x is a polynomial the n coefficient i e of degree n 1 and for power series the nt significant coefficient For polynomials and power series one should rather use polcoeff and for vectors and matri ces the operator Namely if x is a vector then x n represents the n component of x If x is a matrix x m n represents the coefficient of row m and column n of the matrix x m represents the mt row of x and x n represents th
104. is attached to x If x is omitted returns the list of user variables known to the interpreter by order of decreasing priority Highest priority is x which always come first The library syntax is GEN gpolvar GEN x NULL However in library mode this function should not be used for x non NULL since gvar is more appropriate Instead for x a p adic type t_PADIC pis gel x 2 otherwise use long gvar GEN x which returns the variable number of x if it exists NO_VARIABLE otherwise which satisfies the property varncmp NO_VARIABLE v gt 0 for all valid variable number v i e it has lower priority than any variable 87 3 3 Transcendental functions Since the values of transcendental functions cannot be exactly represented these functions will always return an inexact object a real number a complex number a p adic number or a power series All these objects have a certain finite precision As a general rule which of course in some cases may have exceptions transcendental functions operate in the following way e If the argument is either a real number or an inexact complex number like 1 0 I or Pi I but not 2 3 1 then the computation is done with the precision of the argument In the example below we see that changing the precision to 50 digits does not matter because x only had a precision of 19 digits 7 p 15 realprecision 19 significant digits 15 digits displayed x Pi 4 1 0 785398163397448 p 5
105. is disregarded The output v is a vector of vectors where v i j is understood to be in fact V 2t5 i 1 j of a unique big vector V This awkward scheme allows for larger vectors than could be otherwise represented Vk is itself a vector W whose length is the number of ideals of norm k We consider first the case where arch was specified Each component of W corresponds to an ideal m of norm k and gives invariants associated to the ray class field L of bnf of conductor m arch Namely each contains a vector m d r D with the following meaning m is the prime ideal factorization of the modulus d L Q is the absolute degree of L r is the number of real places of L and D is the factorization of its absolute discriminant We set d r D 0 if mis not the finite part of a conductor If arch was omitted all t 2 possible values are taken and a component of W has the form m d1 71 Di de re Dill where m is the finite part of the conductor as above and d ri Di are the invariants of the ray class field of conductor m v where v is the i th Archimedean component ordered by inverse lexicographic order so v 0 0 ve 1 0 0 etc Again we set dj ri D 0 if m v is not a conductor Finally each prime ideal pr p a e f 6 in the prime factorization m is coded as the integer p n f 1 n j 1 where n is the degree of the base field and j is such that pr idealp
106. is never performed on them They get concatenated though The analyzer supplies automatically the quotes you have forgotten and treats Keywords just as normal strings otherwise For instance if you type a b b in Keyword context you will get the string whose contents are ab b In String context on the other hand you would get a2xb All GP functions have prototypes described in Chapter 3 below which specify the types of arguments they expect either generic PARI objects GEN or strings or keywords or unevaluated 46 expression sequences In the keyword case only a very small set of words will actually be meaningful the default function is a prominent example Reference The arguments of the following functions are processed in string context The Str addhelp second argument default second argument error extern plotstring second argument plotterm first argument read and readvec system all the printxxrxz functions all the writexrzrz functions arguments of the following functions are processed as keywords alias default first argument install all arguments but the last trap first argument whatnow 2 9 3 Useful examples The function Str converts its arguments into strings and concatenate them Coupled with eval it is very powerful The following example creates generic matrices into genmat u v s x matrix u v i j eval Str s i j genmat 2 3 genmat 2 3 m 1
107. its standard code e galdata The default polgalois function can only compute Galois groups of polynomials of degree less or equal to 7 Install this package if you want to handle polynomials of degree bigger than 7 and less than 11 e seadata This package contains the database of modular polynomials extracted from the ECHIDNA databases and computed by David R Kohel It is needed by the functions ellap and ellgroup for primes larger than 107 e galpol This package contains the GALPOL database of polynomials defining Galois exten sions of the rationals accessed by galoisgetpol 321 To install package pack you need to fetch the separate archive pack tgz which you can download from the pari server Copy the archive in the PARI toplevel directory then extract its contents these will go to data pack Typing make install installs all such packages 4 3 The GPRC file Copy the file misc gprc dft or gprc dos if you are using GP EXE to HOME gprc Modify it to your liking For instance if you are not using an ANSI terminal remove control characters from the prompt variable You can also enable colors If desired read datadir misc gpalias from the gprc file which provides some common shortcuts to lengthy names fix the path in gpre first Unless you tampered with this via Configure datadir is prefix share pari If you have superuser privileges and want to provide system wide defaults copy your customized gprc file to etc
108. lexically all variables to the smallest possible block Should you forget this in expressions involving such rogue variables the value used will be the one which happens to be on top of the value stack at the time of the call which depends on the whole calling context in a non trivial way This is in general not what you want 2 7 User defined functions The most important thing to understand about user defined functions is that they are ordinary GP objects bound to variables just like any other object Those variables are subject to scoping rules as any other while you can define all your functions in global scope it is usually possible and cleaner to lexically scope your private helper functions to the block of text where they will be needed Whenever gp meets a construction of the form expr argument list and the expression expr evaluates to a function an object of type t_CLOSURE the function is called with the proper arguments For instance constructions like funcs i x are perfectly valid assuming funcs is an array of functions 2 7 1 Defining a function A user function is defined as follows list of formal variables gt seq The list of formal variables is a comma separated list of distinct variable names and allowed to be empty It there is a single formal variable the parentheses are optional This list corresponds to the list of parameters you will supply to your function when calling it In most cases you
109. lexically or dynamically scoped to the block between the declaration and the end of the innermost enclosing scope using a my or local declaration for i 1 10 my x y Z i2 172 temps needed within the loop body Note how the declaration can include optional initial values i2 i 2 in the above Variables for which no explicit default value is given in the declaration are initialized to 0 It would be more natural to initialize them to free variables but this would break backward compatibility To obtain this behavior you may explicitly use the quoting operator my x x y y Z 2 A more complicated example for i 1 3 print main loop my x i local to the outermost loop for j 1 3 my y x72 M local to the innermost loop print y y 2 x When we leave the loops the values of x y i j are the same as before they were started Note that eval is evaluated in the given scope and can access values of lexical variables x 1 my x 0 eval x 2 0 we see the local x scoped to this command line not the global one Variables dynamically scoped using local should more appropriately be called temporary val ues since they are in fact local to the function declaring them and any subroutine called from 36 within In practice you almost certainly want true private variables hence should use almost exclusively my We strongly recommended to explicitly scope
110. lost and no significant digit remains It would be enough to restart part of the computation at a slightly higher precision We now describe error trapping a useful mechanism which alleviates much of the pain in the first situation the break loop debugger and provides satisfactory ways out of the second one the iferr exception handler 2 10 3 Break loop A break loop is a special debugging mode that you enter whenever a user interrupt Control C or runtime error occurs freezing the gp state and preventing cleanup until you get out of the loop By runtime error we mean an error from the evaluator the library or a user error from error not syntax errors When a break loop starts a prompt is issued break gt You can type in a gp command which is evaluated when you hit the lt Return gt key and the result is printed as during the main gp loop except that no history of results is kept Then the break loop prompt reappears and you can type further commands as long as you do not exit the loop If you are using readline the history of commands is kept and line editing is available as usual If you type in a command that results in an error you are sent back to the break loop prompt errors do not terminate the loop To get out of a break loop you can use next break return or type C d EOF any of which will let gp perform its usual cleanup and send you back to the gp prompt Note that C d is slightly dangerous since typing i
111. mandatory argument a in user function This applies to functions defined while strictargs is on Changing strictargs does not affect the behavior of previously defined functions The default value is 0 3 14 40 strictmatch This toggle is either 1 on or 0 off If on unused characters after a sequence has been processed will produce an error Otherwise just a warning is printed This can be useful when you are unsure how many parentheses you have to close after complicated nested loops Please do not use this find a decent text editor instead The default value is 1 3 14 41 threadsize In parallel mode each threads needs its own private stack in which to do its computations see parisize This value determines the size in bytes of the stacks of each threads so the total memory allocated will be parisize nbthreads x threadsize If set to 0 the value used is the same as parisize The default value is 0 312 3 14 42 timer This toggle is either 1 on or 0 off Every instruction sequence in the gp calculator anything ended by a newline in your input is timed to some accuracy depending on the hardware and operating system When timer is on each such timing is printed immediately before the output as follows factor 2727 7 1 time 108 ms this line omitted if timer is 0 1 59649589127497217 1 5704689200685129054721 1 See also and The time measured is the user CPU time not includ
112. minimum of Hxxv y and y v where v x is the index of the first non zero coefficient dirmul1 0 1 0 1 12 O 0 O 1 The library syntax is GEN dirmul GEN x GEN y 105 3 4 18 divisors x Creates a row vector whose components are the divisors of x The factorization of x as output by factor can be used instead By definition these divisors are the products of the irreducible factors of n as produced by factor n raised to appropriate powers no negative exponent may occur in the factorization If n is an integer they are the positive divisors in increasing order The library syntax is GEN divisors GEN x 3 4 19 eulerphi z Euler s totient function of the integer x in other words Z xZ eulerphi 40 1 16 According to this definition we let 0 2 since Z 1 1 this is consistant with znstar 0 we have znstar n no eulerphi n for all n Z The library syntax is GEN eulerphi GEN x 3 4 20 factor z lim General factorization function where x is a rational including integers a complex number with rational real and imaginary parts or a rational function including polyno mials The result is a two column matrix the first contains the irreducibles dividing x rational or Gaussian primes irreducible polynomials and the second the exponents By convention 0 is factored as 0 Q and Q i See factorint for more information about the algorithms used Th
113. navigation bar on the home page or directly at http pari math u bordeaux fr Bugs Further information can be found at that address but to report a configuration problem make sure to include the relevant dif files in the Oxxx directory and the file pari cfg There are a number of mailing lists devoted to PARI GP and most feedback should be directed there Instructions and archives can be consulted at http pari math u bordeaux1 fr lists index html The most important are e pari announce read only to announce major version changes You cannot write to this one but you should probably subscribe e pari dev for everything related to the development of PARI including suggestions tech nical questions or patch submissions Bug reports can be discussed here but as a rule it is better to submit them directly to the BTS e pari users for everything else You may send an email to the last two without being subscribed To subscribe send an message respectively to pari announce request pari math u bordeaux fr pari users request pari math u bordeaux fr pari dev request pari math u bordeaux fr with the word subscribe in the Subject You can also write to us at the address pari math u bordeaux fr but we cannot promise you will get an individual answer If you have used PARI in the preparation of a paper please cite it in the following form BibTeX format manual PARI2 organization The PARI Group title PA
114. needs not parse complicated expressions twice to make sure they are indeed identical Compare vlitj 1 vli j 1 1 gt v i j 1 M i i 3 M i i j 2 gt M i i 3 2 30 Remark Less important but still interesting The and op operators are slightly more efficient a i 1076 0 while i lt a i i 1 time 365 ms i 0 while i lt a i time 352ms For the same reason the shift operators should be preferred to multiplication a 1 lt lt 1075 i 1 while i lt a i i 2 time 1 052 ms i 1 while i lt a i lt lt 1 time 617 ms 2 5 Variables and symbolic expressions In this section we use variable in the standard mathematical sense symbols representing algebraically independent elements used to build rings of polynomials and power series and explain the all important concept of variable priority In the next Section 2 6 we shall no longer consider only free variables but adopt the viewpoint of computer programming and assign values to these symbols bound variables are names associated to values in a given scope 2 5 1 Variable names A valid name starts with a letter followed by any number of keyword characters _ or alphanumeric characters A Za z0 9 The built in function names are reserved and cannot be used see the list with c including the constants Pi Euler Catalan and I y 1 GP names are case sensitive For instance the symbol i is perfectly
115. nf pol Given a number field nf as output by nfinit and a polynomial pol with coefficients in nf defining a relative extension L of nf computes a pseudo basis A J for the maximal order Zr viewed as a Zkx module and the relative discriminant of L This is output as a four element row vector A D d where D is the relative ideal discriminant and d is the relative discriminant considered as an element of nf nf Bey The library syntax is GEN rnfpseudobasis GEN nf GEN pol 3 6 161 rnfsteinitz nf x Given a number field nf as output by nfinit and either a polynomial x with coefficients in nf defining a relative extension L of nf or a pseudo basis x of such an extension as output for example by rnfpseudobasis computes another pseudo basis A T not in HNF in general such that all the ideals of J except perhaps the last one are equal to the ring of integers of nf and outputs the four component row vector A J D d as in rnfpseudobasis The name of this function comes from the fact that the ideal class of the last ideal of J which is well defined is the Steinitz class of the Zx module Zz its image in SKo Zx The library syntax is GEN rnfsteinitz GEN nf GEN x 3 6 162 subgrouplist bnr bound flag 0 bnr being as output by bnrinit or a list of cyclic components of a finite Abelian group G outputs the list of subgroups of G Subgroups are given as HNF left divisors of the SNF matrix corresponding to G If flag 0 defaul
116. o 1 subst 1 x Mat 0 1 KK at top level subst 1 x Mat 0 1 HK O subst forbidden substitution by a non square matrix If x is a power series z must be either a polynomial a power series or a rational function Finally if x is a vector matrix or list the substitution is applied to each individual entry Use the function substvec to replace several variables at once or the function substpol to replace a polynomial expression The library syntax is GEN gsubst GEN x long y GEN z where y is a variable number 3 7 42 substpol x y z Replace the variable y by the argument z in the polynomial expres sion x Every type is allowed for x but the same behavior as subst above apply The difference with subst is that y is allowed to be any polynomial here The substitution is done moding out all components of x recursively by y t where t is a new free variable of lowest priority Then substituting t by z in the resulting expression For instance substpol x 4 x 2 1 x72 y f1 y72 y 1 substpol x 4 x 2 1 x73 y 12 x 2 y x 1 substpol x 4 x72 1 x 1 72 y 13 4 y 6 x y 2 3 y 3 The library syntax is GEN gsubstpol GEN x GEN y GEN z Further GEN gdeflate GEN T long v long d attempts to write T x in the form t 2 where z pol_x v and returns NULL if the substitution fails for instance in the example 2 above 225 3 7 43 substvec z v w v
117. of flag are significant 1 Possibly use a suborder of the maximal order The primes dividing the index of the order chosen are larger than primelimit or divide integers stored in the addprimes table This flag is deprecated the T listP format is more flexible 2 gives also elements The result is a two column matrix the first column giving primitive elements defining these subfields the second giving the corresponding minimal polynomials 199 M polred x 4 8 2 1 1 x 1 1 2 x72 x72 2 1 4 x73 x74 2 x x74 8 minpoly Mod M 2 1 x 4 8 12 x 2 2 The library syntax is polred GEN T flag 0 Also available is GEN polred2 GEN T flag 2 The function polredO is deprecated provided for backward compatibility 3 6 120 polredabs T flag 0 Returns a canonical defining polynomial P for the number field Q X T defined by T such that the sum of the squares of the modulus of the roots i e the T norm is minimal Different T defining isomorphic number fields will yield the same P All T accepted by nfinit are also allowed here e g non monic polynomials or pairs T listP specifying that a non maximal order may be used Warning 1 Using a t_POL T requires fully factoring the discriminant of T which may be very hard The format T listP computes only a suborder of the maximal order and replaces this part of the algorithm by a polynomial time computation In that case the polynomial P is
118. of p and a reduction mod p In particular additions are slow n 1 0 2720 for i 1 10 6 n time 841 ms n Mod 1 2 20 for i 1 10 6 n time 441 ms 7ne 1 for i 1 10 6 n time 328 ms The penalty associated with maintaining p decreases steeply as p increases and updates become very rare But t_INTMODs remain at least 25 more efficient But they do not have denominators 2 3 8 Quadratic numbers t_QUAD This type is used to work in the quadratic order of discrim inant d where d is a non square integer congruent to 0 or 1 modulo 4 The command w quadgen d assigns to w the canonical generator for the integer basis of the order of discriminant d i e w Vd 2 if d 0mod 4 and w 1 Vd 2 if d 1mod4 The name v is of course just a suggestion but corresponds to traditional usage You can use any variable name that you like but quadgen d is always printed as w regardless of the discriminant So beware two t_QUADs can be printed in the same way and not be equal however gp will refuse to add or multiply them for example Since the order is Z wZ any other element can be input as x yx w for some integers x and y In fact you may work in its fraction field Q Vd and use t_FRAC values for x and y 2 3 9 Polmods t_POLMOD Exactly as for intmods to enter z mody where x and y are poly nomials type Mod x y not x4y Note that when y is an irreducible polynomial in one variable
119. of that algebraic number A column vector of results is returned with one component for each complex embedding Therefore the number of components equals the degree of the t_POLMOD modulus e If the argument is an intmod or a p adic at present only a few functions like sqrt square root sqr square log exp powering teichmuller Teichmiiller character and agm arithmetic geometric mean are implemented Note that in the case of a 2 adic number sqr x may not be identical to x x x for example if x 1 0 2 and y 14 0 29 then zxy 1 O 2 while sqr x 1 O 2 Here xxx yields the same result as sqr x since the two operands are known to be identical The same statement holds true for p adics raised to the power n where v n gt 0 88 Remark If we wanted to be strictly consistent with the PARI philosophy we should have x y 4mod 8 and sqr x 4mod 32 when both x and y are congruent to 2 modulo 4 However since intmod is an exact object PARI assumes that the modulus must not change and the result is hence 0 mod 4 in both cases On the other hand p adics are not exact objects hence are treated differently e If the argument is a polynomial a power series or a rational function it is if necessary first converted to a power series using the current series precision held in the default series precision This precision the number of significant terms can be changed using ps or de fault seriesprecision
120. of the image of the matrix zx If flag 5 case as case 4 but x may have polynomial coefficients If flag 8 same as case 0 but x may have polynomial coefficients The library syntax is GEN qf1110 GEN x long flag Also available are GEN 111 GEN x flag 0 GEN 111int GEN x flag 1 and GEN 111kerim GEN x flag 4 3 8 59 qfillgram G flag 0 Same as gf111 except that the matrix G x xx is the Gram matrix of some lattice vectors x and not the coordinates of the vectors themselves In particular G must now be a square symmetric real matrix corresponding to a positive quadratic form not necessarily definite x needs not have maximal rank The result is a unimodular transformation matrix T such that x T is an LLL reduced basis of the lattice generated by the column vectors of x See qf111 for further details about the LLL implementation If flag 0 default assume that G has either exact integral or rational or real floating point entries The matrix is rescaled converted to integers and the behavior is then as in flag 1 If flag 1 assume that G is integral Computations involving Gram Schmidt vectors are approximate with precision varying as needed Lehmer s trick as generalized by Schnorr Adapted from Nguyen and Stehl s algorithm and Stehl s code fp111 1 3 248 flag 4 G has integer entries gives the kernel and reduced image of x flag 5 same as 4 but G may have polynomial coeff
121. of the out of domain argument 3 t_STR an operator op describing the domain error 4 t_GEN the numerical limit l describing the domain error 5 GEN the out of domain argument x The argument satisfies x op l which prevents it from belonging to the function s domain e e MAXPRIME A function using the precomputed list of prime numbers ran out of primes E has one component 1 t_INT the requested prime bound which overflowed primelimit or 0 bound is unknown 285 e e MEM A call to pari_malloc or pari_realloc failed E has no component e e OVERFLOW An object in function s becomes too large to be represented within PARI s hardcoded limits As in 27272710 or exp 1e100 which overflow in 1g and expo E has one component 1 t_STR the function name s e e PREC Function s fails because input accuracy is too low As in floor 1e100 at default accuracy E has one component 1 t_STR the function name s e e STACK The PARI stack overflows E has no component Errors triggered intentionally e e ALARM A timeout generated by the alarm function E has one component t_STR the error message to print e e USER A user error as triggered by error g1 9n E has one component 1 t_VEC the vector of n arguments given to error Mathematical errors e e CONSTPOL An argument of function s is a constant polynomial which does not make sense As in galoisinit Pol 1 E has one component 1 t_
122. of the setting of this default an object can be printed in any of the three formats at any time using the commands a and m and B respectively The default value is 1 prettymatriz 308 3 14 25 parisize gp and in fact any program using the PARI library needs a stack in which to do its computations parisize is the stack size in bytes It is strongly recommended you increase this default using the s command line switch or a gprc if you can afford it Don t increase it beyond the actual amount of RAM installed on your computer or gp will spend most of its time paging In case of emergency you can use the allocatemem function to increase parisize once the session is started The default value is 4M resp 8M on a 32 bit resp 64 bit machine 9 3 14 26 path This is a list of directories separated by colons semicolons in the DOS world since colons are preempted for drive names When asked to read a file whose name is not given by an absolute path does not start with or gp will look for it in these directories in the order they were written in path Here as usual means the current directory and its immediate parent Environment expansion is performed The default value is gp on UNIX systems C C GP on DOS OS 2 and Win dows and otherwise 3 14 27 prettyprinter The name of an external prettyprinter to use when output is 3 alternate prettyprinter Note that the
123. on the elliptic curve corresponding to E The library syntax is GEN elladd GEN E GEN z1 GEN z2 3 5 4 ellak E n Computes the coefficient an of the L function of the elliptic curve E Q i e coef ficients of a newform of weight 2 by the modularity theorem Taniyama Shimura Weil conjecture E must be an ell structure over Q as output by ellinit E must be given by an integral model not necessarily minimal although a minimal model will make the function faster E ellinit 0 11 ellak E 10 12 0 e ellinit 5 4 5 6 not minimal at 5 ellak e 5 wasteful but works 13 3 E ellminimalmodel e now minimal ellak E 5 15 3 If the model is not minimal at a number of bad primes then the function will be slower on those n divisible by the bad primes The speed should be comparable for other n for i 1 10 6 ellak E 5 time 820 ms for i 1 10 6 ellak e 5 5 is bad markedly slower time 1 249 ms for i 1 10 5 ellak E 5 i time 977 ms for i 1 10 5 ellak e 5 i still slower but not so much on average time 1 008 ms The library syntax is GEN akell GEN E GEN n 3 5 5 ellan E n Computes the vector of the first n Fourier coefficients az corresponding to the elliptic curve E The curve must be given by an integral model not necessarily minimal although a minimal model will make the function faster The library syntax is GEN anel1 GEN E long n Also availab
124. provably correct under the GRH a famous result of Bach states that co 6 is fine but it is possible to improve on this algorithmically You may provide a smaller C2 it will be ignored we use the provably correct one you may provide a larger cy than the default value which results in longer computing times for equally correct outputs under GRH The library syntax is GEN quadclassunitO GEN D long flag GEN tech NULL long prec If you really need to experiment with the tech parameter it is usually more convenient to use GEN Buchquad GEN D double c1 double c2 long prec 3 4 71 quaddisc x Discriminant of the quadratic field Q yz where x Q The library syntax is GEN quaddisc GEN x 124 3 4 72 quadgen D Creates the quadratic number w a VD 2 where a 0 if D 0mod 4 a 1 if D 1 mod4 so that 1 w is an integral basis for the quadratic order of discriminant D D must be an integer congruent to 0 or 1 modulo 4 which is not a square The library syntax is GEN quadgen GEN D 3 4 73 quadhilbert D Relative equation defining the Hilbert class field of the quadratic field of discriminant D If D lt 0 uses complex multiplication Schertz s variant If D gt 0 Stark units are used and in rare cases a vector of extensions may be returned whose compositum is the requested class field See bnrstark for details The library syntax is GEN quadhilbert GEN D long prec 3 4 74 quadpoly D v x C
125. return happens 3 13 5 parselect f A flag 0 Selects elements of A according to the selection function f done in parallel If flagis 1 return the indices of those elements indirect selection The function f must not access global variables or variables declared with local and must be free of side effects The library syntax is GEN parselect GEN f GEN A long flag 3 13 6 parsum i a b expr x Sum of expression expr initialized at x the formal parameter going from a to b evaluated in parallel in random order The expression expr must not access global variables or variables declared with local and must be free of side effects parsum i 1 1000 ispseudoprime 2 prime i 1 returns the numbers of prime numbers among the first 1000 Mersenne numbers 3 13 7 parvector N i expr As vector N i expr but the evaluations of expr are done in parallel The expression expr must not access global variables or variables declared with local and must be free of side effects parvector 10 i quadclassunit 27 100 i 1 no computes the class numbers in parallel 303 EMACS 3 14 GP defaults This section documents the GP defaults 3 14 1 TeXstyle The bits of this default allow gp to use less rigid TeX formatting commands in the logfile This default is only taken into account when log 3 The bits of TeXstyle have the following meaning 2 insert right left pairs where appropriate 4 insert discretionary br
126. root of x i e such that Arg sqrt x 7 n r n Intmod a prime and p adics are allowed as arguments If z is present it is set to a suitable root of unity allowing to recover all the other roots If it was not possible z is set to zero In the case this argument is present and no square root exist 0 is returned instead or raising an error sqrtn Mod 2 7 2 1 Mod 4 7 sqrtn Mod 2 7 2 amp z z 12 Mod 6 7 sqrtn Mod 2 7 3 kk at top level sqrtn Mod 2 7 3 4K sqrtn nth root does not exist in gsqrtn sqrtn Mod 2 7 3 amp z 42 0 Z 13 0 The following script computes all roots in all possible cases sqrtnall x n my V r z r2 r sqrtn x n amp z if z error Impossible case in sqrtn if type x t_INTMOD type x t_PADIC r2 rxz n 1 while r2 r r2 z n V vector n V 1 r for i 2 n Vli Vli 1 xz V addhelp sqrtnall sqrtnall x n compute the vector of nth roots of x The library syntax is GEN gsqrtn GEN x GEN n GEN z NULL long prec If lt x isa t_PADIC the function GEN Qp_sqrt GEN x GEN n GEN z is also available 3 3 50 tan x Tangent of x The library syntax is GEN gtan GEN x long prec 97 3 3 51 tanh x Hyperbolic tangent of x The library syntax is GEN gtanh GEN x long prec 3 3 52 teichmuller x Teichm ller character of the p adic number zx i e the unique p 1 th root of
127. searched for libraries You may disable inlining by adding DDISABLE_INLINE to CFLAGS and prevent the use of the volatile keyword with DDISABLE VOLATILE 3 3 Debugging profiling If you also want to debug the PARI library Configure g creates a directory Oxxx dbg containing a special Makefile ensuring that the gp and PARI library built there is suitable for debugging If you want to profile gp or the library using gprof for instance Configure pg will create an Orxx prf directory where a suitable version of PARI can be built The gp binary built above with make all or make gp is optimized If you have run Configure g or pg and want to build a special purpose binary you can cd to the dbg or prf directory and type make gp there You can also invoke make gp dbg or make gp prf directly from the toplevel 3 4 Multiprecision kernel The kernel can be specified via the kernel fully_qualified_kernel_name switch The PARI kernel consists of two levels Level 0 operation on words and Level 1 operation on multi precision integers and reals which can take the following values Level 0 auto as detected none portable C or one of the assembler micro kernels alpha hppa hppa64 ia64 ix86 x86_64 m68k ppc ppc64 sparcv sparcv8_micro sparcv8_super Level 1 auto as detected none native code only or gmp e A fully qualified kernel name is of the form Level0 Level1 the default value being auto auto e A name not containin
128. sine of z The library syntax is GEN gsinh GEN x long prec 3 3 47 sqr x Square of x This operation is not completely straightforward i e identical to g since it can usually be computed more efficiently roughly one half of the elementary multiplications can be saved Also squaring a 2 adic number increases its precision For example 1 0 274 2 1 1 0 275 1 0 274 1 0 274 42 1 0 274 Note that this function is also called whenever one multiplies two objects which are known to be identical e g they are the value of the same variable or we are computing a power x 1 00274 x x 13 1 0 275 1 0 274 74 14 1 0 276 note the difference between 2 and 3 above The library syntax is GEN gsqr GEN x 96 3 3 48 sqrt x Principal branch of the square root of x defined as yx exp logx 2 In particular we have Arg sqrt 1 7 2 7 2 and if x R and z lt 0 then the result is complex with positive imaginary part Intmod a prime p t_PADIC and t_FFELT are allowed as arguments In the first 2 cases t_INTMOD t_PADIC the square root if it exists which is returned is the one whose first p adic digit is in the interval 0 p 2 For other arguments the result is undefined The library syntax is GEN gsqrt GEN x long prec For a t_PADIC x the function GEN Qp_sqrt GEN x is also available 3 3 49 sqrtn z n amp z Principal branch of the nth
129. square of the complex modulus of x real t_QUADs are not supported If x is a polynomial a row or column vector or a matrix norm12 x is defined recursively as norm12 x where x run through the components of x In particular this yields the usual gt gt x resp Y 2 if x is a polynomial or vector resp matrix with complex components norml2 1 2 3 vector 1 14 norml2 1 2 3 4 matrix 2 30 norml2 2 I x 43 5 norml2 1 2 8 4 5 6 recursively defined 4 91 The library syntax is GEN gnorm12 GEN x 244 3 8 50 normlp z p L norm of x sup norm if p is omitted More precisely if x is a scalar normlp x p is defined to be abs z If x is a polynomial a row or column vector or a matrix e if p is omitted normlp x is defined recursively as max norm1p x where x run through the components of x In particular this yields the usual sup norm if x is a polynomial or vector with complex components e otherwise normlp x p is defined recursively as gt normlp aj p In particular this yields the usual gt a if x is a polynomial or vector with complex components v 1 2 3 normlp v vector 1 3 M 1 2 3 4 normlp M matrix 42 4 T 1 1 I x 2 normlp T 3 1 4142135623730950488016887 242096980786 normlp 1 21 3 4 5 61 recursively defined 14 6 normlp v 1 15 6
130. syntax is GEN idealfactor GEN nf GEN x 174 3 6 52 idealfactorback nf f e flag 0 Gives back the ideal corresponding to a factor ization The integer 1 corresponds to the empty factorization If e is present e and f must be vectors of the same length e being integral and the corresponding factorization is the product of the ffit If not and f is vector it is understood as in the preceding case with e a vector of 1s we return the product of the f i Finally f can be a regular factorization as produced by idealfactor nf nfinit y 2 1 idealfactor nf 4 2 y 1 L2 1 11 2 1 1 11 2 5 2 11 1 eS 2 1 1 idealfactorback nf 12 10 4 o 2 f 1 1 e 1 2 idealfactorback nf f e 43 10 4 o 2 idealhnf nf 4 2 y 44 1 If flag is non zero perform ideal reductions idealred along the way This is most useful if the ideals involved are all extended ideals for instance with trivial principal part so that the principal parts extracted by idealred are not lost Here is an example f vector f i ffi transform to extended ideals idealfactorback nf f e 1 76 1 0 O 1 2 1 2 1 1 nffactorback nf 2 7 4 2 The extended ideal returned in 6 is the trivial ideal 1 extended with a principal generator given in factored form We use nffactorback to recover it in standard form The library
131. tex to ASCII program None of these disturb the line you were editing 2 16 GNU Emacs and PariEmacs If you install the PariEmacs package see Appendix A you may use gp as a subprocess in Emacs You then need to include in your emacs file the following lines autoload gp mode pari nil t autoload gp script mode pari nil t autoload gp pari nil t autoload gpman pari nil t setq auto mode alist cons gp gp script mode auto mode alist which autoloads functions from the PariEmacs package and ensures that file with the gp suffix are edited in gp script mode Once this is done under GNU Emacs if you type M x gp where as usual M is the Meta key a special shell will be started launching gp with the default stack size and prime limit You can then work as usual under gp but with all the facilities of an advanced text editor See the PariEmacs documentation for customizations menus etc recall that you can always undo the effect of the preceding keys by hitting C _ 62 Chapter 3 Functions and Operations Available in PARI and GP The functions and operators available in PARI and in the GP PARI calculator are numerous and ever expanding Here is a description of the ones available in version 2 7 0 It should be noted that many of these functions accept quite different types as arguments but others are more restricted The list of acceptable types will be given for each function or cla
132. thanks go to Klaus Peter Nischke who suggested the recursive plotting and forking resizing stuff the graphical window and Ilya Zakharevich who rewrote the graphic code from scratch implementing many new primitives splines clipping Nils Skoruppa and Bill Allombert wrote the Qt and f1tk graphic drivers respectively 272 3 10 3 Functions for PostScript output in the same way that printtex allows you to have a TEX output corresponding to printed results the functions starting with ps allow you to have PostScript output of the plots This will not be identical with the screen output but sufficiently close Note that you can use PostScript output even if you do not have the plotting routines enabled The PostScript output is written in a file whose name is derived from the psfile default pari ps if you did not tamper with it Each time a new PostScript output is asked for the PostScript output is appended to that file Hence you probably want to remove this file or change the value of psfile in between plots On the other hand in this manner as many plots as desired can be kept in a single file 3 10 4 Library mode None of the graphic functions are available within the PARI library you must be under gp to use them The reason for that is that you really should not use PARI for heavy duty graphical work there are better specialized alternatives around This whole set of routines was only meant as a convenient but simple minded visual aid
133. the current position of the virtual cursor and move the virtual cursor to x1 dx yl dy even if it is outside the window 3 10 27 plotrmove w dz dy Move the virtual cursor of the rectwindow w to position x1 dx y1 dy where x1 yl is the initial position of the cursor i e to position dx dy relative to the initial cursor 3 10 28 plotrpoint w dz dy Draw the point x1 dx yl dy on the rectwindow w if it is inside w where xl y1 is the current position of the cursor and in any case move the virtual cursor to position x1 dz yl dy 3 10 29 plotscale w 21 22 y1 y2 Scale the local coordinates of the rectwindow w so that x goes from x1 to 12 and y goes from yl to y2 12 lt x1 and y2 lt yl being allowed Initially after the initialization of the rectwindow w using the function plotinit the default scaling is the graphic pixel count and in particular the y axis is oriented downwards since the origin is at the upper left The function plotscale allows to change all these defaults and should be used whenever functions are graphed 3 10 30 plotstring w x flags 0 Draw on the rectwindow w the String x see Section 2 9 at the current position of the cursor flag is used for justification bits 1 and 2 regulate horizontal alignment left if 0 right if 2 center if 1 Bits 4 and 8 regulate vertical alignment bottom if 0 top if 8 v center if 4 Can insert additional small gap between point a
134. the definition field over its prime field the cardinality of the definition field is thus p g mod the minimal polynomial with reduced integer coefficients of the field generator 2 3 6 Complex numbers t_COMPLEX To enter x iy type x I y That s I not i The letter I stands for y 1 The real and imaginary parts x and y can be of type t_INT t_REAL t_INTMOD t_FRAC or t_PADIC 2 3 7 p adic numbers t_PADIC Typing 0 p k where p and k are integers yields a p adic 0 of accuracy k representing any p adic number whose valuation is gt k To input a general non 0 p adic number write a suitably precise rational or integer approximation and add O p7k to it Note that it is not checked whether p is indeed prime but results are undefined if this is not the case you can work on 10 adics if you want but disasters will happen as soon as you do something non trivial like taking a square root Note that 0 25 is not the same as 0 572 you want the latter For example you can type in the 7 adic number 2x7 1 3 4 7 2 7 2 0 773 exactly as shown or equivalently as 905 7 0 773 If a is a t_PADIC the following member functions are defined a mod returns the modulus p 19 a p returns p Note that this type is available for convenience not for speed internally t_PADICs are stored as p adic units modulo some p Each elementary operation involves updating p multiplying or dividing by powers
135. the model is already minimal the function will run faster The library syntax is long ellrootno GEN E GEN p NULL 3 5 41 ellsearch N This function finds all curves in the elldata database satisfying the con straint defined by the argument N e if Visa character string it selects a given curve e g 11a1 or curves in the given isogeny class e g 11a or curves with given condutor e g 11 e if N is a vector of integers it encodes the same constraints as the character string above according to the ellconvertname correspondance e g 11 0 1 for 11a1 11 0 for 11a and 11 for 11 e if N is an integer curves with conductor N are selected If N is a full curve name e g ilai or 11 0 1 the output format is N a1 2 43 44 46 G where a az a3 a4 06 are the coefficients of the Weierstrass equa tion of the curve and G is a Z basis of the free part of the Mordell Weil group associated to the curve ellsearch 11a3 1 11a3 0 1 1 0 0 O ellsearch 11 0 3 2 11a3 0 1 1 0 0 O 146 If N is not a full curve name then the output is a vector of all matching curves in the above format ellsearch 11a 1 11a1 0 1 1 10 20 0l 11a2 0 1 1 7820 263580 1 11a3 0 1 1 0 0 0113 ellsearch 11b 12 The library syntax is GEN ellsearch GEN N Also available is GEN ellsearchcurve GEN N that only accepts complet
136. the operator is applied componentwise The library syntax is GEN gdivent GEN x GEN y for x y 3 1 7 The expression x y evaluates to the rounded Euclidean quotient of x and y This is the same as x y except for scalar division the quotient is such that the corresponding remainder is smallest in absolute value and in case of a tie the quotient closest to 00 is chosen hence the remainder would belong to y 2 y 2 When z is a vector or matrix the operator is applied componentwise The library syntax is GEN gdivround GEN x GEN y for x y 3 1 8 The expression x y evaluates to the modular Euclidean remainder of x and y which we now define When z or y is a non integral real number r y is defined as x x2 y y Otherwise if y is an integer this is the smallest non negative integer congruent to x modulo y This actually coincides with the previous definition if and only if x is an integer If y is a polynomial this is the polynomial of smallest degree congruent to x modulo y For instance 1 2 3 41 2 70 5 3 2 0 5000000000000000000000000000 1 2 2 3 0 43 1 2 Note that when y is an integer and x a polynomial y is first promoted to a polynomial of degree 0 When is a vector or matrix the operator is applied componentwise The library syntax is GEN gmod GEN x GEN y for x y 3 1 9 The expression xn is powering If the exponent is an integer then exact operations are perfo
137. their types do not match up As in Mod 1 2 Pi E has three components 1 t_STR the operator name op 2 first argument 3 second argument e e PRIORITY Object o in function s contains variables whose priority is incompatible with the expected operation E g Pol x 1 y this raises an error because it s not possible to create a polynomial whose coefficients involve variables with higher priority than the main variable E has four components 1 t_STR the function name s 2 the offending argument o 3 t_STR an operator op describing the priority error 4 t_POL the variable v describing the priority error The argument satisfies variable 1 opvariable v e e VAR The variables of arguments x and y submitted to function s does not match up E g considering the algebraic number Mod t t 2 1 in nfinit x 2 1 E has three component 1 t_STR the function name s 2 t_POL the argument x 3 t_POL the argument y Overflows e e COMPONENT Trying to access an inexistent component in a vector matrix list in a function the index is less than 1 or greater than the allowed length E has four components 1 t_STR the function name 2 t_STR an operator op lt or gt 2 t_GEN a numerical limit bounding the allowed range 3 GEN the index x It satisfies x op l e e DOMAIN An argument is not in the function s domain E has five components 1 t_STR the function name 2 t_STR the mathematical name
138. theory and also many functions dealing with elliptic curves over Q or over local fields 1 5 5 Other functions Quite a number of other functions dealing with polynomials e g finding complex or p adic roots factoring etc power series e g substitution reversion linear algebra e g determinant charac teristic polynomial linear systems and different kinds of recursions are also included In addi tion standard numerical analysis routines like univariate integration using the double exponential method real root finding when the root is bracketed polynomial interpolation infinite series evaluation and plotting are included And now you should really have a look at the tutorial before proceeding 11 12 EMACS Chapter 2 The gp Calculator 2 1 Introduction Originally gp was designed as a debugging device for the PARI system library Over the years it has become a powerful user friendly stand alone calculator The mathematical functions available in PARI and gp are described in the next chapter In the present one we describe the specific use of the gp programmable calculator If you have GNU Emacs and use the PariEmacs package you can work in a special Emacs shell described in Section 2 16 Specific features of this Emacs shell are indicated by an EMACS sign in the left margin 2 1 1 Startup To start the calculator the general command line syntax is gp D key val files where items within bracke
139. this will fail and implicitly exhibit a non trivial factor of the modulus Mod 4 6 7 1 xk at top level Mod 4 6 1 RR T k _ _ impossible inverse modulo Mod 2 6 Here a factor 2 is obtained directly In general take the gcd of the representative and the modulus This is most useful when performing complicated operations modulo an integer N whose factorization is unknown Either the computation succeeds and all is well or a factor d is discovered and the computation may be restarted modulo d or N d For non invertible t_POLMOD this will fail without exhibiting a factor Mod x 2 x 3 x 1 xk at top level Mod x 2 x 3 x 1 kk L T xxx _ _ non invertible polynomial in RgXQ_inv a Mod 3 4 y 3 Mod 1 4 b y 6ty 5 y 4 y 3 y 2 y 1 Mod a b 1 at top level Mod a b 1 RRR kk _7_ impossible inverse modulo Mod 0 4 In fact the latter polynomial is invertible but the algorithm used subresultant assumes the base ring is a domain If it is not the case as here for Z 4Z a result will be correct but chances are an error will occur first In this specific case one should work with 2 adics In general one can try the following approach inversemod a b my m m polsylvestermatrix polrecip a polrecip b m matinverseimage m matid m 1 Polrev vecextract m Str poldegree b variable b inversemod a b 12 Mod 2 4 y 75 Mod 3 4 ry73 Mod 1 4
140. time 670 ms 13 4 03698781490633483156497361352190615794353338591897830587 E 944 suminf i 0 1 i i exp 1 time 110 ms 4 8 39147638 E 1000 faster and more accurate The library syntax is sumalt void E GEN eval void GEN GEN a long prec Also available is sumalt2 with the same arguments flag 1 3 9 21 sumdiv n X expr Sum of expression expr over the positive divisors of n This function is a trivial wrapper essentially equivalent to D divisors n for i 1 D X D il eval expr except that X is lexically scoped to the sumdiv loop If expr is a multiplicative function use sumdivmult 3 9 22 sumdivmult n d expr Sum of multiplicative expression expr over the positive divisors d of n Assume that expr evaluates to f d where f is multiplicative f 1 1 and f ab f a f b for coprime a and b 3 9 23 suminf X a expr infinite sum of expression expr the formal parameter X starting at a The evaluation stops when the relative error of the expression is less than the default precision for 3 consecutive evaluations The expressions must always evaluate to a complex number If the series converges slowly make sure realprecision is low even 28 digits may be too much In this case if the series is alternating or the terms have a constant sign sumalt and sumpos should be used instead p28 suminf i 1 1 7i i Had to hit C C at top level suminf i 1 1
141. two abstract projective Zkx modules N ajw anwn in K P biq bmm in K and a Zx linear map f N gt P given by FO ayu gt gt 0 503 mv This data defines the Zx module M P f N e Any projective Zg module M of finite type in K can be given by a pseudo matrix A 1 e An arbitrary Zg modules of finite type in K with non trivial torsion is given by an integral pseudo matrix A I J 3 6 4 2 Pseudo bases determinant e The pair 4 1 is a pseudo basis of the module it generates if the a are non zero and the Aj are K linearly independent We call n the size of the pseudo basis If A is a relative matrix the latter condition means it is square with non zero determinant we say that it is in Hermite Normal Form HNF if it is upper triangular and all the elements of the diagonal are equal to 1 e For instance the relative integer basis rnf zk is a pseudo basis A I of Zz where A rnf zk 1 is a vector of elements of L which are K linearly independent Most rnf routines return and handle Zx modules contained in L e g Z ideals via a pseudo basis A 1 where A is a relative matrix representing a vector of elements of L in terms of the fixed basis rnf zk 1 e The determinant of a pseudo basis A 1 is the ideal equal to the product of the determinant of A by all the ideals of J The determinant of a pseudo matrix is the determinant of any pseudo basis of the module it generates 153
142. useful additional information 0 2 0 3 kk at top level 0 2 0 3 2k K rae xxx _ _ inconsistent addition t_PADIC t_PADIC Break loop type break to go back to GP prompt break gt E dbg_err ol error inconsistent addition t_PADIC t_PADIC break gt Vec E e_0P 0 2 0 3 Note The debugger is enabled by default and fires up as soon as a runtime error occurs If you do not like this behavior you may disable it by setting the default breakloop to 0 in for gprc A runtime error will send you back to the prompt Note that the break loop is automatically disabled when running gp in non interactive mode i e when the program s standard input is not attached to a terminal Technical Note When you enter a break loop due to a PARI stack overflow the PARI stack is reset so that you can run commands Otherwise the stack would immediately overflow again Still as explained above you do not lose the value of any gp variable in the process 2 10 4 Protecting code The expression iferr statements ERR recovery evaluates and returns the value of statements unless an error occurs during the evaluation in which case the value of recovery is returned As in an if else clause with the difference that statements has been partially evaluated with possible side effects We shall give a lot more details about the ERR argument shortly it is the name of a variable lexically scoped to the recovery expression s
143. using Weierstrass p function and Mazur s classifica tion For this variant E must be an ell The library syntax is GEN elltorsO GEN E long flag Also available is GEN elltors GEN E for elltors E 0 3 5 47 ellweilpairing P Q m Computes the Weil pairing of the two points of m torsion P and Q on the elliptic curve E The library syntax is GEN ellweilpairing GEN E GEN P GEN Q GEN m 3 5 48 ellwp w z x flag 0 Computes the value at z of the Weierstrass gp function attached to the lattice w as given by ellperiods It is also possible to directly input w w1 wa or an elliptic curve E as given by ellinit w E omega w ellperiods 1 1 ellwp w 1 2 2 6 8751858180203728274900957798105571978 E ellinit 1 1 ellwp E 1 2 4 3 9413112427016474646048282462709151389 One can also compute the series expansion around z 0 E ellinit 1 0 ellwp E x implicitly at default seriesprecision 15 x7 2 1 5 xx 2 1 75 xx 6 2 4875 x710 O x714 ellwp E x 0 x 12 explicit precision 16 x7 2 1 5 x 2 1 75 x76 O x79 Optional flag means 0 default compute only p z 1 compute z o z The library syntax is GEN ellwpO GEN w GEN z NULL long flag long prec For flag 0 we also have GEN ellwp GEN w GEN z long prec and GEN ellwpseries GEN E long v long precdl for the power series in variable v 148 3 5 49 ellzeta w z 1 Comput
144. vector gen orders with the same meaning as for gal gen and gal orders Hence gen is a vector of permutations generating the subgroup and orders is the relatives orders of the generators The cardinal of a subgroup is the product of the relative orders Such subgroup can be used instead of a Galois group in the following command galoisisabelian galoissubgroups galoisexport and galoisidentify To get the subfield fixed by a subgroup sub of gal use galoisfixedfield gal sub 1 The library syntax is GEN galoissubgroups GEN G 3 6 45 idealadd nf x y Sum of the two ideals x and y in the number field nf The result is given in HNF K nfinit x 2 1 a idealadd K 2 x 1 ideal generated by 2 and 1 I 4 2 2 1 o 1 pr idealprimedec K 5 1 a prime ideal above 5 idealadd K a pr coprime as expected 4 1 0 0 1 This function cannot be used to add arbitrary Z modules since it assumes that its arguments are ideals b Mat 1 0 idealadd K b b only square t_MATs represent ideals idealadd non square t_MAT in idealtyp c 2 0 2 O idealadd K c c non sense 6 2 0 0 2 d 1 0 0 2 idealadd K d d non sense ht 1 0 0 1 In the last two examples we get wrong results since the matrices c and d do not correspond to an ideal the Z span of their columns as usual interpreted as coordinates with respect to the integer basis
145. want to assign a function to a variable immediately as in R x y gt sqrt x 2 y 2 sq x gt x72 or equivalently x gt x72 but it is quite possible to define a priori short lived anonymous functions The trailing semicolon is not part of the definition but as usual prevents gp from printing the result of the evaluation i e the function object The construction f list of formal variables seq is available as an alias for f list of formal variables gt seq Using that syntax it is not possible to define anonymous functions obviously and the above two examples become R x y sqrt x 2 y 2 sq x x72 The semicolon serves the same purpose as above preventing the printing of the resulting function object compare sq x x72 no output 37 sq x x 2 MN print the result a function object 12 x gt x 2 Of course the sequence seg can be arbitrarily complicated in which case it will look better written on consecutive lines with properly scoped variables f x X15 Lig my to ti MN variables lexically scoped to the function body Note that the following variant would also work f x Tis e my to ti AN variables lexically scoped to the function body the first newline is disregarded due to the preceding sign and the others because of the enclosing braces The my statements can actually occur anywhere within the function body scoping t
146. we A 103 CONT TACO s 4 6283462224944 104 contfracpngn esposa 104 continued fraction 103 CONVOl Lera we ae hw ee 224 Coppersmith 127 COPE seisa 6 4 02 Bete Some ae ee 104 COPED eu ene pati cd Bh vce a 104 COTE 2 he soe dee God a we RE d 104 Goredis e 64 as 24 4 a Kee 105 COTEdISCO mesas we ee ARK eG 105 COVEdIsc mie Ae hee ae A oe we A 105 COS ema dd RHE Ge BEES bdo 92 GOSH 2 ese ee bleed dw BO we E 92 GOTAN 2 4 48a da oe eee aS He 92 CPU times where 313 CYC ia ae e he eae de 132 155 D Gatadir gt e s 3 bee eee ee ee 305 dbg_down 51 278 279 DELS o ocos moa ato mo ee 8 a al 279 GDS UD canoa cad eee e ete e 51 279 ADE ea d se bods 51 58 279 debug 0 57 110 305 debugfiles 57 305 debugmem 57 305 decodemodule 158 decomposition into squares 246 Dedekind sum 0 127 Dedekind 92 168 203 214 215 deep recursion 004 42 def factor_add_primes 306 def factor_proven 306 def new_galois_format 308 def prompt cont rocosos 310 default precision 9 A A SO eee 47 291 AefaulbO 5 648265 4 682669 2 amp 4 291 327 defaults Dd DT Genom oir he eee Ee eS 80 denominator 33 79 Geplin seos ces Oe ee r Ee BS 232 ASC e ara A Sek eee 215 dSTIV ID esa a a RS oS 257 derivnum
147. weights corresponding to negative t nh for N lt n lt 1 or may be empty but not always if p t is an odd function implicitly we would have tab 6 tab 4 and tab 7 tab 5 The library syntax is GEN intnuminit GEN a GEN b long m long prec 3 9 12 intnuminitgen t a b ph m 0 flag 0 Initialize tables for integrations from a to b using abscissas ph t and weights ph t Note that there is no equal sign after the variable name t since t always goes from oo to 00 but it is ph t which goes from a to b and this is not checked If flag 1 or 2 multiply the reserved table length by 4flag to avoid corresponding error The library syntax is intnuminitgen void E GEN eval void GEN GEN a GEN b long m long flag long prec 3 9 13 intnumromb X a b expr flag 0 Numerical integration of expr smooth in Ja bl with respect to X Suitable for low accuracy if expr is very regular e g analytic in a large region and high accuracy is desired try intnum first Set flag 0 or omit it altogether when a and b are not too large the function is smooth and can be evaluated exactly everywhere on the interval a b If flag 1 uses a general driver routine for doing numerical integration making no particular assumption slow flag 2 is tailored for being used when a or b are infinite One must have ab gt 0 and in fact if for example b 00 then it is preferable to have a as large a
148. x Returns the absolute trace of z The library syntax is GEN nftrace GEN nf GEN x 3 6 96 nfeltval nf x pr Given an element x in nf and a prime ideal pr in the format output by idealprimedec computes their the valuation at pr of the element x The same result could be obtained using idealval nf x pr since x would then be converted to a principal ideal but it would be less efficient The library syntax is long nfval GEN nf GEN x GEN pr 3 6 97 nffactor nf 7T Factorization of the univariate polynomial T over the number field nf given by nfinit T has coefficients in nf i e either scalar polmod polynomial or column vector The factors are sorted by increasing degree The main variable of nf must be of lower priority than that of T see Section 2 5 3 However if the polynomial defining the number field occurs explicitly in the coefficients of T as modulus of a t_POLMOD or as a t_POL coefficient its main variable must be the same as the main variable of T For example nf nfinit y 2 1 nffactor nf x 2 y OK nffactor nf x 2 Mod y y 2 1 OK nffactor nf x 2 Mod z z72 1 WRONG NN ND 189 It is possible to input a defining polynomial for nf instead but this is in general less efficient since parts of an nf structure will then be computed internally This is useful in two situations when you do not need the nf elsewhere or when you cannot compute the field discriminant due to integer
149. x M selects the non 0 entries in M If the latter is a t_MAT we extract the matrix of non 0 columns Note that removing entries instead of selecting them just involves replacing the selection function f with its negation select x gt isprime x vector 50 i1 172 1 The library syntax is genselect void E long fun void GEN GEN a Also avail able is GEN genindexselect void E long fun void GEN GEN a corresponding to flag 1 3 12 34 setrand n Reseeds the random number generator using the seed n No value is returned The seed is either a technical array output by getrand or a small positive integer used to generate deterministically a suitable state array For instance running a randomized computation starting by setrand 1 twice will generate the exact same output The library syntax is void setrand GEN n 3 12 35 system str str is a string representing a system command This command is executed its output written to the standard output this won t get into your logfile and control returns to the PARI system This simply calls the C system command 299 3 12 36 trap e rec seg THIS FUNCTION IS OBSOLETE use iferr which has a nicer and much more powerful interface For compatibility s sake we now describe the obsolete function trap This function tries to evaluate seq trapping runtime error e that is effectively preventing it from aborting computations in the usual way the recovery
150. x long prec 3 3 37 hyperu a b x U confluent hypergeometric function with parameters a and b The pa rameters a and b can be complex but the present implementation requires x to be positive The library syntax is GEN hyperu GEN a GEN b GEN x long prec 3 3 38 incgam s z g Incomplete gamma function S e t 1 dt extended by analytic con tinuation to all complex z s not both 0 The relative error is bounded in terms of the precision of s the accuracy of x is ignored when determining the output precision When g is given assume that g T s For small x this will speed up the computation The library syntax is GEN incgamO GEN s GEN x GEN g NULL long prec Also avail able is GEN incgam GEN s GEN x long prec 3 3 39 incgamc s x Complementary incomplete gamma function The arguments x and s are complex numbers such that s is not a pole of T and z s 1 is not much larger than 1 otherwise the convergence is very slow The result returned is Je ee dt The library syntax is GEN incgamc GEN s GEN x long prec 3 3 40 lambertw y Lambert W function solution of the implicit equation ze y for y gt 0 The library syntax is GEN glambertW GEN y long prec 94 3 3 41 Ingamma 2 Principal branch of the logarithm of the gamma function of x This function is analytic on the complex plane with non positive integers removed and can have much larger arguments than gamma itself For x a power series
151. x ulong lim The obsolete function GEN factorO GEN x long lim is kept for backward compatibility 108 3 4 21 factorback f e Gives back the factored object corresponding to a factorization The integer 1 corresponds to the empty factorization If e is present e and f must be vectors of the same length e being integral and the corre sponding factorization is the product of the ffi If not and f is vector it is understood as in the preceding case with e a vector of 1s we return the product of the fli Finally f can be a regular factorization as produced with any factor command A few examples factor 12 1 2 2 3 1 factorback 12 12 factorback 2 3 2 11 273 371 13 12 factorback 5 2 3 4 30 The library syntax is GEN factorback2 GEN f GEN e NULL Also available is GEN fac torback GEN f case e NULL 3 4 22 factorcantor z p Factors the polynomial x modulo the prime p using distinct degree plus Cantor Zassenhaus The coefficients of z must be operation compatible with Z pZ The result is a two column matrix the first column being the irreducible polynomials dividing x and the second the exponents If you want only the degrees of the irreducible polynomials for example for computing an L function use factormod z p 1 Note that the factormod algorithm is usually faster than factorcantor The library syntax is GEN factcantor GEN x GEN p 3 4 23 factorff
152. x and y that is the integer SN a and y 2 i Negative numbers behave 2 adically i e the result is the 2 adic limit of bitand x Yn where Zn and y are non negative integers tending to x and y respectively The result is an ordinary integer possibly negative bitand 5 3 1 1 bitand 5 3 12 3 bitand 5 3 3 7 The library syntax is GEN gbitand GEN x GEN y Also available is GEN ibitand GEN x GEN y which returns the bitwise and of x and y two integers 3 2 20 bitneg z n 1 bitwise negation of an integer x truncated to n bits n gt 0 that is the integer n 1 Y not x 2 i 0 The special case n 1 means no truncation an infinite sequence of leading 1 is then represented as a negative number See Section 3 2 19 for the behavior for negative arguments The library syntax is GEN gbitneg GEN x long n 3 2 21 bitnegimply z y Bitwise negated imply of two integers x and y or not x gt y that is the integer y a andnot y 2 See Section 3 2 19 for the behavior for negative arguments The library syntax is GEN gbitnegimply GEN x GEN y Also available is GEN ibitnegim ply GEN x GEN y which returns the bitwise negated imply of x and y two integers 3 2 22 bitor z y bitwise inclusive or of two integers x and y that is the integer 5 or y 2 See Section 3 2 19 for the behavior for negative arguments The library syntax is GEN gbitor GEN x
153. you a specific error message In short the types nf lt bn lt bnr are ordered each function requires a minimal type to work properly but you may always substitute a larger type The data types corresponding to the structures described above are rather complicated Thus as we already have seen it with elliptic curves GP provides member functions to retrieve data from these structures once they have been initialized of course The relevant types of number fields are indicated between parentheses bid bnr bid ideal structure bnf bnr baf Buchmann s number field clgp bnr bnf classgroup This one admits the following three subclasses cyc cyclic decomposition SNF gen generators no number of elements diff bnr bnf nf the different ideal codiff bnr bnf nf the codifferent inverse of the different in the ideal group disc bnr bnf nf discriminant fu bnr bnf fundamental units index bnr bnf nf index of the power order in the ring of integers mod bnr modulus nf bnr bnf nf number field pol bnr bnf nf defining polynomial ri bnr bnf nf the number of real embeddings r2 bnr bnf nf the number of pairs of complex embeddings reg bnr bnf regulator roots bnr bnf nf roots of the polynomial generating the field sign bnr bnf nf signature rl r2 t2 bnr bnf nf the Tz matrix see nfinit tu bnr bnf a generator
154. zero index giving a component number a negative index means we start counting from the end e a range of the form a b where a and b are indexes as above Any of a and b can be omitted in this case we take as default values a 1 and b 1 i e the first and last components respectively We then extract all components in the interval a b in reverse order if b lt a In addition if the first character in the string is the complement of the given set of indices is taken 253 If z is not omitted x must be a matrix y is then the row specifier and z the column specifier where the component specifier is as explained above v la b c d el vecextract v 5 mask 1 la c vecextract v 4 2 1 component list 72 d b al vecextract v 2 4 interval 13 b c dl vecextract v 1 3 interval reverse order 14 e d c vecextract v 2 complement 5 la c d el vecextract matid 3 2 76 0 1 0 0 0 1 The range notations v i j and v 7i for t_VEC or t_COL and M i j k 1 and friends for t_MAT implement a subset of the above in a simpler and faster way hence should be preferred in most common situations The following features are not implemented in the range notation e reverse order e omitting either a or bin a b The library syntax is GEN extractO GEN x GEN y GEN z NULL 3 8 74 vecsearch v x cmpf Determi
155. 0 o 1 0 2 concat x y 13 1 0 o 1 2 0 o 2 matconcat x y 4 1 0 0 1 2 0 0 2 To concatenate vectors sideways i e to obtain a two row or two column matrix use Mat instead or matconcat x 1 2 y 3 4 concat x y 3 1 2 3 4 Mat x y 44 1 2 3 4 matconcat x y 15 1 2 3 4 Concatenating a row vector to a matrix having the same number of columns will add the row to the matrix top row if the vector is x i e comes first and bottom row otherwise The empty matrix is considered to have a number of rows compatible with any operation in particular concatenation Note that this is not the case for empty vectors or 230 If y is omitted x has to be a row vector or a list in which case its elements are concatenated from left to right using the above rules concat 1 2 3 4 1 1 2 3 4 a 1 21 3 4 concat a ADA 1 3 2 4 concat 1 2 3 4 5 6 13 1 2 5 3 4 6 concat 7 8 1 2 3 4 5 i 5 7 2 3 4 6 8 1 2 3 4 The library syntax is GEN concat GEN x GEN y NULL GEN concat1 GEN x isashortcut for concat x NULL 3 8 4 forqfvec v q b expr q being a square and symmetric matrix representing a positive definite quadratic form evaluate expr for all vector v such that q v lt b The formal variable v runs through all such vectors in turn
156. 0 30 40 amp v v 13 2 2 The library syntax is GEN vecmax0 GEN x GEN v NULL Also available is GEN vec max GEN x 3 1 19 vecmin z amp v If x is a vector or a matrix returns the smallest entry of x otherwise returns a copy of x Error if x is empty If v is given set it to the index of a smallest entry indirect minimum when z is a vector If x is a matrix set v to coordinates i j such that xfi j is a smallest entry This is ignored if x is not a vector or matrix vecmin 10 20 30 40 1 30 vecmin 10 20 30 40 amp v v 12 3 vecmin 10 20 30 40 amp v v 13 2 1 The library syntax is GEN vecmin0 GEN x GEN v NULL Also available is GEN vecmin GEN x 3 1 20 Comparison and Boolean operators The six standard comparison operators lt lt gt gt are available in GP The result is 1 if the comparison is true 0 if it is false The operator is quite liberal for instance the integer 0 a 0 polynomial and a vector with 0 entries are all tested equal The extra operator tests whether two objects are identical and is much stricter than objects of different type or length are never identical For the purpose of comparison t_STR objects are strictly larger than any other non string type two t_STR objects are compared using the standard lexicographic order GP accepts lt gt as a synonym for On the other hand is definitely not
157. 0 realprecision 57 significant digits 50 digits displayed sin x 12 0 7071067811865475244 Note that even if the argument is real the result may be complex e g acos 2 0 or acosh 0 0 See each individual function help for the definition of the branch cuts and choice of principal value e If the argument is either an integer a rational an exact complex number or a quadratic number it is first converted to a real or complex number using the current precision held in the default realprecision This precision the number of decimal digits can be changed using p or default realprecision After this conversion the computation proceeds as above for real or complex arguments In library mode the realprecision does not matter instead the precision is taken from the prec parameter which every transcendental function has As in gp this prec is not used when the argument to a function is already inexact Note that the argument prec stands for the length in words of a real number including codewords Hence we must have prec gt 3 Some accuracies attainable on 32 bit machines cannot be attained on 64 bit machines for parity reasons For example the default gp accuracy is 28 decimal digits on 32 bit machines corresponding to prec having the value 5 but this cannot be attained on 64 bit machines e If the argument is a polmod representing an algebraic number then the function is evaluated for every possible complex embedding
158. 098 E 39 A sample alias file misc gpalias is provided with the standard distribution The library syntax is void aliasO const char newsym const char sym 3 12 5 allocatemem s 0 This special operation changes the stack size after initialization x must be a non negative integer If x gt 0 a new stack of at least x bytes is allocated We may allocate more than x bytes if x is way too small or for alignment reasons the current formula is max 16 1 16 500032 bytes If x 0 the size of the new stack is twice the size of the old one The old stack is discarded 289 Warning This function should be typed at the gp prompt in interactive usage or left by itself at the start of batch files It cannot be used meaningfully in loop like constructs or as part of a larger expression sequence e g allocatemem x 1 This will not set x In fact all loops are immediately exited user functions terminated and the rest of the sequence following allocatemem is silently discarded as well as all pending sequences of instructions We just go on reading the next instruction sequence from the file we re in or from the user In particular we have the following possibly unexpected behavior in read file gp x 1 were file gp contains an allocatemem statement the x 1 is never executed since all pending instructions in the current sequence are discarded The technical reason is that this routine moves the st
159. 1 in Q y li The first two are straightforward but the last one may not be what you want y is treated here as a numerical parameter not as a polynomial variable If the main variables are the same it is allowed to mix t_POL and t_POLMODs The result is the expected t_POLMOD For instance x Mod x x72 1 1 Mod 2 x x72 1 2 3 10 Polynomials t_POL Type the polynomial in a natural way not forgetting to put a x between a coefficient and a formal variable 1 2xx 3 x72 1 3 x 2 2 x 1 This assumes that x is still a free variable x 1 1 Qkx 3x72 42 6 generates an integer not a polynomial It is good practice to never assign values to polynomial variables to avoid the above problem but a foolproof construction is available using x instead of x x is a constant evaluating to the free variable with name x independently of the current value of x x 1 1 2 x 3x x 2 3 1 2xx 3x72 x x 1 2xx 3 x72 14 1 2xx 3x72 You may also use the functions Pol or Polrev Pol 1 2 3 Pol creates a polynomial in x by default fi x72 2xx 3 Polrev 1 2 3 2 3 x72 2x x 1 Pol 1 2 3 y we use y safer than y 21 3 y 2 2xy 3 The latter two are much more efficient constructors than an explicit summation the latter is quadratic in the degree the former linear for i 1 1074 Polrev vector 100 i i time 124ms
160. 1 bnf being the number field data output by bnfinit and x being an ideal this function tests whether the ideal is principal or not The result is more complete than a simple true false answer and solves general discrete logarithm problem Assume the class group is Z d Z g where the generators g and their orders d are respectively given by bnf gen and bnf cyc The routine returns a row vector e t where e is a vector of exponents 0 lt e lt di and t is a number field element such that t 9 For given g i e for a given bnf the e are unique and t is unique modulo units In particular x is principal if and only if e is the zero vector Note that the empty vector which is returned when the class number is 1 is considered to be a zero vector of dimension 0 K bnfinit y 2 23 K cyc 42 3 K gen 3 2 0 O 11 a prime ideal above 2 P idealprimedec K 3 1 a prime ideal above 3 v bnfisprincipal K P 5 2 3 4 1 4 idealmul K v 2 idealfactorback K K gen v 1 76 3 0 o 1 160 idealhnf K P 7 1 The binary digits of flagmean e 1 If set outputs e t as explained above otherwise returns only e which is much easier to compute The following idiom only tests whether an ideal is principal is_principal bnf x bnfisprincipal bnf x 0 e 2 It may not be possible to recover t given the initial accuracy to which bnf was com
161. 1 22801763489 Uses checkpointing and a naive O n algorithm The library syntax is GEN prime long n 120 3 4 59 primepi x The prime counting function Returns the number of primes p p lt zx primepi 10 hi 4 primes 5 42 2 3 5 7 11 primepi 10 11 3 4118054813 Uses checkpointing and a naive O x algorithm The library syntax is GEN primepi GEN x 3 4 60 primes n Creates a row vector whose components are the first n prime numbers Returns the empty vector for n lt 0 A t_VEC n a b is also allowed in which case the primes in a b are returned primes 10 the first 10 primes 41 2 3 5 7 11 13 17 19 23 29 primes 0 29 the primes up to 29 42 2 3 5 7 11 13 17 19 23 29 primes 15 30 3 17 19 23 29 The library syntax is GEN primesO GEN n 3 4 61 qfbclassno D flag 0 Ordinary class number of the quadratic order of discriminant D In the present version 2 7 0 a O D algorithm is used for D gt 0 using Euler product and the functional equation so D should not be too large say D lt 108 for the time to be reasonable On the other hand for D lt 0 one can reasonably compute qfbclassno D for D lt 107 since the routine uses Shanks s method which is in O D For larger values of D see quadclassunit If flag 1 compute the class number using Euler products and the functional equation However it is in O D
162. 1 2 3 4 setsearch L 4 12 6 setsearch L 2 3 0 This is faster than the vecsort command since the list is sorted in place no copy is made No value returned If flag is non zero suppresses all repeated coefficients The library syntax is void listsort GEN L long flag 3 8 12 matadjoint M flag 0 adjoint matrix of M i e a matrix N of cofactors of M satisfying M x N det M x Id M must be a non necessarily invertible square matrix of dimension n If flag is 0 or omitted we try to use Leverrier Faddeev s algorithm which assumes that n invertible If it fails or flag 1 compute T charpoly M independently first and return 1 T x T 0 x evaluated at M a 1 2 3 3 4 5 6 7 8 Mod 1 4 2 Mod 1 4 Mod 2 4 Mod 3 4 Mod 3 4 Mod 0 4 Mod 1 4 Mod 2 4 Mod 3 4 Mod 0 4 Both algorithms use O n operations in the base ring and are usually slower than computing the characteristic polynomial or the inverse of M directly The library syntax is GEN matadjointO GEN M long flag Also available are GEN adj GEN x flag 0 and GEN adjsafe GEN x flag 1 3 8 13 matcompanion z The left companion matrix to the non zero polynomial x The library syntax is GEN matcompanion GEN x 233 3 8 14 matconcat v Returns a t_MAT built from the entries of v which may be a t_VEC con catenate horizontally a t_COL concatenate vertically or a t_MAT concatenate verti
163. 1 used to contain an obsolete check number the number of roots of unity and a generator bnf tu the fundamental units bnf fu bnf 9 is a 3 element row vector used in bnfisprincipal only and obtained as follows Let D UWV obtained by applying the Smith normal form algorithm to the matrix W bnf 1 and let U be the reduction of U modulo D The first elements of the factorbase are given in terms of bnf gen by the columns of U with Archimedean component ga let also GD be the Archimedean components of the generators of the principal ideals defined by the bnf gen i bnf cyc i Then bnf 9 Ur Ja GDa bnf 10 is by default unused and set equal to 0 This field is used to store further information about the field as it becomes available which is rarely needed hence would be too expensive to compute during the initial bnfinit call For instance the generators of the principal ideals bnf gen i bnf cyc i during a call to bnrisprincipal or those corresponding to the relations in W and B when the bnf internal precision needs to be increased The library syntax is GEN bnfinitO GEN P long flag GEN tech NULL long prec Also available is GEN Buchall GEN P long flag long prec corresponding to tech NULL where flag is either 0 default or nf_FORCE insist on finding fundamental units The function GEN Buchall_param GEN P double c1 double c2 long nrpid long flag long prec gives direct access to the tec
164. 1 4 The PARI philosophy The basic principles which govern PARI is that operations and functions should firstly give as exact a result as possible and secondly be permitted if they make any kind of sense In this respect we make an important distinction between exact and inexact objects by definition types t_REAL t_PADIC or t_SER are imprecise A PARI object having one of these imprecise types anywhere in its tree is inexact and exact otherwise No loss of accuracy rounding error is involved when dealing with exact objects Specifically an exact operation between exact objects will yield an exact object For example dividing 1 by 3 does not give 0 333 but the rational number 1 3 To get the result as a floating point real number evaluate 1 3 or 0 1 3 Conversely the result of operations between imprecise objects although inexact by nature will be as precise as possible Consider for example the addition of two real numbers x and y The accuracy of the result is a priori unpredictable it depends on the precisions of x and y on their sizes and also on the size of x y From this data PARI works out the right precision for the result Even if it is working in calculator mode gp where there is a notion of default precision its value is only used to convert exact types to inexact ones In particular if an operation involves objects of different accuracies some digits will be dis regarded by PARI It is a common source of e
165. 1 bo ao b1 a1 bn an Note that in this case one usually has by 1 If n gt 0 is present returns all convergents from po qo up to Pn qn All convergents if x is too small to compute the n 1 requested convergents a contfrac Pi 20 1 3 7 15 1 292 1 1 1 2 1 3 1 14 2 1 1 2 2 2 2 contfracpngn a 3 42 3 22 333 355 1 7 106 113 contfracpnqn a 7 43 3 22 333 355 103993 104348 208341 312689 1 7 106 113 33102 33215 66317 99532 The library syntax is GEN contfracpnqn GEN x long n also available is GEN pnqn GEN x for n 1 104 3 4 13 core n flag 0 If n is an integer written as n df with d squarefree returns d If flag is non zero returns the two element row vector d f By convention we write 0 0 x 17 so core 0 1 returns 0 1 The library syntax is GEN coreO GEN n long flag Also available are GEN core GEN n flag 0 and GEN core2 GEN n flag 1 3 4 14 coredisc n flag 0 A fundamental discriminant is an integer of the form t 1 mod 4 or 4t 8 12mod 16 with t squarefree i e 1 or the discriminant of a quadratic number field Given a non zero integer n this routine returns the unique fundamental discriminant d such that n df f a positive rational number If flag is non zero returns the two element row vector d f If n is congruent to 0 or 1 modulo 4 f is an integer and a half integer otherwise By convention cored
166. 1 x 1 time 340 ms 5 5 98 E 85 8 08 E 85 I better than 1 intlaplaceinv x 5 1 1 x m 1 1 time 680 ms 16 1 09 E 106 0 E 104 I perfect fast intlaplaceinv x 10 1 1 x 1 time 340 ms AT 4 36 E 106 0 E 102 I perfect fastest but why sig 10 intlaplaceinv x 100 1 1 x 1 time 330 ms 47 1 07 E 72 3 2 E 72 I too far now The library syntax is intlaplaceinv void E GEN eval void GEN GEN sig GEN z GEN tab long prec 3 9 8 intmellininv X sig z expr tab Numerical integration of 2ir lexpr X z with respect to X on the line R X sig in other words inverse Mellin transform of the function corresponding to expr at the value z sig is coded as follows Either it is a real number equal to the abscissa of integration and then the integrated is assumed to decrease exponentially fast of the order of exp t when the imaginary part of the variable tends to oo Or it is a two component vector o a where is as before and either a 0 for slowly decreasing functions or a gt 0 for functions decreasing like exp at such as gamma products Note that it is not necessary to choose the exact value of a and that a 1 equivalent to sig alone is usually sufficient tab is as in intnun 259 As all similar functions this function is provided for the convenience of the user who could use intnum directly However it is in
167. 105 perfect intnum x 0 oo 1 sin 2 x x Pi 2 oops wrong k 4 0 07 intnum x O o00 2 I sin 2 x x Pi 2 5 0 E 105 perfect intnum x 0 oo I sin x 3 x Pi 4 76 0 0092 bad sin x 3 3 sin x sin 3 x 4 AT 0 x717 We may use the above linearization and compute two oscillating integrals with infinite endpoints loo I and oo 3 I respectively or notice the obvious change of variable and reduce to the single integral 5 J sin x x du We finish with some more complicated examples intnum x 0 loo 1 1 cos x x 2 Pi 2 1 0 0004 bad intnum x 0 1 1 cos x x 2 intnum x 1 oo 1 x 2 intnum x 1 oo 1 cos x x 2 Pi 2 12 2 18 E 106 OK 263 intnum x 0 oo 1 sin x 3 exp x 0 3 3 5 45 E 107 OK intnum x 0 o0o0 I sin x 3 exp x 0 3 14 1 33 E 89 lost 16 decimals Try higher m m intnumstep 15 7 the value of m actually used above tab intnuminit 0 oo I m 1 try m one higher intnum x 0 oo sin x 3 exp x tab 0 3 76 5 45 E 107 OK this time Warning Like sumalt intnum often assigns a reasonable value to diverging integrals Use these values at your own risk For example intnum x O oo I x 2 sin x 1 2 0000000000 Note the formula ie sin x ax dx cos rs 2 T 1 s a priori valid only for 0 lt
168. 2 is invertible and check whether squaring the truncated power series for the square root yields the original input The library syntax is long issquareall GEN x GEN n NULL Also available is long is square GEN x Deprecated GP specific functions GEN gissquare GEN x and GEN gissquare all GEN x GEN pt return gen_0O and gen_1 instead of a boolean value 3 4 45 issquarefree x True 1 if x is squarefree false 0 if not Here x can be an integer or a polynomial The library syntax is long issquarefree GEN x 3 4 46 istotient z amp N True 1 if x p n for some integer n false 0 if not istotient 14 41 0 istotient 100 42 0 If N is given set N n as well istotient 4 amp n 41 1 n 92 10 The library syntax is long istotient GEN x GEN N NULL 3 4 47 kronecker z y Kronecker symbol x y where x and y must be of type integer By definition this is the extension of Legendre symbol to Z x Z by total multiplicativity in both arguments with the following special rules for y 0 1 or 2 e 1 0 1 if x 1 and 0 otherwise e x 1 1 if x gt 0 and 1 otherwise e 1 2 0 if x is even and 1 if z 1 1mod8 and 1 if z 3 3 mod 8 The library syntax is long kronecker GEN x GEN y 117 3 4 48 lcm x y Least common multiple of x and y i e such that lem z y x gcd x y abs x x y If y is omitted and x is a vector returns the lem of all comp
169. 3 1 0 o 1 More often than not a principal ideal yields the unit ideal as above This is a quick and dirty way to check if ideals are principal but it is not a necessary condition a non trivial result does not prove that the ideal is non principal For guaranteed results see bnfisprincipal which requires the computation of a full bnf structure If the input is an extended ideal I s the output is J sa r this way one can keep track of the principal ideal part idealred K P 1 25 LLL 0 O 11 2 11 meaning that P is generated by 2 1 The number field element in the extended part is an algebraic number in any form or a factorization matrix in terms of number field elements not ideals In the latter case elements stay in factored form which is a convenient way to avoid coefficient explosion see also idealpow Technical note The routine computes an LLL reduced basis for the lattice J equipped with the quadratic form ri r2 ell 2 tal 1 where as usual the c are the real and complex embeddings and e 1 resp 2 for a real resp complex place The element a is simply the first vector in the LLL basis The only reason you may want to try to change some directions and set some v 4 0 is to randomize the elements found for a fixed ideal which is heuristically useful in index calculus algorithms like bnfinit and bnfisprincipal Even more technical note In fact the above is a white lie
170. 32 136 137 140 146 280 elldivpol 6 e240 rosas 136 elleisnum 136 137 ell ta eee daa wk 137 149 OLLETOM sp ds ee ST ellgenerators 132 137 ellglobalred 137 138 6lIgrOUp 22 26 2 eas es 135 138 139 ClLIBTOUpO fA uch dica is 139 ellheegner 139 ellheight c o2c0o be Pee Has 139 ellheight0 139 ellheightmatriX 139 ellidentify 132 140 ellinit 130 132 137 140 141 ellisoncurve 141 Ts woe bes ae oe ee 141 ELILI ooa rase Be de ae ees 132 133 elllocalred av fee edea RE 142 SLLLOS 24 52 oR pews ea ee eee 142 elllseries 142 ellminimalmodel 137 142 143 ellmodulareqn 143 elilmul cos o oo 143 144 146 ELITES a sae oe ee oe a er 144 llord r lt 4 56 4 845 be BERS 144 145 ellordinate 84 145 ellperiods 130 136 145 147 148 ellpointtoz soe soss redeat pas 145 OLIPOW 24 24 406 folei eas po ae eo 146 ellrandom 2 qq p s 85 ellr ObtmO ys oy ae 146 ellsearch 132 146 147 ellsearchcurve 147 ellsigma s o sacs Aa e046 Pe noe a 4 147 CllSUD soassa eroa eA i aata 147 elltaniyama 147 elltatepairing 147 CLITOTS a a Ok a ee we 147 148 lltorsO ee 2 ange 4 ee ae a 148 ellweilpairing 148 CLUWD 04 w dae ble de ae we eo 148 CllwpO a ete ae fee e
171. 34 x 8 7453176 x76 13950764 x 4 5596840 x72 46225 T1 polredabs T T2 polredbest T norml2 polroots T1 norml2 polroots T2 3 88 0000000 120 000000 200 sizedigit poldisc T1 sizedigit poldisc T2 4 75 67 The library syntax is GEN polredabsO GEN T long flag Instead of the above hardcoded numerical flags one should use an or ed combination of e nf_PARTIALFACT possibly use a suborder of the maximal order without attempting to certify the result e nf_ORIG return P a where Mod a P is a root of T e nf_RAW return P b where Mod b T is a root of P The algebraic integer b is the raw result produced by the small vectors enumeration in the maximal order P was computed as the characteristic polynomial of Mod b T Mod a P as in nf_ORIG is obtained with modreverse e nf_ADDZK if r is the result produced with some of the above flags of the form P or P c return r zk where zk is a Z basis for the maximal order of Q X P e nf_ALL return a vector of results of the above form for all polynomials of minimal T gt norm 3 6 121 polredbest T flag 0 Finds a polynomial with reasonably small coefficients defining the same number field as T All T accepted by nfinit are also allowed here e g non monic polynomials nf bnf T Z_K_basis Contrary to polredabs this routine runs in polynomial time but it offers no guarantee as to the minimality of its result
172. 445 4525 be ES es 185 POPE hi Ge Biot eG wee Oo es 251 Perl 202 56 5 ot oe ig Be oo be Be 54 Perl sor g fe a wists a a 34 PETMtONUM veces a eae 83 PI is igen eds hoes amp Phau See fh a we te 89 PLOG s 6 2 ag ao oe HS alee Fle tae A 213 PLOTDOR va soros a o ee eS 273 PLOtCIAP soseo e a ee a o 273 plotcolor ss 220244 e 273 306 PLOUCOPY se a s ute Hee dew ee A 273 plotcursor 244 46 bens Sb ee aS 200 PLlOtdrawW cos oa ca bee ede ee eS 273 PLOCH soo yaw Bae doe ae oo A 64 274 Plothraw s sose osag em e d dow 4 ee as 275 plothsizes 275 276 PLOTIDIE secos ew eee a 275 Pl AA 276 plotlines 220 lt 246 240 fe ea 4 276 plotlinetype 276 PLOUMOVE spo ani a ee bee a p os 276 PlOtPOINtS s sows aa ee E ws 276 plotpointsize s ed suea consi 276 plotpointtype ooo ooo 276 PLOTEDOX sm ios sor 276 plotrecth e saara ma 2174 211 plotrecthraw 2i plottline aires a PPa a 277 P O TITMOVE secs a 277 plotrpoint io cr 20 plotscale ususarios rd 274 277 plotstring s re rerata 47 277 plotterm es s a e gobi 4448 ee das 47 A Bat eee eh Se es 104 pointell s sses eite be ee bh pips 149 DOUET o e eee a ae a 64 Pol ag ae ae a k a ad 22 13 T4 Pol ecse ee be eae netii n rae 155 polchebyshev 219 polchebyshevi 219 224 polchebyshev2 219 polchebyshev_eval 219 POLCOCEE ima ios ac eR 78 219 polcoeffO sr te Pea rras 219
173. 456789 1 45 Other bases that 10 are not supported Note that the sum of bits in n is returned by hammingweight The library syntax is GEN sumdigits GEN n 3 4 86 zncoppersmith P N X B NY N being an integer and P Z X finds all integers x with x lt X such that gcd N P x gt B using Coppersmith s algorithm a famous application of the LLL algorithm X must be smaller than exp log B deg P log N for B N this means X lt N ces P Some z larger than X may be returned if you are very lucky The smaller B or the larger X the slower the routine will be The strength of Coppersmith method is the ability to find roots modulo a general composite N if N is a prime or a prime power polrootsmod or polrootspadic will be much faster We shall now present two simple applications The first one is finding non trivial factors of N given some partial information on the factors in that case B must obviously be smaller than the largest non trivial divisor of N setrand 1 to make the example reproducible p nextprime random 10730 q nextprime random 10730 N px q pO p 10 20 assume we know 1 p gt 10 29 2 the last 19 digits of p p1 zncoppersmith 10719 x pO N 10712 10729 result in 10ms 1 35023733690 gcd p1 1 10719 pO N p 12 1 and we recovered p faster than by trying all possibilities lt 1012 The second application is an attack on RSA with low exp
174. 5 824 re pri 251 252 Setintersect 252 SOL SSO 5 ise A wa eee es 252 SCtLMINUS sis e es KO 252 Setrand kos je ee ke Do a ee 84 292 299 setsearch 233 252 253 S tUNION seo soe ewe ee ew ee 253 Shanks SQUFOF 99 110 Shanks 4 2 0 24 454 8 1281 122 123 SHIELD ven gee ia Ron ee 2 69 ShHiftmul p a0 2 Osis eae el es 69 SIMA osos nos ba aes 105 125 SION os Row a Be RR ee ee ew S 69 Sit oA ge kh AA ek 69 155 255 Signunits 162 Simplify sora eresia 56 85 86 311 SIN 235 4 559 R408 ia Ss 96 SUNN fod BOG A Be He Art we eo 96 SUZCDY UC ein ese ere Guile Se eae 86 Sizedigit co cooomom corras 86 Smith normal form 155 159 161 183 196 242 282 NOF se oca pate e soe A he Se as 157 SOLVO oe csere bowled eae ew ew ee 267 SOMME ope teh ee ee a oe ee we A 267 SOpath crespe aa Be PEG ew 311 SOE a We so di athe ee She de Re de Ee i 96 BOCE ee 4 conser A 96 SQrtint 2 63 6 2846556 Pea e ee 126 SOTEN 00 500 SAAS OA BAe eee SM 96 SQUUNIOU si oe dad 126 Statki 2 ei eck we ba ae eS 58 308 312 Stacksize 2 2 28 eee ee ee ee 43 Stark units 126 167 Startup secorre 2 guide amp As is e 58 Steinitz class 213 Stirling number 126 Stirling recreere ta dras Ge x 126 Stirlingi i wae BS he deacons a Wise dM 4 126 Stirling 2 sits ae ase we Bs GSE ae Se 126 SEP esmero 46 47 75 SUrchY sassa ke ee ALG yaaa 75
175. 6 72 matalgtobasis nf x nf being a number field in nfinit format and x a row or column vector or matrix apply nfalgtobasis to each entry of x The library syntax is GEN matalgtobasis GEN nf GEN x 3 6 73 matbasistoalg nf x nf being a number field in nfinit format and x a row or column vector or matrix apply nfbasistoalg to each entry of x The library syntax is GEN matbasistoalg GEN nf GEN x 183 3 6 74 modreverse z Let z Mod A T be a polmod and Q be its minimal polynomial which must satisfy deg Q deg T Returns a reverse polmod Mod B Q which is a root of T This is quite useful when one changes the generating element in algebraic extensions Mod x x73 x 1 v ub w modreverse v 2 Mod x 2 4 x 1 x73 5 x72 4 x 1 u which means that z 52 4x 1 is another defining polynomial for the cubic field Q u Qla 1 x 1 Qlr 2 5a 4a 1 Q v and that u gt v 4v 1 gives an explicit isomorphism From this it is easy to convert elements between the A u Q u and B v Q v representations A u72 24u 3 subst lift A x w 13 Mod x72 3 x 3 x73 5 x 2 4 x 1 B v2 v 1 subst 1ift B x v 4 Mod 26xx 2 31 x 26 x 3 x 1 If the minimal polynomial of z has lower degree than expected the routine fails u Mod x73 9 x x74 10 x72 1 modreverse u xxx modreverse domain error in modrev
176. 7i i kK paca suminf user interrupt after 10min 20 100 ms sumalt i 1 1 i i Log 2 time O ms 1 2 524354897 E 29 The library syntax is suminf void E GEN eval void GEN GEN a long prec 268 3 9 24 sumnum X a sig expr tab flag 0 Numerical summation of expr the variable X taking integer values from ceiling of a to 00 where expr is assumed to be a holomorphic function f X for R X gt 0 The parameter R is coded in the argument sig as follows it is either e a real number O Then the function f is assumed to decrease at least as 1 X at infinity but not exponentially e a two component vector o a where o is as before a lt 1 The function f is assumed to decrease like X In particular a lt 2 is equivalent to no a at all e a two component vector lo a where is as before a gt 0 The function f is assumed to decrease like exp aX In this case it is essential that a be exactly the rate of exponential decrease and it is usually a good idea to increase the default value of m used for the integration step In practice if the function is exponentially decreasing sumnum is slower and less accurate than sumpos or suminf so should not be used The function uses the intnum routines and integration on the line R s The optional argument tab is as in intnum except it must be initialized with sumnuminit instead of intnuminit When tab is not precomputed
177. 857142857142857 1428571429 42 1 7 bestappr Pi sqrt 2 x 1073 3 355 113 x 1393 985 By definition a b is the best rational approximation to x if ba a lt vx ul for all integers u v with 0 lt v lt B Which implies that n d is a convergent of the continued fraction of x e If x is a t_INTMOD modulo N or a t_PADIC of precision N p this function performs rational modular reconstruction modulo N The routine then returns the unique rational number a b in coprime integers a lt N 2B and b lt B which is congruent to x modulo N Omitting B amounts to choosing it of the order of y N 2 If rational reconstruction is not possible no suitable a b exists returns bestappr Mod 18526731858 11710 1 1 7 bestappr Mod 18526731858 11720 42 bestappr 3 5 3 5 2 5 3 3 5 4 575 3x5 6 O 5 7 2 1 3 In most concrete uses B is a prime power and we performed Hensel lifting to obtain x 100 The function applies recursively to components of complex objects polynomials vectors If rational reconstruction fails for even a single entry return The library syntax is GEN bestappr GEN x GEN B NULL 3 4 5 bestapprPade z B Using variants of the extended Euclidean algorithm returns a rational function approximation a b to x whose denominator is limited by B if present If B is omitted return the best approximation affordable given the input acc
178. A 188 nfalgtobasis 183 184 nfbasis 185 186 187 189 194 309 nfbasistoalg 183 186 nfcertify 186 187 194 200 MEMSCANG ee sinsa Bw Hark HL OS 187 nidis gh wae are be 187 189 309 NEGAV cosa oe a ee ee 188 NGAIVEUC e so we ari a we Re ee Boe a 188 MiGivModpY s o gt e Bw de ee A 188 DEV soe e a Boe ee ae a 188 NEST Cad Me moco axes anta we ig 188 NE6ITdIV so onea SRS SRE HS HS ad 188 334 nfeltdiveuc 24444 4464 9 266 4 45 188 nfeltdivmodpr 188 Hfeltdivrem cui ote GU ee Se eS 188 nteltmod era 2024 2 e620 s 188 nfeltmul se s be eb ee cross 188 nfeltmulmodpr 188 nfeltnorm 2 44 664548 264 65 188 NPCLCPOW doo soy we a Dae oe es 188 nfeltpowmodpr 189 nfeltreduce venom rei a os 189 nfeltreducemodpr 189 ds we ad eee hee wai on eR Ge we 189 nteltval 5 2 oe Oa e ale hae amp 189 nffactor s o so wex 108 168 189 190 195 nffactorback 152 175 190 nifactormod sc se cr bp aaea 190 Nipaloisapplys eases g ew ewe s 190 nfgaloisconj 169 191 D NDTIDOTE rena s Gree ss HS de 192 nfhilbertO 2 0 66 bee eee ee ae 192 MERGE sd Gog Se K Hoke Se oo ee 192 240 nfihnimod e 2 a a Yea e a eee he 5 192 nfinit 151 170 192 194 199 200 201 309 DEILCC ela re 194 nfinitall 2 524 2 45 64 2 4 od 4 194 nfinitred ss cos hee senses 194 MEIMACVEM age e g eh ee a Se ea e i 194 OLIVA as 189 nfisidea
179. A Tt Bbittese so ca ee Bw a ss 78 BDIEXOS caso mias pa Oe pipes 78 EDOUMAC es Pe big ee ee ee als 104 Ed i gh pao Rh Sa He alk ee ae 9 113 Bedekt 2 2 64 na ees He Foe ee oe 114 BCdeXtO vir ek new eG ae eee 101 114 Geil cea nde he eee we Eee eae 78 BCL come Gp eae a ma o ae aop aew i 104 BCE wpa A a aed gs 104 ECON TP sie d wba ew bee Ben Ge an 79 GCOS sA s kia SD eA aS 92 BCOSh 2 245 6 be ee EG eee RE 92 COOTAD 2448474 Gaon he go oes 92 ECUV OL cos vans See be ee des 86 gdeflate 2 25 wae ea wa eas 225 EdiVo pere e dine eB help dow He alk Set ag 65 gdivent escocesa 66 diventres 4 268 485 eb eee eG 68 gdiyro nd ostia doo ed es 66 gen member function 155 GEN es aes a es 8 BOL 2 bed we ee be eee ee 132 Senapply e sks Borie ee Ba a A 291 generic MATIX surta EE we Soe a 47 genindexselect 299 Genrand o e rosas a dk Pod ere o 85 genselect aor ms woy aos i aeaa E ee 299 GENTOS T 25 2 eee ag ada a 75 Senus2red 2g Ca ew 149 151 gens rad ore ed ek ans a a 89 Serfc cop occ ee ee es 92 getabstime s s rooe sirat sos 291 292 SOLE ie acara ica e e a de ad 291 Setheap AA ek serea prd 292 POULT ANG ais a m Pose ne ge wi a E 84 292 getstack s aosa ea doa we OSE ore awe A 292 get time co be ees eee ee es 292 oval e e oo 4 Go be Gee oe 217 BOXP a seeped ag eee ee eee eo 93 SOXPpml ios sae wae we We ee 93 BLOOT s oc a Wes Be odie dd 80 B TAC 2 wk ne hak DEG a 80 ESAMA oc oid de RG gS ae
180. As 1 27 vecthetanullk 98 tuFFELT 2 2 nea eA ee es Y 19 vecthetanullk_tau 98 COPRAC 2 ae dhe eae aa 1 18 VECTOR 2 6 e a ee Rm ae te 8 ELINT 5 y Gk ork bh e da amp fy ke VECtOr asii SRR 23 256 TOINTMOD o cai Hh Gok ne at BG Be Y 18 yectorsmall ss ws ee e a 256 NSLS tee ee aa A wh ee es Bs 7 26 VECLOIV s oeo moa Pe ew wwe e O a 23 256 MAT ers ra anina aa a a he ars 7 24 version number 58 E PADICC lt gt encara Be ee bs 7 19 version 300 a re ee a re Tol i eyewear hace eee Pe Beles 61 t_POLMOD 000 7 20 QED s oreu bby oe be oe bee od 7 2 Ww COMER corre Weare oe eee a4 T 22 WAITING A 301 t_QUAD ee ee ee 7 20 WEDOL i iire ie ok GO ie ed WS ee ws 98 t_REAL eee ee ee 7 17 WED TO ae ro eroe rs 98 TERFRAG mom a oed e T 22 weberf eee ee ee 98 t_SER ooo T 22 WEDOLLEL 4 Gk Sie Gee e a a 98 t_STR oe 7 26 WEDET Z oane av oee He eH ek Ra a a 98 t_VEC 2 aaa arara 7 23 Weierstrass g function 149 t_VECSMALL ooo 7 26 Weierstrass equation 130 Weil curve o 147 U WHATKOW o ses 6 a G4 ee Ee a 47 301 ulimit 2 ce aaa a aaa 43 WHILE t n a 2 wie dane eS es 287 WHINING s gn ae es Heal 44 Sw eo 300 write 2 2 eae 47 55 58 301 302 Until cis eo Se oe a oe ee e a 287 Writel o eee ee ee 302 user defined functions 37 Wwritebin oo 302 WEITCTCK 4a w e ba ww ee A eS 302 V
181. B2 generates CI K we are done User supplied c are only used to compute initial guesses for the bounds B and the algorithm increases them until one can prove under GRH that S B2 generates CI K A uniform result of Bach says that cz 12 is always suitable but this bound is very pessimistic and a direct algorithm due to Belabas Diaz Friedman is used to check the condition assuming GRH The default values are Cy Co 0 When c is equal to 0 the algorithm takes it equal to cz 156 nrpid is the maximal number of small norm relations associated to each ideal in the factor base Set it to 0 to disable the search for small norm relations Otherwise reasonable values are between 4 and 20 The default is 4 Warning Make sure you understand the above By default most of the bnf routines depend on the correctness of the GRH In particular any of the class number class group structure class group generators regulator and fundamental units may be wrong independently of each other Any result computed from such a bnf may be wrong The only guarantee is that the units given generate a subgroup of finite index in the full unit group You must use bnfcertify to certify the computations unconditionally Remarks You do not need to supply the technical parameters under the library you still need to send at least an empty vector coded as NULL However should you choose to set some of them they must be given in the requested order For example
182. Comp 51 note that Silverman s height is twice ours If flag 1 use Tate s 4 algorithm If flag 2 use Mestre s AGM algorithm The latter converges quadratically and is much faster than the other two The library syntax is GEN ellheightO GEN E GEN x long flag long prec Also avail able is GEN ghell GEN E GEN x long prec flag 2 139 3 5 23 ellheightmatrix E x x being a vector of points this function outputs the Gram matrix of x with respect to the N ron Tate height in other words the i j component of the matrix is equal to ellbil E x i x j The rank of this matrix at least in some approximate sense gives the rank of the set of points and if x is a basis of the Mordell Weil group of E its determinant is equal to the regulator of E Note that this matrix should be divided by 2 to be in accordance with certain normalizations E is assumed to be integral given by a minimal model The library syntax is GEN mathell GEN E GEN x long prec 3 5 24 ellidentify E Look up the elliptic curve E defined by an arbitrary model over Q in the elldata database Return N M G C where N is the curve name in Cremona s elliptic curve database M is the minimal model G is a Z basis of the free part of the Mordell Weil group E Q and C is the change of coordinates change suitable for ellchangecurve The library syntax is GEN ellidentify GEN E 3 5 25 ellinit z D 1 Initialize an ell structure associated t
183. EN q assumes that q is positive definite and is a little faster returns NULL if a vector with negative norm occurs non positive matrix or too many rounding errors 3 8 55 qfisom G H fl G H being square and symmetric matrices with integer entries repre senting positive definite quadratic forms return an invertible matrix S such that G SHS This defines a isomorphism between the corresponding lattices Since this requires computing the mini mal vectors the computations can become very lengthy as the dimension grows See qfisominit for the meaning of fl G can also be given by an qfisominit structure which is preferable if several forms H need to be compared to G This function implements an algorithm of Plesken and Souvignier following Souvignier s im plementation The library syntax is GEN qfisom0 GEN G GEN H GEN fl NULL Also available is GEN qfisom GEN G GEN H GEN fl where G is a vector of zm and H is a zm 3 8 56 qfisominit G f1 G being a square and symmetric matrix with integer entries represent ing a positive definite quadratic form return an isom structure allowing to compute isomorphisms between G and other quadratic forms faster The interface of this function is experimental and will likely change in future release If present the optional parameter ff must be a t_VEC with two components It allows to specify the invariants used which can make the computation faster or slower The components are
184. EN x long flag See also GEN kerint GEN x flag 0 which is a trivial wrapper around ZM_111 ZM_111 x 0 99 LLL_KER 0 99 LLL_INPLACE Remove the outermost ZM_111 if LLL reduction is not desired saves time 3 8 35 matmuldiagonal x d Product of the matrix x by the diagonal matrix whose diagonal entries are those of the vector d Equivalent to but much faster than x matdiagonal d The library syntax is GEN matmuldiagonal GEN x GEN d 3 8 36 matmultodiagonal x y Product of the matrices x and y assuming that the result is a diagonal matrix Much faster than xx y in that case The result is undefined if x y is not diagonal The library syntax is GEN matmultodiagonal GEN x GEN y 3 8 37 matpascal n q Creates as a matrix the lower triangular Pascal triangle of order x 1 i e with binomial coefficients up to x If q is given compute the q Pascal triangle i e using g binomial coefficients The library syntax is GEN matqpascal long n GEN q NULL Also available is GEN mat pascal GEN x 3 8 38 matqr M flag 0 Returns Q R the QR decomposition of the square invertible matrix M with real entries Q is orthogonal and R upper triangular If flag 1 the orthogonal matrix is returned as a sequence of Householder transforms applying such a sequence is stabler and faster than multiplication by the corresponding Q matrix More precisely if Q R matqr M q r matgr M 1 then r R and mathousehol
185. EN x long prec 3 3 24 besselk nu x K Bessel function of index nu and argument z The library syntax is GEN kbessel GEN nu GEN x long prec 3 3 25 besseln nu x N Bessel function of index nu and argument zx The library syntax is GEN nbessel GEN nu GEN x long prec 3 3 26 cos x Cosine of x The library syntax is GEN gcos GEN x long prec 3 3 27 cosh x Hyperbolic cosine of x The library syntax is GEN gcosh GEN x long prec 3 3 28 cotan x Cotangent of x The library syntax is GEN gcotan GEN x long prec 3 3 29 dilog x Principal branch of the dilogarithm of x i e analytic continuation of the power series log z 0 5 1 n2 The library syntax is GEN dilog GEN x long prec 3 3 30 eint1 x n Exponential integral 2 a dt incgam 0 x where the latter expression extends the function definition from real x gt 0 to all complex x 4 0 If n is present we must have x gt 0 the function returns the n dimensional vector eint1 x einti nx Contrary to other transcendental functions and to the default case n omitted the values are correct up to a bounded absolute rather than relative error 107n where n is precision if x is a t_REAL and defaults to realprecision otherwise In the most important application to the computation of L functions via approximate functional equations those values appear as weights in long sums and small individual relative errors are less useful than con
186. FOF Pollard Rho ECM and MPQS stages and has an early exit option for the functions moebius and the integer function underlying issquarefree This machinery relies on a fairly strong probabilistic primality test see ispseudoprime but you may also set default factor_proven 1 to ensure that all tentative factorizations are fully proven This should not slow down PARI too much unless prime numbers with hundreds of decimal digits occur frequently in your application 3 4 2 Orders in finite groups and Discrete Logarithm functions The following functions compute the order of an element in a finite group ellorder the rational points on an elliptic curve defined over a finite field fforder the multiplicative group of a finite field znorder the invertible elements in Z nZ The following functions compute discrete logarithms in the same groups whenever this is meaningful elllog fflog znlog All such functions allow an optional argument specifying an integer N representing the order of the group The order functions also allows any non zero multiple of the order with a minor loss of efficiency That optional argument follows the same format as given above e t_INT the integer N e t_MAT the factorization fa factor N e t_VEC this is the preferred format and provides both the integer N and its factorization in a two component vector N fal 99 When the group is fixed and many orders or discrete logarithms will be comput
187. Finite field element type t_COMPLEX T i Complex numbers type t_PADIC Qp p adic numbers type t_QUAD Qfw Quadratic Numbers where Z w Z 2 type t_POLMOD T X P Polmods polynomials modulo P T X type t_POL T X Polynomials type t_SER T X Power series finite Laurent series type t_RFRAC TX Rational functions in irreducible form type t_VEC T Row i e horizontal vectors type t_COL T Column i e vertical vectors type t_MAT Mm n T Matrices type t_LIST T Lists type t_STR Character strings type t_CLOSURE Functions type t_ERROR Error messages and where the types T in recursive types can be different in each component The first nine basic types from t_INT to t_POLMOD are called scalar types because they essentially occur as coefficients of other more complicated objects Type t_POLMOD is used to define algebraic extensions of a base ring and as such is a scalar type In addition there exist types t_QFR and t_QFI for integral binary quadratic forms and the in ternal type t_VECSMALL The latter holds vectors of small integers whose absolute value is bounded T by 2 resp 263 on 32 bit resp 64 bit machines They are used internally to represent permu tations polynomials or matrices over a small finite field etc Every PARI object called GEN in the sequel belongs to one of these basic types Let us have a closer look 1 3 1 Integers and reals They are of arbitrary and varying length each n
188. HNF basis of the lattice Z N ImQq 4 matrixqz A 1 ha 8 5 4 3 Lo 1 matrixqz A 2 5 2 1 1 0 o 1 The library syntax is GEN matrixqz0 GEN A GEN p NULL 3 8 42 matsize x x being a vector or matrix returns a row vector with two components the first being the number of rows 1 for a row vector the second the number of columns 1 for a column vector The library syntax is GEN matsize GEN x 242 3 8 43 matsnf X flag 0 If X is a singular or non singular matrix outputs the vector of elementary divisors of X i e the diagonal of the Smith normal form of X normalized so that dn deco ssa di The binary digits of flag mean 1 complete output if set outputs U V D where U and V are two unimodular matrices such that UXV is the diagonal matrix D Otherwise output only the diagonal of D If X is not a square matrix then D will be a square diagonal matrix padded with zeros on the left or the top 2 generic input if set allows polynomial entries in which case the input matrix must be square Otherwise assume that X has integer coefficients with arbitrary shape 4 cleanup if set cleans up the output This means that elementary divisors equal to 1 will be deleted i e outputs a shortened vector D instead of D If complete output was required returns U V D so that U XV D holds If this flag is set X is allowed to be of the form vector of elementary d
189. If P is a polynomial in this list a is any root of P and K Q a then a is the sum of a uniformizer and a lift of a generator of the residue field of K in particular the powers of a generate the ring of p adic integers of K If flag 1 replace each polynomial P by a vector P e f d c where e is the ramification index f the residual degree d the valuation of the discriminant and c the number of conjugate fields If flag 2 only return the number of extensions in a fixed algebraic closure Krasner s formula which is much faster The library syntax is GEN padicfieldsO GEN p GEN N long flag Also available is GEN padicfields GEN p long n long d long flag which computes extensions of Q of degree n and discriminant pl 3 7 10 polchebyshev n flag 1 a x Returns the nt Chebyshev polynomial of the first kind Tn flag 1 or the second kind U flag 2 evaluated at a x by default Both series of polynomials satisfy the 3 term relation Pr 1 22Pn Pn 1 and are determined by the initial conditions Uy To 1 T x U 2a In fact T nU _1 and for all complex numbers z we have T cos z cos nz and U 1 cos z sin nz sin z If n gt 0 then these polynomials have degree n For n lt 0 T is equal to T n and U is equal to U_9_y In particular U_ 0 The library syntax is GEN polchebyshev_eval long n long flag GEN a NULL Also available are GEN polchebyshev lon
190. If s is a scalar this gives a constant power series in v with precision d If s is a polynomial the polynomial is truncated to d terms if needed Ser 1 7 5 1 1 O y75 Ser x 2 5 2 x 2 O x 7 T polcyclo 100 13 x 40 x 30 x 20 x710 1 Ser T x 11 4 1 x710 O x711 The function is more or less equivalent with multiplication by 1 O v in theses cases only faster If s is a vector on the other hand the coefficients of the vector are understood to be the coefficients of the power series starting from the constant term as in Polrev x and the precision d is ignored in other words in this case we convert t_VEC t_COL to the power series whose 74 significant terms are exactly given by the vector entries Finally if s is already a power series in v we return it verbatim ignoring d again If d significant terms are desired in the last two cases convert truncate to t_POL first v 1 2 3 Ser v t 7 5 1 2xt 3 t 2 0 t73 3 terms 7 is ignored Ser Polrev v t t 7 6 1 2 t 3 xt72 0 t77 s 1 x 0 x 2 Ser s x 7 17 1 x 0 x72 2 terms 7 ignored Ser truncate s x 7 8 1 x 0 x 7 The warning given for Pol also applies here this is not a substitution function The library syntax is GEN gtoser GEN s long v 1 long precdl where vis a variable number 3 2 10 Set x Converts x into a set i e into a row vect
191. If you are using GMP tune it first then PARI Make sure you tune PARI on the machine that will actually run your computations Do not use a heavily loaded machine for tunings You may speed up the compilation by using a parallel make env MAKE make j4 Configure tune graphic lib enables a particular graphic library The default is X11 on most platforms but PARI can use Qt f1tk ps or win32 GDI time function chooses a timing function The default usually works fine however you can use a different one that better fits your needs PARI can use getrusage clock_gettime times or 316 ftime as timing functions Not all timing functions are available on all platforms The three first functions give timings in terms of CPU usage of the current task approximating the complexity of the algorithm The last one ftime gives timings in terms of absolute wall clock time Moreover the clock_gettime function is more precise but much slower at the time of this writing than getrusage or times The remaining options are specific to parallel programming We provide an Introduction to parallel GP programming in the file doc parallel dvi and to multi threaded libpari programs in Appendix D Beware that these options change the library ABI mt engine specify the engine used for parallel computations Supported value are e single default no parallellism e pthread use POSIX threads This is well suited for multi core systems S
192. L long prec 3 6 32 dirzetak nf b Gives as a vector the first b coefficients of the Dedekind zeta function of the number field nf considered as a Dirichlet series The library syntax is GEN dirzetak GEN nf GEN b 3 6 33 factornf x t Factorization of the univariate polynomial x over the number field defined by the univariate polynomial t x may have coefficients in Q or in the number field The algorithm reduces to factorization over Q Trager s trick The direct approach of nffactor which uses van Hoeij s method in a relative setting is in general faster The main variable of t must be of lower priority than that of x see Section 2 5 3 However if non rational number field elements occur as polmods or polynomials as coefficients of x the variable of these polmods must be the same as the main variable of t For example factornf x 2 Mod y y 2 1 y 2 1 factornf x 2 y y 2 1 these two are OK factornf x 2 Mod z z 2 1 y72 1 kk at top level factornf x 2 Mod z z kK xxx factornf inconsistent data in rnf function factornf x 2 z y 2 1 eK at top level factornf x 2 z y 2 1 WRK Tee factornf incorrect variable in rnf function The library syntax is GEN polfnf GEN x GEN t 3 6 34 galoisexport gal flag gal being be a Galois group as output by galoisinit export the underlying permutation group as a string suitable for no flags or flag 0 GAP or flag 1 M
193. N amp GEN L long s char n variable e p to supply realprecision usually long prec in the argument list P to supply series precision usually long precdl We also have special constructs for optional arguments and default values e DG optional GEN NULL if omitted e D amp optional GEN NULL if omitted e Dn optional variable 1 if omitted For instance the prototype corresponding to long issquareall GEN x GEN n NULL is 1GD amp 293 Caution This function may not work on all systems especially when gp has been compiled statically In that case the first use of an installed function will provoke a Segmentation Fault this should never happen with a dynamically linked executable If you intend to use this function please check first on some harmless example such as the one above that it works properly on your machine The library syntax is void gpinstall const char name const char code const char gpname const char lib 3 12 22 kill sym Restores the symbol sym to its undefined status and deletes any help mes sages associated to sym using addhelp Variable names remain known to the interpreter and keep their former priority you cannot make a variable less important by killing it Pz y 1 y f1 1 kill y y restored to undefined status 12 y variable 43 x y z but the variable name y is still known with y gt z For the sa
194. N D computes the class number of an imaginary quadratic field by counting reduced forms an O D algorithm 3 4 62 qfbcompraw x y composition of the binary quadratic forms x and y without reduction of the result This is useful e g to compute a generating element of an ideal The result is undefined if x and y do not have the same discriminant The library syntax is GEN qfbcompraw GEN x GEN y 3 4 63 qfbhclassno x Hurwitz class number of x where x is non negative and congruent to 0 or 3 modulo 4 For 2 gt 5 10 we assume the GRH and use quadclassunit with default parameters The library syntax is GEN hclassno GEN x 122 3 4 64 qfbnucomp z y L composition of the primitive positive definite binary quadratic forms x and y type t_QFI using the NUCOMP and NUDUPL algorithms of Shanks a la Atkin L is any positive constant but for optimal speed one should take L D where D is the common discriminant of x and y When x and y do not have the same discriminant the result is undefined The current implementation is straightforward and in general slower than the generic routine since the latter takes advantage of asymptotically fast operations and careful optimizations The library syntax is GEN nucomp GEN x GEN y GEN L Also available is GEN nudupl GEN x GEN L when z y 3 4 65 qfbnupow x n n th power of the primitive positive definite binary quadratic form x using Shanks s NUCOMP and NUDUPL algorith
195. NT amp amp isprime p does not evaluate isprime p if p is not an integer e Priority 2 assignment lvalue expr The result of x y is the value of the expression y which is also assigned to the variable x This assignment operator is right associative This is not the equality test operator a statement like x 1 is always true i e non zero and sets x to 1 the equality test would be x 1 The right hand side of the assignment operator is evaluated before the left hand side It is crucial that the left hand side be an lvalue there it avoids ambiguities in expressions like 1 x 1 The latter evaluates as 1 x 1 not as 1 x 1 even though the priority of is lower than the priority of 1 x is not an lvalue If the expression cannot be parsed in a way where the left hand side is an lvalue raise an error x 1 1 Hook unused characters x 1 1 a kkk op where op is any binary operator among V lt lt or gt gt composed assignment lvalue op expr The expression x op y assigns x op y to x and returns the new value of x The result is not an lvalue thus x 2 3 is invalid These assignment operators are right associative X Xx x t x 2 1 3x x e Priority 1 gt function definition vars gt expr returns a function object of type t_CLOSURE Remark Use the op operators as often as possible since they make complex assignments more legible one
196. OBSOLETE use complex approximations to the roots and an integral LLL The result is not guaranteed to be complete some conjugates may be missing a warning is issued if the result is not proved complete especially so if the corresponding polynomial has a huge index and increasing the default precision may help This variant is slow and unreliable don t use it If flag 4 use galoisinit very fast but only applies to most Galois fields If the field is Galois with weakly super solvable Galois group see galoisinit return the complete list of automorphisms else only the identity element If present d is assumed to be a multiple of the least common denominator of the conjugates expressed as polynomial in a root of pol This routine can only compute Q automorphisms but it may be used to get K automorphism for any base field K as follows rnfgaloisconj nfK R K automorphisms of L K X R my polabs N R Mod 1 nfK pol convert coeffs to polmod elts of K polabs rnfequation nfK R N nfgaloisconj polabs R Q automorphisms of L select the ones that fix K select s gt subst R variable R Mod s R 0 N K nfinit y 2 7 rnfgaloisconj K x 4 y x 3 3 x 2 y x 1 AN K automorphisms of L The library syntax is GEN galoisconjO GEN nf long flag GEN d NULL long prec Use directly GEN galoisconj GEN nf GEN d corresponding to flag 0 the others only have historical interest
197. P where listP is as in nfbasis a vector of pairwise coprime integers usually distinct primes a factorization matrix or a single integer In that case the function returns the discriminant of an order whose basis is given by nfbasis T listP which need not be the maximal order and whose valuation at a prime entry in listP is the same as the valuation of the field discriminant In particular if 1istP is p for a prime p we can return the p adic discriminant of the maximal order of Z X T as a power of p as follows padicdisc T p p valuation nfdisc T p p nfdisc x 2 6 1 24 padicdisc x 2 6 2 12 8 padicdisc x 2 6 3 187 13 3 The library syntax is nfdisc GEN T listP NULL Also available is GEN nfbasis GEN T GEN d GEN listP NULL which returns the order basis and where d receives the order discriminant 3 6 82 nfeltadd nf x y Given two elements x and y in nf computes their sum x y in the number field nf The library syntax is GEN nfadd GEN nf GEN x GEN y 3 6 83 nfeltdiv nf x y Given two elements x and y in nf computes their quotient x y in the number field nf The library syntax is GEN nfdiv GEN nf GEN x GEN y 3 6 84 nfeltdiveuc nf x y Given two elements x and y in nf computes an algebraic integer q in the number field nf such that the components of x qy are reasonably small In fact this is functionally identical to round nfdiv nf x y
198. Print the address of the argument in hexadecimal as if by x e 7 A is written No argument is converted The complete conversion specification is Examples printf floor d field width 3 3d with sign 3d n Pi 1 2 floor 3 field width 3 1 with sign 2 printf 5g 5g 5g n 123 123 456 123456789 123 00 0 26974 1 2346 e8 printf 2 58 2 58 42 5s n P PARI PARIGP P PARI PARIG min field width and precision given by arguments x 23 y 1 x printf x 06 2f y t 0 f n x 6 2 y x 23 00 y 00 04 minimum fields width 5 pad left with zeroes for i 2 5 printf 05d n 10 1 00100 01000 10000 100000 don t truncate fields whose length is larger than the minimum width printf 2f 06 2f Pi Pi 296 3 14 3 14 All numerical conversions apply recursively to the entries of vectors and matrices print 4d 1 2 31 1 2 3 printf 5 2f mathilbert 3 1 00 0 50 0 33 0 50 0 33 0 25 0 33 0 25 0 20 Technical note Our implementation of printf deviates from the C89 and C99 standards in a few places e whenever a precision is missing the current realprecision is used to determine the number of printed digits C89 use 6 decimals after the radix character e in conversion style e we do not impose that the exponent has at least two digits we never write a sign in the exponent 0 is printed in a specia
199. RI GP Version 2 7 0 year 2012 address Bordeaux note available from tt http pari math u bordeaux fr 323 In any case if you like this software we would be indebted if you could send us an email message giving us some information about yourself and what you use PARI for Good luck and enjoy 324 Index Some Word refers to PARI GP concepts SomeWord is a PARI GP keyword SomeWord is a generic index entry A la a gos eases Gogo Be Se ee a 130 Qe 5 sek A dhe eee a GTS ace eee nae 130 AS Sen ae ede at ee Ga aes oe erie ae e 130 Ge bate tn a ase wider se GTA dies 130 AO Se k eae a Ge E ae A er ee 130 Abelian extension 203 211 ODS 2 8 feu a bh sh ae oO a 89 ACGUTAC 20 6 alk ek 6 Ee ae a eS Kos 9 AOS su es a A ee oe a E 90 COSH 4 5 a he a a EO eS em ES 90 addhelp es o c coe aa Re ae 47 287 addprimes 100 110 186 194 195 199 306 adj 2a Gee ea eee ba ek wane a we 233 adjoint matrix 233 Bd Sal O gre he ee a Se a ae he ee F 233 AGM 2 Baie gori dw Soe el we de ek 90 akell e gos oss ae Gi a o ee BS A 133 alarm js 36 5 a Seth SO ee BS 286 287 algdep 23 5 eee ea sos 228 229 algdepO 224446 bese Fae eG es 229 algebraic dependence 228 251 algebraic number 02044 igl algtobasis s1 sp ane i oe Be a 184 alias esca Bel ak ey Ad ete Gg 47 288 alias iria aoe bh we Bee Gee i 289 allocatemem 289 297 309 alternating series
200. Returns normalized periods W1 W2 generating the same lattice such that T W W2 has positive imaginary part and lies in the standard fundamental domain for SL2 Z If flag 1 the function returns W1 W2 71 72 where 7 and 72 are the quasi periods associated to W 1 W2 satisfying 171 Wa n2W 2ir The output of this function is meant to be used as the first argument given to ellwp ellzeta ellsigma or elleisnum Quasi periods are needed by ellzeta and ellsigma only The library syntax is GEN ellperiods GEN w long flag long prec 3 5 38 ellpointtoz E P If E C C A is a complex elliptic curve A E omega computes a complex number z well defined modulo the lattice A corresponding to the point P i e such that P pa z pr z satisfies the equation y 42 gox 9s where g2 g3 are the elliptic invariants If E is defined over R and P E R we have more precisely 0 lt R t lt w1 and 0 lt S t lt 3 w2 where w1 w2 are the real and complex periods of E E ellinit 0 1 P 2 3 z ellpointtoz E P 2 3 5054552633136356529375476976257353387 ellwp E z 3 2 0000000000000000000000000000000000000 ellztopoint E z P 4 6 372367644529809109 E 58 7 646841173435770930 E 57 ellpointtoz E 0 the point at infinity 45 0 If E Q has multiplicative reduction then E Q is analytically isomorphic to Q q Tate curve for some p adic intege
201. S 229 Catalan isda on bbe BOR ee Ea ee 89 Cl Bae eae eee eA Bee eS i 78 centerlift 78 81 character string 26 CROTAC N es madera ee a 152 character 162 164 166 characteristic polynomial 229 characteristic 78 CHAarpoOly ses tee oros 229 charpolyO sise sra 34 4 ee ee a4 229 Chebyshev is acs ice as a e ew aii 219 CHINESE foc wea is a hs ade ee i a 102 Chinese ula eS SS SS OR Es 102 ClASSNO 254 4 be lt lt ooss osos 122 CLASSMOD epa ee o Ge a 122 CIGD coso oo poe a oea De Re Se 155 GEISP ginas aoe a BOR Be ee Se eS 54 CMdtool su cae be ha ead ee fies 310 CMP ene We de ee Hee a 67 75 233 252 cmp_universal serasi red ta 68 Gode WOrdS 6 4 ceo a g A 78 COQIEE saas ann ewe e a ee a 155 COL E eS Bore ee bE SS 23 70 COLO S 224836 6 Ge eee Oo es 304 COMTE ba ea a we eee Zas TL column vector 004 T23 comparison operators 70 compatible 304 completion e s oo se se na e eer das 60 complex number 7 8 19 COMPO eo an A 79 component s a soora s a neka 78 283 composition r o 122 compositum sooo 197 COMPOST s eseo A ee we 198 compositum2 198 COMPTESS 24 sorre radasta 58 CONCA k pka dee eee ON 45 230 231 CONnCAtd 12 wee acs bad ee BAS ee 231 CON ie Ged Dae a a OS eed 79 CON VCC ur e a a a 79 content 93 10 103 113 COML TAC ceda o a e ee
202. STR the function name s e e _COPRIME Function s expected coprime arguments and did receive x y which were not E has three component 1 t_STR the function name s 2 the argument x 3 the argument y e e INV Tried to invert a non invertible object x in function s E has two components 1 t_STR the function name s 2 the non invertible x If Mod a b is a t_INTMOD and a is not 0 mod b this allows to factor the modulus as gcd a b is a non trivial divisor of b e e_TRREDPOL Function s expected an irreducible polynomial and did receive T which was not As in nfinit x 2 1 E has two component 1 t_STR the function name s 2 t_POL the polynomial x e e MISC Generic uncategorized error E has one component t_STR the error message to print e e MODULUS moduli x and y submitted to function s are inconsistent As in nfalgtobasis nfinit t73 2 Mod t t 2 1 E has three component 1 t_STR the function s 2 the argument x 3 the argument z e e NEGVAL An argument of function s is a power series with negative valuation which does not make sense As in cos 1 x E has one component 1 t_STR the function name s e e PRIME Function s expected a prime number and did receive p which was not As in idealprimedec nf 4 E has two component 1 t_STR the function name s 2 the argument Pp e e ROOTSO An argument of function s is a zero polynomial and we need to consider its r
203. TOFVEC es esa 4 Fat Se ee gee eS a 282 TEAC wee ae oe eee Hane eee ee a 80 free variable 31 Dl era a A 155 fundamental units 125 155 158 PUG Ube 2 ara n AE 155 G Babs osuna a Ta a 90 BaCO seara apear a ee ee ee G 90 gacosH sorg osi eg a e A 90 Gadd spa as a a 64 galdata awis 2 0 a eee eee 198 GALOS e erae RR eee 44 329 Galois 160 190 191 197 198 210 282 galolS apply 2 6 255426 eee ws 191 galoisconj e a s s ota Sb Boe Ba 192 galoisconjO screny senere ss 192 galoisexport 168 169 170 galoisfixedfield 169 282 galoisgetpol 169 galoisidentify 169 170 galoisinit 168 169 170 171 192 galoisisabelian 171 galoisisnormal 171 galoisnbpol 169 galoispermtopol 171 galoissubcyclo 167 171 172 224 282 galoissubfields 168 172 197 galoissubgroups 172 173 GAMMA s oe dat o aa ee ee a S 93 gamma function 93 gammah vs bias bee eee hee EA 94 o co oe oe Gwe Se oh oe eS 90 BASIN eee vane eee ee ee be ees 90 Basinh 2625 ea Pee dia wae ee eas 90 SALADO i ps hee Bet es He elk det a s 90 Gatanh esea nowe apaa a gi CASS orse rooe de ba gate A t 243 gaussmodulo 243 gaussmodulo2 243 pbitand wie wee eae bee oe 77 EDITION a e q eae Bae ee ee ae TF gbitnegimply Val EDITOS is adi ede a oa e
204. The library syntax is GEN nfdiveuc GEN nf GEN x GEN y 3 6 85 nfeltdivmodpr nf x y pr Given two elements x and y in nf and pr a prime ideal in modpr format see nfmodprinit computes their quotient y modulo the prime ideal pr The library syntax is GEN nfdivmodpr GEN nf GEN x GEN y GEN pr This function is normally useless in library mode Project your inputs to the residue field using nf_to_Fq then work there 3 6 86 nfeltdivrem nf x y Given two elements x and y in nf gives a two element row vector q r such that x qy r q is an algebraic integer in nf and the components of r are reasonably small The library syntax is GEN nfdivrem GEN nf GEN x GEN y 3 6 87 nfeltmod nf x y Given two elements x and y in nf computes an element r of nf of the form r x qy with q and algebraic integer and such that r is small This is functionally identical to x nfmul nf round nfdiv nf x y y The library syntax is GEN nfmod GEN nf GEN x GEN y 3 6 88 nfeltmul nf x y Given two elements x and y in nf computes their product x y in the number field nf The library syntax is GEN nfmul GEN nf GEN x GEN y 3 6 89 nfeltmulmodpr nf x y pr Given two elements x and y in nf and pr a prime ideal in modpr format see nfmodprinit computes their product x x y modulo the prime ideal pr The library syntax is GEN nfmulmodpr GEN nf GEN x GEN y GEN pr This function is normally useless in library mode Projec
205. UCI em e a ale es e al a 13 2 2 The penera 2 p IDput lines io a ean Pe A A A iii SY 15 2 3 The PART types ap A RO as AAA ath IS AA ee cha 17 DA GP Operators eco hk wey esd BON SG Bod el ee A A a 27 2 5 Variables and symbolic expressions 2 2 ee a 31 2 6 Variables and Scope aoaaa ee 34 24 User defined functions epica oe ee eS ee ie A ew Rae ned dos Ds 37 2 8 Member functions lt sa sa aioe ee 44 2 9 Strings and Keywords i eoi dasa Se ee ck aa ee a 45 2 10 Errors and error recovery 6 0 a 47 2 11 Interfacing GP with other languages e 54 2412 Defaults late hae a a a AI A eel 54 2 13 Simple metacommands aa ee 55 2 14 The preferences M m sn aiar uaua a E atat aa a T A aua ORE aA aala a a o A n s 59 2 15 Using readline s ciudad kor BE a ido bce EE acah iio cd 61 2 16 GNU Emacs and PariEmacs a 62 Chapter 3 Functions and Operations Available in PARI and GP 63 3 1 Standard monadic or dyadic operators ee ee 65 3 2 Conversions and similar elementary functions or commands 71 3 3 Transcendental finctions s 26s os asd ere i ee aaa 88 3 4 Arithmetic functions ooa 99 3 5 Functions related to elliptic curves a 130 3 6 Functions related to general number fields ooa a a 151 3 7 Polynomials and power series 2 ee 215 3 8 Vectors matrices linear algebra and sets o oo o a a a a 228 3 9 Sums products integrals and similar functions
206. URE f select returns the elements x of A such that f x is non zero In other words f is seen as a selection function returning a boolean value select x gt isprime x vector 50 i i72 1 1 2 5 17 37 101 197 257 401 577 677 1297 1601 select x gt x lt 100 2 2 5 17 371 returns the primes of the form i 1 for some i lt 50 then the elements less than 100 in the preceding result The select function also applies to a matrix A seen as a vector of columns i e it selects columns instead of entries and returns the matrix whose columns are the selected ones 298 Remark For v a t_VEC t_COL t_LIST or t_MAT the alternative set notations g x x lt v f x x x lt v f x g x x lt v are available as shortcuts for apply g select f Vec v select f Vec v apply g Vec v respectively x x lt vector 50 i i 2 1 isprime x 1 2 5 17 37 101 197 257 401 577 677 1297 1601 If flag 1 this function returns instead the indices of the selected elements and not the elements themselves indirect selection V vector 50 i i172 1 select x gt isprime x V 1 12 Vecsmal1 1 2 4 6 10 14 16 20 24 26 36 40 vecextract V 13 2 5 17 37 101 197 257 401 577 677 1297 1601 The following function lists the elements in Z NZ invertibles N select x gt gcd x N 1 1 N Finally select x gt
207. User s Guide to PARI GP version 2 7 0 The PARI Group Institut de Math matiques de Bordeaux UMR 5251 du CNRS Universit Bordeaux 1 351 Cours de la Lib ration F 33405 TALENCE Cedex FRANCE e mail pari math u bordeaux fr Home Page http pari math u bordeaux fr Copyright 2000 2014 The PARI Group Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies Permission is granted to copy and distribute modified versions or translations of this manual under the conditions for verbatim copying provided also that the entire resulting derived work is distributed under the terms of a permission notice identical to this one PARI GP is Copyright 2000 2014 The PARI Group PARI GP is free software you can redistribute it and or modify it under the terms of the GNU General Public License as published by the Free Software Foundation It is distributed in the hope that it will be useful but WITHOUT ANY WARRANTY WHATSOEVER Table of Contents Chapter 1 Overview of the PARI systeM o eee eee 5 Ldelittrod ction rt AA A AA A e dd 5 1 2 Multiprecision kernels Portability Layos ar e A a ade pra 6 13 The PART types ta a A A A A A aS 7 TA The PART philosophy cita E AOS A as 9 1 5 Operations and functions a 10 Chapter 2 The gp Calculator o eee ee een ee 13 Dil tro
208. We do not use exactly but a rescaled rounded variant which gets us faster and simpler LLLs There s no harm since we are not using any theoretical property of a after all except that it belongs to I and is expected to be small The library syntax is GEN idealredO GEN nf GEN I GEN v NULL 182 3 6 69 idealstar nf J flag 1 Outputs a bid structure necessary for computing in the finite abelian group G Zx I Here nf is a number field and J is a modulus either an ideal in any form or a row vector whose first component is an ideal and whose second component is a row vector of r 0 or 1 Ideals can also be given by a factorization into prime ideals as produced by idealfactor This bid is used in ideallog to compute discrete logarithms It also contains useful information which can be conveniently retrieved as bid mod the modulus bid clgp G as a finite abelian group bid no the cardinality of G bid cyc elementary divisors and bid gen generators If flag 1 default the result is a bid structure without generators If flag 2 as flag 1 but including generators which wastes some time If flag 0 only outputs Zx 1 as an abelian group i e as a 3 component vector h d g h is the order d is the vector of SNF cyclic components and g the corresponding generators The library syntax is GEN idealstarO GEN nf GEN I long flag Instead the above hard coded numerical flags one should rather us
209. X valuation 86 van Hoeij Bee ee ee A 107 168 Xadic_lindep Soe fee cep fe te Sa ey eine ee a 232 variable priority 20 32 zia ooo 78 WHable SCOPES e Visa e avisas 34 xima ooo 78 Variable cepo le a 20 31 Xlmlooooooooo ooo 78 variable s d s a whe A ae 4 32 87 xm 78 Vel 2b ea aa den a 23 26 76 Z vecbinome 102 VECS DCL se sa sedata ate g a puk i 92 Zassenhaus o 109 218 vecextract 240 253 ZOrent 2 din vg tess sita y G 267 VECMA 244 24 53 ee Ba ee Ta 69 70 Zell eses E A ee ee as 146 vecmax0 como mo he ee ee 70 NR 9 VECMIN cie a ha a Be ae a 70 ZOYTOPadio pes a eee eS eee aS 215 VECINO oia ae die 4 Bw et OO 70 ZOO seen we BESS Se oe eS 215 340 Zea ceo A oe A ee A 98 Zeta ula ss Got eae ere A tee Se ee 214 Zetakinit peris ee Re EH gds 215 Zk A RA ed oe Sore anes LH i 155 ZKS e E abr ce ae eral eee 155 ZMidet cessa bebe ae m enw ae a 235 ZM ga s8 s o oc coa os ee ee ae ee 243 zncoppersmith 127 128 znlog 111 128 129 142 179 ZNOVAGSL iio ae ve eB Sh a a we csc 129 ZOPrimro0t gt 26g ee ewe es trad 129 ZNSCAN oror rona a ee ae ee 129 LPA Shoe a Meh ke de ee Se oe 219 341
210. _MAT the alternative set notations g x x lt v x x lt v f x g x x lt v are available as shortcuts for apply g select f Vec v select f Vec v apply g Vec v respectively L List Mod 1 3 Mod 2 4 lift x x lt L 12 1 2 The library syntax is genapply void E GEN fun void GEN GEN a 3 12 7 default key val Returns the default corresponding to keyword key If val is present sets the default to val first which is subject to string expansion first Typing default or Md yields the complete default list as well as their current values See Section 2 12 for an introduction to GP defaults Section 3 14 for a list of available defaults and Section 2 13 for some shortcut alter natives Note that the shortcuts are meant for interactive use and usually display more information than default The library syntax is GEN default0 const char key NULL const char val NULL 3 12 8 errname Returns the type of the error message E as a string The library syntax is GEN errname GEN E 3 12 9 error str x Outputs its argument list each of them interpreted as a string then interrupts the running gp program returning to the input prompt For instance error n n is not squarefree 3 12 10 extern str The string str is the name of an external command i e one you would type from your UNIX shell prompt This command is immediately run an
211. a and the generator chosen for the absolute extension is not the same as for the relative one If this happens one can of course go on working but we advise to change the relative polynomial so that its root becomes 8 ka Typical GP instructions would be P a k rnfequation K pol 1 if k pol subst pol x x k Mod y K pol L rnfinit K pol rnf 12 is by default unused and set equal to 0 This field is used to store further information about the field as it becomes available which is rarely needed hence would be too expensive to compute during the initial rnfinit call The library syntax is GEN rnfinit GEN nf GEN pol 3 6 150 rnfisabelian nf T T being a relative polynomial with coefficients in nf return 1 if it defines an abelian extension and 0 otherwise K nfinit y 2 23 rnfisabelian K x73 3 x y 12 1 The library syntax is long rnfisabelian GEN nf GEN T 3 6 151 rnfisfree bnf x Given bnf as output by bnfinit and either a polynomial x with coefficients in bnf defining a relative extension L of bnf or a pseudo basis x of such an extension returns true 1 if L bnf is free false 0 if not The library syntax is long rnfisfree GEN bnf GEN x 3 6 152 rnfisnorm 7 a flag 0 Similar to bnfisnorm but in the relative case T is as output by rnfisnorminit applied to the extension L K This tries to decide whether the element a in K is the norm of some z in the extension L K T
212. a function having no mandatory argument you can even omit the a standalone foo would be enough though we do not recommend it for your scripts for the sake of clarity In defining GP syntax we strove to put optional arguments at the end of the argument list of course since they would not make sense otherwise and in order of decreasing usefulness so that most of the time you will be able to ignore them Finally an optional argument between braces followed by a star like x x means that any number of such arguments possibly none can be given This is in particular used by the various print routines Flags A flag is an argument which rather than conveying actual information to the routine instructs it to change its default behavior e g return more or less information All such flags are optional and will be called flag in the function descriptions to follow There are two different kind of flags e generic all valid values for the flag are individually described If flag is equal to 1 then e binary use customary binary notation as a compact way to represent many toggles with just one integer Let po pn be a list of switches i e of properties which take either the value 0 or 1 the number 2 2 40 means that p3 and ps are set that is set to 1 and none of the others are that is they are set to 0 This is announced as The binary digits of flag mean 1 po 2 pi 4 p2 and so on using th
213. a kill function allowing to reuse symbols This is no longer the case There is now no distinction between variable and function names we have PARI objects functions of type t_CLOSURE or more mundane mathematical entities like t_INT etc and variables bound to them There is nothing wrong with the following sequence of assignments f 1 assigns the integer 1 to f fa 1 210 1 a function with a constant value 12 O gt 1 1 x2 now holds a polynomial 13 x72 f x x72 and now a polynomial function 14 x gt x 2 g fun fun Pi a function taking a function as argument g cos 6 1 000000000000000000000000000 Previously used names can be recycled as above you are just redefining the variable The previous definition is lost of course Important technical note Built in functions are a special case since they are read only you cannot overwrite their default meaning and they use features not available to user functions in particular pointer arguments In the present version 2 7 0 it is possible to assign a built in function to a variable or to use a built in function name to create an anonymous function but some special argument combinations may not be available issquare 9 te si 1 e 42 3 g issquare g 9 14 1 7 g 9 e pointers are not implemented for user functions xxx unexpected g 9 ke xk Ree 2 7 2 Function call Default arguments You may no
214. a priori no longer canonical and it may happen that it does not have minimal T norm The routine attempts to certify the result independently of this order computation as per nfcertify we try to prove that the order is maximal if it fails the routine returns 0 instead of P In order to force an output in that case as well you may either use polredbest or polredabs 16 or polredabs T nfbasis T listP In all three cases the result is no longer canonical Warning 2 Apart from the factorization of the discriminant of T this routine runs in polynomial time for a fixed degree But the complexity is exponential in the degree this routine may be exceedingly slow when the number field has many subfields hence a lot of elements of small T gt norm If you do not need a canonical polynomial the function polredbest is in general much faster it runs in polynomial time and tends to return polynomials with smaller discriminants The binary digits of flag mean 1 outputs a two component row vector P a where P is the default output and Mod a P is a root of the original T 4 gives all polynomials of minimal T norm of the two polynomials P x and P lt only one is given 16 Possibly use a suborder of the maximal order without attempting to certify the result as in Warning 1 we always return a polynomial and never 0 The result is a priori not canonical T x 16 136xx 14 6476 x712 141912xx 10 15133
215. a synonym for it is the assignment statement The standard boolean operators inclusive or amp amp and and not are also available 70 3 2 Conversions and similar elementary functions or commands Many of the conversion functions are rounding or truncating operations In this case if the argu ment is a rational function the result is the Euclidean quotient of the numerator by the denomi nator and if the argument is a vector or a matrix the operation is done componentwise This will not be restated for every function 3 2 1 Col x n Transforms the object x into a column vector The dimension of the resulting vector can be optionally specified via the extra parameter n If n is omitted or 0 the dimension depends on the type of x the vector has a single component except when z is e a vector or a quadratic form in which case the resulting vector is simply the initial object considered as a row vector e a polynomial or a power series In the case of a polynomial the coefficients of the vector start with the leading coefficient of the polynomial while for power series only the significant coefficients are taken into account but this time by increasing order of degree In this last case Vec is the reciprocal function of Pol and Ser respectively e a matrix the column of row vector comprising the matrix is returned e a character string a vector of individual characters is returned In the last two case
216. ack so temporary objects created during the current expression evaluation are not correct anymore In particular byte compiled expressions which are allocated on the stack To avoid accessing obsolete pointers to the old stack this routine ends by a longjmp Remark If the operating system cannot allocate the desired x bytes a loop halves the allocation size until it succeeds allocatemem 5 10710 Warning not enough memory new stack 50000000000 Warning not enough memory new stack 25000000000 Warning not enough memory new stack 12500000000 kk Warning new stack size 6250000000 5960 464 Mbytes 3 12 6 apply f A Apply the t_CLOSURE f to the entries of A If A is a scalar return f A If A is a polynomial or power series apply f on all coefficients If A is a vector or list return the elements f x where x runs through A If A is a matrix return the matrix whose entries are the f A i j apply x gt x 2 1 2 3 4 1 1 4 9 16 apply x gt x 2 1 2 3 4 12 1 4 9 16 apply x gt x7 2 4 x 2 3 x 2 13 16 x 2 9 x 4 Note that many functions already act componentwise on vectors or matrices but they almost never act on lists in this case apply is a good solution L List Mod 1 3 Mod 2 4 1ift L k at top level lift L KK PSSS lift incorrect type in lift apply lift L 12 List 1 2 290 Remark For v a t_VEC t_COL t_LIST or t
217. agma The following example compute the index of the underlying abstract group in the GAP library G galoisinit x 6 108 s galoisexport G 12 Group 1 2 3 4 5 6 1 4 2 6 3 5 extern echo IdGroup s gap q 43 6 1 galoisidentify G 4 6 1 This command also accepts subgroups returned by galoissubgroups To import a GAP permutation into gp for galoissubfields for instance the following GAP function may be useful PermToGP function p n return Permuted 1 n p end 168 gap gt p 1 26 2 5 3 17 4 32 6 9 7 11 8 24 10 13 12 15 14 27 16 22 18 28 19 20 21 29 23 31 25 30 gap gt PermToGP p 32 26 5 17 32 2 9 11 24 6 13 7 15 10 27 12 22 3 28 20 19 29 16 31 8 30 1 14 18 21 25 23 4 The library syntax is GEN galoisexport GEN gal long flag 3 6 35 galoisfixedfield gal perm flag v y gal being be a Galois group as output by galoisinit and perm an element of gal group a vector of such elements or a subgroup of gal as returned by galoissubgroups computes the fixed field of gal by the automorphism defined by the permutations perm of the roots gal roots P is guaranteed to be squarefree modulo gal p If no flags or flag 0 output format is the same as for nfsubfield returning P x such that P is a polynomial defining the fixed field and x is a root of P expressed as a polmod in gal pol If flag
218. aid to be weakly super solvable if there exists a normal series 1 Ho lt 4 H lt lt Hn lt 9 Ay such that each H is normal in G and for i lt n each quotient group H H is cyclic and either H G then G is super solvable or G H is isomorphic to either A4 or S4 In practice almost all small groups are WKSS the exceptions having order 36 1 exception 48 2 56 1 60 1 72 5 75 1 80 1 96 10 and gt 108 This function is a prerequisite for most of the galoisxxzx routines For instance P x76 108 G galoisinit P L galoissubgroups G vector L i galoisisabelian L i 1 vector L i galoisidentify L i The output is an 8 component vector gal gal 1 contains the polynomial pol gal pol gal 2 is a three components vector p e q where p is a prime number gal p such that pol totally split modulo p e is an integer and q p gal mod is the modulus of the roots in gal roots gal 3 is a vector L containing the p adic roots of pol as integers implicitly modulo gal mod gal roots gal 4 is the inverse of the Vandermonde matrix of the p adic roots of pol multiplied by gal 5 gal 5 is a multiple of the least common denominator of the automorphisms expressed as polynomial in a root of pol 170 gal 6 is the Galois group G expressed as a vector of permutations of L gal group gal 7 is a generating subset S s1 5 of G expressed as a vect
219. alt with by a preprocessor e comments are removed This applies to all text surrounded by as well as to everything following on a given line e lines starting with if boolean are treated as comments if boolean evaluates to false and read normally otherwise The condition can be negated using either if not or if If the rest of the current line is empty the test applies to the next line same behavior as under gp Only three tests can be performed EMACS true if gp is running in an Emacs or TeXmacs shell see Section 2 16 READL true if gp is compiled with readline support see Section 2 15 VERSION op number where op is in the set gt lt lt gt and number is a PARI version number of the form Major Minor patch where the last two components can be omitted i e 1 is understood as version 1 0 0 This is true if gp s version number satisfies the required inequality 2 14 1 2 Commands After preprocessing the remaining lines are executed as sequence of ex pressions as usual separated by if necessary Only two kinds of expressions are recognized e default value where default is one of the available defaults see Section 2 12 which will be set to value on actual startup Don t forget the quotes around strings e g for prompt or help e read some_GP file where some_GP_file is a regular GP script this time which will be read just before gp prompts you for commands but after initializi
220. alue of p cannot be modified within seq forprime p 2 10 p kk at top level forprime p 2 10 p kK mses prime index read only was changed to 281 3 11 13 forstep X a b s seq Evaluates seq where the formal variable X goes from a to b in increments of s Nothing is done if s gt 0 and a gt b or if s lt 0 and a lt b s must be in R or a vector of steps s1 5n In the latter case the successive steps are used in the order they appear in s forstep x 5 20 2 4 print x 5 7 11 13 17 19 3 11 14 forsubgroup H G bound seq Evaluates seg for each subgroup H of the abelian group G given in SNF form or as a vector of elementary divisors If bound is present and is a positive integer restrict the output to subgroups of index less than bound If bound is a vector containing a single positive integer B then only subgroups of index exactly equal to B are computed The subgroups are not ordered in any obvious way unless G is a p group in which case Birkhoff s algorithm produces them by decreasing index A subgroup is given as a matrix whose columns give its generators on the implicit generators of G For example the following prints all subgroups of index less than 2 in G Z 2Zg1 x Z 2Z qo G 2 2 forsubgroup H G 2 print H 1 1 1 2 2 1 1 O 1 1 The last one for instance is generated by 91 91 g2 This routine is intended to treat huge groups whe
221. am could not compute it When p was not specified V is the vector of all V for all considered p Notes about Namikawa Ueno types e A lower index is denoted between braces for instance I 2 II 5 means I_2 II 5 e If K and K are Kodaira symbols for singular fibers of elliptic curves K K m and K K m are the same e K K 1 is K K a in the notation of Namikawa Ueno e The figure 2I1_0 m in Namikawa Ueno page 159 must be denoted by 21_0 m 1 The library syntax is GEN genus2red GEN Q GEN P GEN p NULL 3 6 Functions related to general number fields In this section can be found functions which are used almost exclusively for working in general number fields Other less specific functions can be found in the next section on polynomials Functions related to quadratic number fields are found in section Section 3 4 Arithmetic functions 3 6 1 Number field structures Let K Q X T a number field Zg its ring of integers T Z X is monic Three basic number field structures can be associated to K in GP e nf denotes a number field i e a data structure output by nfinit This contains the basic arithmetic data associated to the number field signature maximal order given by a basis nf zk discriminant defining polynomial T etc e bnf denotes a Buchmann s number field i e a data structure output by bnfinit This contains nf and the deeper invariants of the field units U K class gr
222. ame is omitted uses name If lib is omitted all symbols known to gp are available this includes the whole of libpari so and possibly others such as libc so Most importantly install gives you access to all non static functions defined in the PARI library For instance the function GEN addii GEN x GEN y adds two PARI integers and is not directly accessible under gp it is eventually called by the operator of course install addii GG addii 1 2 hi 3 It also allows to add external functions to the gp interpreter For instance it makes the function system obsolete install system vs sys omitted sys 1ls gp gp c gp h gp_rl c This works because system is part of libc so which is linked to gp It is also possible to compile a shared library yourself and provide it to gp in this way use gp2c or do it manually see the modules_build variable in pari cfg for hints Re installing a function will print a warning and update the prototype code if needed However it will not reload a symbol from the library even if the latter has been recompiled Prototype We only give a simplified description here covering most functions but there are many more possibilities The full documentation is available in libpari dvi see prototype e First character i 1 v return type int long void Default GEN e One letter for each mandatory argument in the same order as they appear in the argument list G GE
223. amming weight of x Otherwise x must be of type t_POL t_VEC t_COL t_VECSMALL or t_MAT and the function returns the number of non zero coefficients of x hammingweight 15 hi 4 hammingweight x 100 2 x 1 12 3 hammingweight Mod 1 2 2 Mod 0 3 13 2 hammingweight matid 100 14 100 The library syntax is long hammingweight GEN x 3 2 36 imag x Imaginary part of x When zx is a quadratic number this is the coefficient of w in the canonical integral basis 1 w The library syntax is GEN gimag GEN x 3 2 37 length x Length of x x is a shortcut for length x This is mostly useful for e vectors dimension 0 for empty vectors e lists number of entries 0 for empty lists e matrices number of columns e character strings number of actual characters without trailing 0 should you expect it from C char a string hl 8 8 2 1 42 3 80 0 3 0 matrix 2 5 44 5 L List 1 2 3 4 HL 75 4 The routine is in fact defined for arbitrary GP types but is awkward and useless in other cases it returns the number of non code words in x e g the effective length minus 2 for integers since the t_INT type has two code words The library syntax is long glength GEN x 3 2 38 lift x v If v is omitted lifts intmods from Z nZ in Z p adics from Q to Q as truncate and polmods to polynomials Otherwise lifts only polmods whose modulus ha
224. ample the debugger reports at least 3 enclosed frames last innermost is the body of user function f the body of g and the top level global scope In fact the for loop in g s body defines an extra frame since there exist variables scoped to the loop body 2 10 2 Error recovery It is annoying to wait for a program to finish and find out the hard way that there was a mistake in it like the division by 0 above sending you back to the prompt First you may lose some valuable intermediate data Also correcting the error may not be obvious you might have to change your program adding a number of extra statements and tests to narrow down the problem A different situation still related to error recovery is when you actually foresee that some error may occur are unable to prevent it but quite capable of recovering from it given the chance Examples include lazy factorization where you knowingly use a pseudo prime N as if it were prime you may then encounter an impossible situation but this would usually exhibit a factor of N enabling you to refine the factorization and go on Or you might run an expensive computation at low precision to guess the size of the output hence the right precision to use You can then 49 encounter errors like precision loss in truncation e g when trying to convert 1E1000 known to 28 digits of accuracy to an integer or division by 0 e g inverting 0E1000 when all accuracy has been
225. are welcome to improve this help script or write new ones and we would like to know about it so that we may include them in future distributions By the way outside of gp you can give more than one keyword as argument to gphelp 2 13 2 A comment Everything between the stars is ignored by gp These comments can span any number of lines 2 13 3 MM A one line comment The rest of the line is ignored by gp 2 13 4 Na n Prints the object number n n in raw format If the number n is omitted print the latest computed object 2 13 5 Ac Prints the list of all available hardcoded functions under gp not including opera tors written as special symbols see Section 2 4 More information can be obtained using the metacommand see above For user defined functions member functions see u and um 2 13 6 Md Prints the defaults as described in the previous section shortcut for default see Section 3 12 7 2 13 7 Me n Switches the echo mode on 1 or off 0 If n is explicitly given set echo to n 2 13 8 g n Sets the debugging level debug to the non negative integer n 2 13 9 gf n Sets the file usage debugging level debugfiles to the non negative integer n 2 13 10 gm n Sets the memory debugging level debugmem to the non negative integer n 2 13 11 h m n Outputs some debugging info about the hashtable If the argument is a number n outputs the contents of cell n Ranges can be given in the form m n
226. ariables retain their former priority but are set to the new value In particular reading such a session log into a clean session will restore all variables exactly as they were in the original one Just as a regular input file a binary file can be compressed using gzip provided the file name has the standard gz extension In the present implementation the binary files are architecture dependent and compatibility with future versions of gp is not guaranteed Hence binary files should not be used for long term storage also they are larger and harder to compress than text files The library syntax is void gpwritebin const char filename GEN x NULL 3 12 45 writetex filename str As write in T X format 3 13 Parallel programming These function are only available if PARI was configured using Configure mt Two multithread interfaces are supported e POSIX threads e Message passing interface MPI As arule POSIX threads are well suited for single systems while MPI is used by most clusters However the parallel GP interface does not depend on the chosen multithread interface a properly written GP program will work identically with both 302 3 13 1 parapply f x Parallel evaluation of f on the elements of x The function f must not access global variables or variables declared with local and must be free of side effects parapply factor 27256 1 27193 1 factors 2258 1 and 219 1 in parallel
227. ashes with loop indices are likely Note Typing um will output all user defined member functions Member function names A valid name starts with a letter followed by any number of keyword characters _ or alphanumeric characters A Za z0 9 The built in member function names are reserved and cannot be used see the list with Finally names starting with e or E followed by a digit are forbidden due to a clash with the floating point exponent notation we understand 1 e2 as 100 000 not as extracting member e2 of object 1 2 9 Strings and Keywords 2 9 1 Strings GP variables can hold values of type character string internal type t_STR This section describes how they are actually used as well as some convenient tricks automatic concate nation and expansion keywords valid in string context As explained above the general way to input a string is to enclose characters between quotes This is the only input construct where whitespace characters are significant the string will contain the exact number of spaces you typed in Besides you can escape characters by putting a just before them the translation is as follows e lt Escape gt n lt Newline gt t lt Tab gt 45 For any other character x x is expanded to x In particular the only way to put a into a string is to escape it Thus for instance a would produce the string whose content is a This is definitely not the same thing as t
228. associative unless mentioned otherwise An expression is an lvalue if something can be assigned to it The name comes from left value to the left of a operator e g x or v 1 are lvalues but x 1 is not e Priority 14 as in x small is used to indicate to the GP2C compiler that the variable on the left hand side always contains objects of the type specified on the right hand side here a small integer in order to produce more efficient or more readable C code This is ignored by GP e Priority 13 is the function call operator If f is a closure and args is a comma separated list of arguments possibly empty f args evaluates f on those arguments e Priority 12 and unary postfix if x is an lvalue assigns the value x 1 to x then returns the new value of x This corresponds to the C statement x there is no prefix operator in GP x does the same with x 1 These operators are not associative i e x is invalid since x is not an lvalue e Priority 11 member unary postfix member extracts member from structure x see Section 2 8 is the selection operator x i returns the i th component of vector x x i j x j and x i respectively return the entry of coordinates i j the j th column and the i th row of matrix x If the assignment operator immediately follows a sequence of selections it assigns its right hand side to the selected component E g x 1 1 0 is valid
229. at 23 1 2 3 4 5 6 2 3 16 Lists t_LIST Lists can be input directly as in List 1 2 3 4 but in most cases one creates an empty list then appends elements using listput a List listput a 1 listput a 2 Ta 12 List 1 21 Elements can be accessed directly as with the vector types described above 2 3 17 Strings t_STR To enter a string enclose it between double quotes like this this is a string The function Str can be used to transform any object into a string 26 2 3 18 Small vectors t_VECSMALL This is an internal type used to code in an efficient way vectors containing only small integers such as permutations Most gp functions will refuse to operate on these objects 2 3 19 Functions t_CLOSURE We will explain this at length in Section 2 7 For the time being suffice it to say that functions can be assigned to variables as any other object and the following equivalent basic forms are available to create new ones f x y gt x72 y 2 f x y x 2 y 2 2 3 20 Error contexts t_ERROR An object of this type is created whenever an error occurs it contains some information about the error and the error context Usually an appropriate error is printed immediately the computation is aborted and GP enters the break loop 1 0 1 1 xxx at top level 1 0 1 1 k Ai _ _ division by a non invertible object Break loop type break to go back to the GP pro
230. at po ee RS ee 87 OVER cia ee ka Som a OR OE a h 87 ZOA ose caa ei ea ERE Re 2 21 98 Szetak vicario a 215 gzetakall nio hg bee ds e 215 BZIP 665k eG eH ee eS 58 302 H Hadamard product 224 hammingweight 80 127 Hbesselt o a ra gon a d SE cde dera a 91 hbessel2 2 435 a6 hee ee e 91 helassno esos csod ea He sa 122 DEAD e acco s mas RAY ee he i 58 Helps fu eee got ee we Rye 4 hee a 307 Hermite normal form 153 176 192 237 239 Hermite a sose p ehe sor e a pigia 221 BESS im Eee He ae es 237 Hilbert class field 125 Hilbert Matrik sec 4 4 a ae args 237 Hilbert symbol 114 192 hilb 6rt i orri a Heb ed eeud desc 114 histfile sese rean orereta es 307 history lt o cos sa ke PA a ee ea a 47 HIS SIZE poe e sye da ala e 16 307 A e ae e e oe oe 239 hnfadd ise aau ieaie awe OG e a 239 DAEMON ecc eee Swe ee eS So 239 hnfmodid o o 239 Householder transform 239 241 Hurwitz class number 122 hyper 2346282454 86 4 6 94 331 Laa ia ee em ee 19 89 UDESSEL o sos p srona as 91 IDIDADA x bo eor Ake e a Re ee ar ibitnegimply 77 IDICOF soem eG se EO eae bo wee ring D CXOT s o ee RE RS eee e a 78 ideal extended 152 177 179 ideal list o ee 153 ideal s ak eae eae ee AS See es 152 idealadd aoaaa aaa 173 idealaddtoone 173 idealaddtooneO 174 Ide alappE subia e
231. atadjointO 233 Matalgtobasis 183 333 Matbasistoalg 183 Matcompanion 233 Matconcat so poe enese Go 230 233 235 matdet pigura eae eros 235 Matdetint s ss eke eee eee eae 235 matdiagonal 235 Matelgel 2 ee eee Re nra 236 matfrobenius 237 Maths Parl se cias ee A es 54 mathell es sae heey ss sae 140 Mathes8 eae eee eho ede sm 237 Matha LbSrb cosacos ese Bow Bk es wh 237 Mathnf 2355426 44554 2 oe 228 237 MathnfO 2 2 4 o eee ee ee eee at 239 Mathnimod e ema ose ae Gh ec ay E R a 239 mathnfimodid occiso 239 mathouseholder 239 Mabld gais g Ean a ie de e da 239 Matimage 2 o ooo 100 239 matinag eos epe do a Se ee A 239 matimagecompl 240 matindexrank 240 matintersect 240 matinverseimage 240 matisdiagonal 240 Matker soa eee eee ee ee ew 240 matker sooo ee a ee ee ae ee he 241 Matkerint ames 64 2644 4 241 matkerintO 241 matmuldiagonal 241 matmultodiagonal 241 matpascal o 241 matgpascal 241 MAtdL a weeds gos a Bw i a es pS 241 MatTrank wig vig aR Pade Ke He aS 241 MAIX cy vee Sas ee RS es ae 7 8 24 47 M T RZ 24608 de eae ae be wd 24 242 MatYVixqZ o cose cee eee e ee eens 242 MatrixqzZ0 4 234565 e 0e pee 242 MaAtSiZer iia we ae a a a os
232. ate the points e 512 no _X_ticks plot no x ticks e 1024 no_Y_ticks plot no y ticks e 2048 Same _ticks plot all ticks with the same length e 4096 Complex is a parametric plot but where each member of expr is considered a complex number encoding the two coordinates of a point For instance ploth X 0 2 Pi exp I X Complex ploth X 0 2 Pi 1 1 X exp I X Complex will draw respectively a circle and a circle cut by the line y z 3 10 13 plothraw listz listy flag 0 Given liste and listy two vectors of equal length plots in high precision the points whose x y coordinates are given in listx and listy Automatic positioning and scaling is done but with the same scaling factor on x and y If flag is 1 join points other non 0 flags toggle display options and should be combinations of bits 2 k gt 3 as in ploth 3 10 14 plothsizes flag 0 Return data corresponding to the output window in the form of a 6 component vector window width and height sizes for ticks in horizontal and vertical directions this is intended for the gnuplot interface and is currently not significant width and height of characters If flag 0 sizes of ticks and characters are in pixels otherwise are fractions of the screen size 275 3 10 15 plotinit w x y flag 0 Initialize the rectwindow w destroying any rect objects you may have already drawn in w The virtual cursor is set to 0 0 The rectwindow siz
233. ation of the ideals x and y in the number field nf the result is the ideal product in HNF If either x or y are extended ideals their principal part is suitably updated i e multiplying J t J u yields I J tu multiplying J and J u yields I J ul nf nfinit x 2 1 idealmul nf 2 x 1 12 4 2 0 2 idealmul nf 2 x x 1 extended ideal ideal 74 4 2 0 2 x idealmul nf 2 x x 1 x two extended ideals 5 4 2 0 2 1 0 If flag is non zero reduce the result using idealred The library syntax is GEN idealmul0 GEN nf GEN x GEN y long flag See also GEN idealmul GEN nf GEN x GEN y flag 0 and GEN idealmulred GEN nf GEN x GEN y flag 4 0 3 6 62 idealnorm nf x Computes the norm of the ideal x in the number field nf The library syntax is GEN idealnorm GEN nf GEN x 3 6 63 idealnumden nf z Returns A B where A B are coprime integer ideals such that x A B in the number field nf nf nfinit x 2 1 idealnumden nf x 1 2 12 1 0 O 1 2 1 O 1 The library syntax is GEN idealnumden GEN nf GEN x 179 3 6 64 idealpow nf x k flag 0 Computes the k th power of the ideal x in the number field nf k Z If x is an extended ideal its principal part is suitably updated i e raising 7 t to the k th power yields 1 t If flag is non zero reduce the result using idealred throughout the binary powering process
234. atisfies 0 lt r lt y if x and y are polynomials then q and r are polynomials such that degr lt deg y and the leading terms of r and x have the same sign rounded Euclidean quotient for integers rounded towards 00 when the exact quotient would be a half integer lt lt gt gt left and right binary shift By definition x lt lt n 2x2 ifn gt 0 and truncate 12 otherwise Right shift is defined by x gt gt n x lt lt n e Priority 5 addition subtraction e Priority 4 lt gt lt gt the usual comparison operators returning 1 for true and 0 for false For instance x lt 1 returns 1 if x lt 1 and 0 otherwise lt gt test for exact inequality test for exact equality t_QFR having the same coefficients but a different distance component are tested as equal 29 test whether two objects are identical component wise This is stricter than for instance the integer 0 a 0 polynomial or a vector with O entries are all tested equal by but they are not identical e Priority 3 amp amp logical and logical inclusive or Any sequence of logical or and and operations is evaluated from left to right and aborted as soon as the final truth value is known Thus for instance x test 1 x will never produce an error since test 1 x is not even evaluated when the first test is true hence the final truth value is true Similarly type p t_I
235. backslash character X at the end of the line that you are typing just before hitting lt Return gt This tells gp that what you will write on the next line is the physical continuation of what you have just written In other words it makes gp forget your newline character You can type a anywhere It is interpreted as above only if apart from ignored whitespace characters it is immediately followed by a newline For example you can type 73 4 instead of typing 3 4 The second one is a variation on the first and is mostly useful when defining a user function see Section 2 7 since an equal sign can never end a valid expression gp disregards a newline immediately following an a 123 hi 123 The third one is in general much more useful and uses braces and An opening brace signals that you are typing a multi line command and newlines are ignored until you type a closing brace There are two important but easily obeyed restrictions first braces do not nest second inside an open brace close brace pair all input lines are concatenated suppressing any newlines Thus all newlines should occur after a semicolon a comma or an operator for clarity s sake never split an identifier over two lines in this way For instance the following program 16 would silently produce garbage since this is interpreted as a bb c which assigns the value of c to both bb and a It should have been written
236. be a square matrix defining a congruence subgroup of the ray class group corresponding to bnr the trivial congruence subgroup if omitted This function returns for each character x of the ray class group which is trivial on H the value at s 1 or s 0 of the abelian L function associated to x For the value at s 0 the function returns in fact for each x a vector ry cy where L s x c s O s near 0 The argument flag is optional its binary digits mean 1 compute at s 0 if unset or s 1 if set 2 compute the primitive L function associated to x if unset or the L function with Euler factors at prime ideals dividing the modulus of bnr removed if set that is Ls s x where S is the set of infinite places of the number field together with the finite prime ideals dividing the modulus of bnr 3 return also the character if set K bnfinit x 2 229 bnr bnrinit K 1 1 bnrL1 bnr returns the order and the first non zero term of L s x at s 0 where x runs through the characters of the class group of K Q v229 Then bnr2 bnrinit K 2 1 bnrL1 bnr2 2 returns the order and the first non zero terms of Lg s x at s 0 where x runs through the characters of the class group of K and S is the set of infinite places of K together with the finite prime 2 Note that the ray class group modulo 2 is in fact the class group so bnrL1 bnr2 0 returns the same answer as bnrL1 bnr 0 This function will fail with t
237. being a vector of monomials of degree 1 variables w a vector of expressions of the same length replace in the expression x all occurrences of v by wi The substitutions are done simultaneously more precisely the v are first replaced by new variables in x then these are replaced by the w substvec x y x y y x 1 ly x substvec x y x y y x y 12 ly x y not Ly 2 y The library syntax is GEN gsubstvec GEN x GEN v GEN w 3 7 44 sumformal f v formal sum of the polynomial expression f with respect to the main variable if v is omitted with respect to the variable v otherwise it is assumed that the base ring has characteristic zero In other words considering f as a polynomial function in the variable v returns F a polynomial in v vanishing at 0 such that F b F a sum 1 f v sumformal n i n 1 1 2xn 2 1 2 n f n n 3 n 2 1 F gumformal f n f 1 f a 13 1 4 n 4 5 6xn 3 3 4 n 2 7 6 n sum n 1 2000 f n subst F n 2000 44 1 sum n 1001 2000 f n subst F n 2000 subst F n 1000 5 1 sumformal x 2 x y y 2 y 16 y x 2 1 2 xy72 1 2 y x 1 3 y73 1 2xy72 1 6 y x 2 y x sumformal y sumformal y 2 7 1 The library syntax is GEN sumformal GEN f long v 1 where v is a variable number 3 7 45 taylor x t d seriesprecision Taylor expansion around 0 of x with re
238. ber field associated to bnf so as to be able to compute the Dedekind zeta and lambda functions respectively zetak 1 and zetak x 1 at the current real precision If you do not need the bnfinit data somewhere else you may call it with an irreducible polynomial instead of a bnf it will call bnfinit itself The result is a 9 component vector v whose components are very technical and cannot really be used except through the zetak function This function is very inefficient and should be rewritten It needs to computes millions of coef ficients of the corresponding Dirichlet series if the precision is big Unless the discriminant is small it will not be able to handle more than 9 digits of relative precision For instance zetakinit x78 2 needs 440MB of memory at default precision This function will fail with the message x bnrL1 overflow in zeta_get_NO need too many primes if the approximate functional equation requires us to sum too many terms if the discriminant of the number field is too large The library syntax is GEN initzeta GEN bnf long prec 3 7 Polynomials and power series We group here all functions which are specific to polynomials or power series Many other functions which can be applied on these objects are described in the other sections Also some of the functions described here can be applied to other types 3 7 1 O p e If p is an integer greater than 2 returns a p adic 0 of precision e In all other
239. bfields if d is null or omitted The result is a vector of subfields each being given by g h where g is an absolute equation and h expresses one of the roots of g in terms of the root x of the polynomial defining nf This routine uses J Kliiners s algorithm in the general case and B Allombert s galoissubfields when nf is Galois with weakly supersolvable Galois group The library syntax is GEN nfsubfields GEN pol long d 3 6 117 polcompositum P Q flag 0 P and Q being squarefree polynomials in Z X in the same variable outputs the simple factors of the tale Q algebra A Q X Y P X Q Y The factors are given by a list of polynomials R in Z X associated to the number field Q X R and sorted by increasing degree with respect to lexicographic ordering for factors of equal degrees Returns an error if one of the polynomials is not squarefree Note that it is more efficient to reduce to the case where P and Q are irreducible first The routine will not perform this for you since it may be expensive and the inputs are irreducible in most applications anyway In this case there will be a single factor R if and only if the number fields defined by P and Q are disjoint Assuming P is irreducible of smaller degree than Q for efficiency it is in general much faster to proceed as follows nf nfinit P L nffactor nf Q 1 vector L i rnfequation nf L i to obtain the same result If you are only intere
240. bs nf pol flag 0 THIS FUNCTION IS OBSOLETE use rnf polredbest instead Relative version of polredabs Given a monic polynomial pol with coefficients in nf finds a simpler relative polynomial defining the same field The binary digits of flag mean The binary digits of flag correspond to 1 add information to convert elements to the new representation 2 absolute polynomial instead of relative 16 possibly use a suborder of the maximal order More precisely 0 default return P 1 returns P a where P is the default output and a a t_POLMOD modulo P is a root of pol 2 returns Pabs an absolute instead of a relative polynomial Same as but faster than rnfequation nf rnfpolredabs nf pol 3 returns Pabs a b where Pabs is an absolute polynomial as above a b are t_POLMOD modulo Pabs roots of nf pol and pol respectively 16 possibly use a suborder of the maximal order This is slower than the default when the relative discriminant is smooth and much faster otherwise See Section 3 6 120 Warning In the present implementation rnfpolredabs produces smaller polynomials than rnf polred and is usually faster but its complexity is still exponential in the absolute degree The function rnfpolredbest runs in polynomial time and tends to return polynomials with smaller discriminants The library syntax is GEN rnfpolredabs GEN nf GEN pol long flag 3 6 159 rnfpolredbest nf pol flag 0 Relative version of polre
241. by removing all prime factors lt primelimit Don t use this supply an explicit bound factor N bound which avoids relying on an unpredictable global variable The default value is 500k 3 14 29 prompt A string that will be printed as prompt Note that most usual escape sequences are available there Ne for Esc n for Newline for Time expansion is performed This string is sent through the library function strftime on a Unix system you can try man strftime at your shell prompt This means that constructs have a special meaning usually related to the time and date For instance 4H hour 24 hour clock and M minute 00 59 use hh to get a real If you use readline escape sequences in your prompt will result in display bugs If you have a relatively recent readline see the comment at the end of Section 3 14 3 you can brace them with special sequences and and you will be safe If these just result in extra spaces in your prompt then you ll have to get a more recent readline See the file misc gprc dft for an example Caution PariEmacs needs to know about the prompt pattern to separate your input from previous gp results without ambiguity It is not a trivial problem to adapt automatically this regular expression to an arbitrary prompt which can be self modifying See PariEmacs s documentation The default value is 3 14 30 prompt cont A string that will be printed to prompt for continuati
242. cally close to a square root of 1 The final example is more problematic but there are natural maps from Z 15Z and Z 10Z to Z 5Z and the computation takes place there 1 5 Operations and functions The available operations and functions in PARI are described in detail in Chapter 3 Here is a brief summary 1 5 1 Standard arithmetic operations Of course the four standard operators exist We emphasize once more that division is as far as possible an exact operation 4 divided by 3 gives 4 3 In addition to this operations on integers or polynomials like Euclidean division Euclidean remainder exist for integers computes the quotient such that the remainder has smallest possible absolute value There is also the exponentiation operator when the exponent is of type integer otherwise it is considered as a transcendental function Finally the logical operators not prefix operator amp amp and operator or operator exist giving as results 1 true or O false 1 5 2 Conversions and similar functions Many conversion functions are available to convert between different types For example floor ceiling rounding truncation etc Other simple functions are included like real and imaginary part conjugation norm absolute value changing precision or creating an intmod or a polmod An ambiguity arises with square matrices PARI always considers that you want to do com ponentwise functi
243. cally each column and concatenate vertically the resulting matrices The entries of v are always considered as matrices they can themselves be t_VEC seen as a row matrix a t_COL seen as a column matrix a t_MAT or a scalar seen as an 1 x 1 matrix A 1 2 3 4 B 5 6 C 7 8 D 9 matconcat A B horizontal 1 1 2 5 3 4 6 matconcat A C vertical Yo 1 2 3 4 7 8 matconcat A B C DJ block matrix 13 1 2 5 3 4 6 7 8 9 If the dimensions of the entries to concatenate do not match up the above rules are extended as follows e each entry v of v has a natural length and height 1 x 1 for a scalar 1 x n for a t_VEC of length n n x 1 for a t_COL m x n for an m x n t_MAT e let H be the maximum over j of the lengths of the v j let Lj be the maximum over i of the heights of the v The dimensions of the i j th block in the concatenated matrix are H x Lj e a scalar s v j is considered as s times an identity matrix of the block dimension min H L e blocks are extended by 0 columns on the right and 0 rows at the bottom as needed matconcat 1 2 3 4 5 6 horizontal 44 1 2 4 o 3 5 0 O 6 matconcat 1 2 3 4 5 6 vertical 5 1 0 0 2 3 0 4 5 6 matconcat B C A D block matrix 76 5 0 7 8 6 0 0 0 234 1 29 0 3 40 9 U 1 2 3 4 V 1 2 3 4 5 6 7 8 9 matconcat mat
244. cases returns a power series zero with precision given by ev where v is the X adic valuation of p with respect to its main variable The library syntax is GEN ggrando GEN zeropadic GEN p long e for a p adic and GEN zeroser long v long e for a power series zero in variable v 3 7 2 bezoutres A B v Deprecated alias for polresultantext The library syntax is GEN polresultantextO GEN A GEN B long v 1 where visa variable number 215 3 7 3 deriv z v Derivative of x with respect to the main variable if v is omitted and with respect to v otherwise The derivative of a scalar type is zero and the derivative of a vector or matrix is done componentwise One can use x as a shortcut if the derivative is with respect to the main variable of x By definition the main variable of a t_POLMOD is the main variable among the coefficients from its two polynomial components representative and modulus in other words assuming a polmod represents an element of R X T X the variable X is a mute variable and the derivative is taken with respect to the main variable used in the base ring R The library syntax is GEN deriv GEN x long v 1 where v is a variable number 3 7 4 diffop z v d n 1 Let v be a vector of variables and d a vector of the same length return the image of x by the n power 1 if n is not given of the differential operator D that assumes the value d i on the variable v i The value of D on a scalar typ
245. characters while the M ones operate on words The Meta key might be called A1t on some keyboards will display a black diamond on most others and can safely be replaced by Esc in any case Typing any ordinary key inserts text where the cursor stands the arrow keys enabling you to move in the line There are many more movement commands which will be familiar to the Emacs user for instance C a C e will take you to the start end of the line M b M f move the cursor backward forward by a word etc Just press the lt Return gt key at any point to send your command to gp All the commands you type at the gp prompt are stored in a history a multiline command being saved as a single concatenated line The Up and Down arrows or C p C n will move you through the history M lt M gt sending you to the start end of the history C r C s will start an incremental backward forward search You can kill text C k kills till the end of line M d to the end of current word which you can then yank back using the C y key M y will rotate the kill ring C _ will undo your last changes incrementally M r undoes all changes made to the current line C t and M t will transpose the character word preceding the cursor and the one under the cursor Keeping the M key down while you enter an integer a minus sign meaning reverse behavior gives an argument to your next readline command for instance M C k will kill text back to the start of line If you prefe
246. cipal ideals Ideal operations suitably update the principal part when it makes sense in a multiplicative context e g using idealmul on J t J u we obtain JJ tu When it does not make sense the extended part is silently discarded e g using idealadd with the above input produces I J The principal part t in an extended ideal may be represented in any of the above forms and also as a factorization matrix in terms of number field elements not ideals possibly the empty matrix representing 1 In the latter case elements stay in factored form or famat for factorization matrix which is a convenient way to avoid coefficient explosion To recover the conventional expanded form try nffactorback but many functions already accept famats as input for instance ideallog so expanding huge elements should never be necessary 3 6 3 Finite abelian groups A finite abelian group G in user readable format is given by its Smith Normal Form as a pair h d or triple h d g Here h is the cardinality of G d is the vector of elementary divisors and gi is a vector of generators In short G j lt n Z d Z gi with dn da d and d h This information can also be retrieved as G no G cyc and G gen e a character on the abelian group Z d Z g is given by a row vector x a1 n such that x 9 exp 2ia gt ayn d e given such a structure a subgroup H is input as a square matrix in HNF whose column
247. ck idea of what is available by looking at the gp reference card refcard dvi or refcard ps In case of need you can refer to the complete function description in Chapter 3 How to get the latest version Everything can be found on PARI s home page http pari math u bordeaux fr From that point you may access all sources some binaries version information the complete mailing list archives frequently asked questions and various tips All threaded and fully searchable How to report bugs Bugs are submitted online to our Bug Tracking System available from PARI s home page or directly from the URL http pari math u bordeaux fr Bugs Further instructions can be found on that page 1 2 Multiprecision kernels Portability The PARI multiprecision kernel comes in three non exclusive flavors See Appendix A for how to set up these on your system various compilers are supported but the GNU gcc compiler is the definite favourite A first version is written entirely in ANSI C with a C compatible syntax and should be portable without trouble to any 32 or 64 bit computer having no drastic memory constraints We do not know any example of a computer where a port was attempted and failed In a second version time critical parts of the kernel are written in inlined assembler At present this includes e the whole ix86 family Intel AMD Cyrix starting at the 386 up to the Xbox gaming console including the Opteron 64 bit proce
248. cks whether the environment variable GPRC is set On Unix this can be done with something like GPRC my dir anyname export GPRC in sh syntax for instance in your profile setenv GPRC my dir anyname in csh syntax in your login or cshrc file env GPRC my dir anyname gp on the command line launching gp If so the file named by GPRC is the gprc e If GPRC is not set and if the environment variable HOME is defined gp then tries HOME gprc on a Unix system HOMEAgprc txt on a DOS OS 2 or Windows system e If no gpre was found among the user files mentioned above we look for etc gpre for a system wide gprc file you will need root privileges to set up such a file yourself e Finally we look in pari s datadir for a file named gprc on a Unix system gprc txt on a DOS OS 2 or Windows system If you are using our Windows installer this is where the default preferences file is written d Note that on Unix systems the gprc s default name starts with a and thus is hidden to regular ls commands you need to type 1s a to list it 60 2 15 Using readline This very useful library provides line editing and contextual completion to gp You are en couraged to read the readline user manual but we describe basic usage here A too short introduction to readline In the following C stands for the Control key combined with another and the same for M with the Meta key generally C combinations act on
249. counted with multiplicity factor 392 1 2 3 7 2 bigomega 392 2 5 3 2 omega 392 3 2 without multiplicity The library syntax is long bigomega GEN x 101 3 4 8 binomial z y binomial coefficient EN Here y must be an integer but x can be any PARI object The library syntax is GEN binomial GEN x long y The function GEN binomialuu ulong n ulong k is also available and so is GEN vecbinome long n which returns a vector v with n 1 components such that v k 1 binomial n k for k from 0 up to n 3 4 9 chinese z y If x and y are both intmods or both polmods creates with the same type a z in the same residue class as x and in the same residue class as y if it is possible chinese Mod 1 2 Mod 2 3 1 Mod 5 6 chinese Mod x x 2 1 Mod x 1 x 2 1 12 Mod 1 2 x 2 x 1 2 x74 1 This function also allows vector and matrix arguments in which case the operation is recursively applied to each component of the vector or matrix chinese Mod 1 2 Mod 1 3 Mod 1 5 Mod 2 7 13 Mod 1 10 Mod 16 21 For polynomial arguments in the same variable the function is applied to each coefficient if the polynomials have different degrees the high degree terms are copied verbatim in the result as if the missing high degree terms in the polynomial of lowest degree had been Mod 0 1 Since the latter behavior is usually not the desired one we propose to co
250. ction 3 4 2 the preferred format for this parameter is Lord factor ord where ord is the order of g It may be set as a side effect of calling ffprimroot If no o is given assume that g is a primitive root The result is undefined if e does not exist This function uses e a combination of generic discrete log algorithms see znlog e a cubic sieve index calculus algorithm for large fields of degree at least 5 e Coppersmith s algorithm for fields of characteristic at most 5 t ffgen ffinit 7 5 111 o fforder t 12 5602 not a primitive root fflog t710 t 3 10 fflog t710 t o 74 10 g ffprimroot t ko order is 16806 bundled with its factorization matrix 6 16806 2 1 3 1 2801 1 fforder g o 7 16806 fflog g 10000 g o 8 10000 The library syntax is GEN fflog GEN x GEN g GEN o NULL 3 4 30 ffnbirred q n fl 0 Computes the number of monic irreducible polynomials over F of degree exactly n flag 0 or omited or at most n flag 1 The library syntax is GEN ffnbirredO GEN q long n long f1 Also available are GEN ffnbirred GEN q long n for flag 0 and GEN ffsumnbirred GEN q long n for flag i 3 4 31 fforder z 0 Multiplicative order of the finite field element x If o is present it represents a multiple of the order of the element see Section 3 4 2 the preferred format for this parameter is N factor N where N is t
251. ction computes the relative norm of x as an ideal of K in HNF The library syntax is GEN rnfidealnormrel GEN rnf GEN x 3 6 146 rnfidealreltoabs rnf x Let rnf be a relative number field extension L K as output by rnfinit and let x be a relative ideal given as a Zk module by a pseudo matrix 4 1 This function returns the ideal x as an absolute ideal of L Q in the form of a Z basis given by a vector of polynomials modulo rnf po1 The reason why we do not return the customary HNF in terms of a fixed Z basis for Zz is precisely that no such basis has been explicitly specified On the other hand if you already computed an absolute nf structure Labs associated to L then xabs rnfidealreltoabs L x xLabs mathnf matalgtobasis Labs xabs computes a traditional HNF xLabs for x in terms of the fixed Z basis Labs zk The library syntax is GEN rnfidealreltoabs GEN rnf GEN x 208 3 6 147 rnfidealtwoelt rnf x rnf being a relative number field extension L K as output by rnfinit and x being an ideal of the relative extension L K given by a pseudo matrix gives a vector of two generators of x over Zr expressed as polmods with polmod coefficients The library syntax is GEN rnfidealtwoelement GEN rnf GEN x 3 6 148 rnfidealup rnf x Let rnf be a relative number field extension L K as output by rnfinit and let x be an ideal of K This function returns the ideal zZz as an absolute ideal of L Q in the form of a Z basis giv
252. d The library syntax is GEN gmin GEN x GEN y 3 1 15 shift x n Shifts z componentwise left by n bits if n gt 0 and right by n bits if n lt 0 May be abbreviated as x lt lt n or x gt gt n A left shift by n corresponds to multiplication by 2 A right shift of an integer x by n corresponds to a Euclidean division of x by 2 with a remainder of the same sign as x hence is not the same in general as 112 The library syntax is GEN gshift GEN x long n 3 1 16 shiftmul x n Multiplies x by 2 The difference with shift is that when n lt 0 ordinary division takes place hence for example if x is an integer the result may be a fraction while for shifts Euclidean division takes place when n lt 0 hence if x is an integer the result is still an integer The library syntax is GEN gmul2n GEN x long n 3 1 17 sign x sign 0 1 or 1 of x which must be of type integer real or fraction The library syntax is GEN gsigne GEN x 69 3 1 18 vecmax z amp v If x is a vector or a matrix returns the largest entry of x otherwise returns a copy of x Error if x is empty If v is given set it to the index of a largest entry indirect maximum when z is a vector If x is a matrix set v to coordinates i j such that x i j is a largest entry This flag is ignored if x is not a vector or matrix vecmax 10 20 30 40 1 40 vecmax 10 20 30 40 amp v v 42 4 vecmax 10 2
253. d its output fed into gp just as if read from a file 3 12 11 externstr str The string str is the name of an external command i e one you would type from your UNIX shell prompt This command is immediately run and its output is returned as a vector of GP strings one component per output line 3 12 12 getabstime Returns the time in milliseconds elapsed since gp startup This provides a reentrant version of gettime my t getabstime print Time getabstime t The library syntax is long getabstime 291 3 12 13 getenv s Return the value of the environment variable s if it is defined otherwise return 0 The library syntax is GEN gp_getenv const char s 3 12 14 getheap Returns a two component row vector giving the number of objects on the heap and the amount of memory they occupy in long words Useful mainly for debugging purposes The library syntax is GEN getheap 3 12 15 getrand Returns the current value of the seed used by the pseudo random number generator random Useful mainly for debugging purposes to reproduce a specific chain of compu tations The returned value is technical reproduces an internal state array and can only be used as an argument to setrand The library syntax is GEN getrand 3 12 16 getstack Returns the current value of top avma i e the number of bytes used up to now on the stack Useful mainly for debugging purposes The library syntax is long getstack
254. d you should leave it alone 158 The components of a bnf or sbnf are technical and never used by the casual user In fact never access a component directly always use a proper member function However for the sake of completeness and internal documentation their description is as follows We use the notations ex plained in the book by H Cohen A Course in Computational Algebraic Number Theory Graduate Texts in Maths 138 Springer Verlag 1993 Section 6 5 and subsection 6 5 5 in particular bnf 1 contains the matrix W i e the matrix in Hermite normal form giving relations for the class group on prime ideal generators p 1 lt i lt r bnf 2 contains the matrix B i e the matrix containing the expressions of the prime ideal factorbase in terms of the p It is an r x c matrix bnf 3 contains the complex logarithmic embeddings of the system of fundamental units which has been found It is an r 72 x r r2 1 matrix bnf 4 contains the matrix M of Archimedean components of the relations of the matrix WIB bnf 5 contains the prime factor base i e the list of prime ideals used in finding the relations bnf 6 used to contain a permutation of the prime factor base but has been obsoleted It contains a dummy 0 bnf 7 or bnf nf is equal to the number field data nf as would be given by nfinit bnf 8 is a vector containing the classgroup bnf clgp as a finite abelian group the regulator bnf reg a
255. d2 di da P Q where P is of order d and P Q generates the curve Caution It is not guaranteed that Q has order dz which in the worst case requires an expensive discrete log computation Only that ellweilpairing E P Q d1 has order da E ellinit 0 11 y 2 x 3 0 x 1 defined over Q ellgroup E 7 12 6 2 AX Z 6 x Z 2 non cyclic E ellinit 0 1 Mod 1 11 AM defined over F_11 ellgroup E no need to repeat 11 4 12 ellgroup E 11 MX but it also works 5 12 ellgroup E 13 ouch inconsistent input kk at top level ellgroup E 13 xk ellgroup inconsistent moduli in Rg_to_Fp 11 13 ellgroup E 7 1 6 12 6 2 Mod 2 7 Mod 4 7 Mod 4 7 Mod 4 7 138 If E is defined over Q we allow singular reduction and in this case we return the structure of the group of non singular points satisfying Ens Fp p Gp E ellinit 0 5 ellgroup E 5 1 2 5 5 Mod 4 5 Mod 2 5 ellap E 5 13 0 additive reduction at 5 E ellinit 0 1 0 35 0 ellgroup E 5 1 5 4 4 Mod 2 5 Mod 2 5 ellap E 5 6 1 split multiplicative reduction at 5 ellgroup E 7 1 7 8 8 Mod 3 7 Mod 5 7 11 ellap E 7 18 1 non split multiplicative reduction at 7 The library syntax is GEN ellgroupO GEN E GEN p NULL long flag Also available is GEN el
256. dbest Given a monic polynomial pol with coefficients in nf finds a simpler relative polynomial P defining the same field As opposed to rnfpolredabs this function does not return a smallest canonical polynomial with respect to some measure but it does run in polynomial time The binary digits of flag correspond to 1 add information to convert elements to the new representation 2 absolute polynomial instead of relative More precisely 0 default return P 1 returns P a where P is the default output and a a t_POLMOD modulo P is a root of pol 2 returns Pabs an absolute instead of a relative polynomial Same as but faster than rnfequation nf rnfpolredbest nf pol 3 returns Pabs a b where Pabs is an absolute polynomial as above a b are t_POLMOD modulo Pabs roots of nf pol and pol respectively 212 K nfinit y 3 2 pol x72 x y y72 P a rnfpolredbest K pol 1 P 43 x 2 x Mod y 1 y 3 2 a 4 Mod Mod 2 y 2 3 y 4 y 3 2 x Mod y 2 2 y 2 y 3 2 x 2 x Mod y 1 y 3 2 subst K pol y a 5 0 Pabs a b rnfpolredbest K pol 3 Pabs AT x 6 3x75 5 x 3 3 x 1 a 18 Mod x 2 x 1 x 6 3x x 5 5x x 3 3x x 1 b 9 Mod 2 x 5 5 x 4 3 x 3 10 x 2 5 x 5 x 6 3 xx 5 5 x 3 3x x 1 subst K pol y a 110 0 substvec pol x y a b 111 0 The library syntax is GEN rnfpolredbest GEN nf GEN pol long flag 3 6 160 rnfpseudobasis
257. dealapprO GEN nf GEN x long flag 3 6 48 idealchinese nf x y x being a prime ideal factorization i e a 2 by 2 matrix whose first column contains prime ideals and the second column integral exponents y a vector of elements in nf indexed by the ideals in x computes an element b such that Up b yp gt Up z for all prime ideals in x and vp b gt 0 for all other p The library syntax is GEN idealchinese GEN nf GEN x GEN y 3 6 49 idealcoprime nf x y Given two integral ideals x and y in the number field nf returns a 6 in the field such that x is an integral ideal coprime to y The library syntax is GEN idealcoprime GEN nf GEN x GEN y 3 6 50 idealdiv nf x y flag 0 Quotient x y of the two ideals x and y in the number field nf The result is given in HNF If flag is non zero the quotient x y is assumed to be an integral ideal This can be much faster when the norm of the quotient is small even though the norms of x and y are large The library syntax is GEN idealdivO GEN nf GEN x GEN y long flag Also available are GEN idealdiv GEN nf GEN x GEN y flag 0 and GEN idealdivexact GEN nf GEN x GEN y flag 1 3 6 51 idealfactor nf x Factors into prime ideal powers the ideal x in the number field nf The output format is similar to the factor function and the prime ideals are represented in the form output by the idealprimedec function i e as 5 element vectors The library
258. default tex2mail looks much nicer than the built in beautified format output 2 The default value is tex2mail TeX noindent ragged by_par 3 14 28 primelimit gp precomputes a list of all primes less than primelimit at initialization time and can build fast sieves on demand to quickly iterate over primes up to the square of primelimit These are used by many arithmetic functions usually for trial division purposes The maximal value is 232 2049 resp 264 2049 on a 32 bit resp 64 bit machine but values beyond 108 allowing to iterate over primes up to 1016 do not seem useful Since almost all arithmetic functions eventually require some table of prime numbers PARI guarantees that the first 6547 primes up to and including 65557 are precomputed even if prime limit is 1 This default is only used on startup changing it will not recompute a new table Deprecated feature primelimit was used in some situations by algebraic number theory func tions using the nf __PARTIALFACT flag nfbasis nfdisc nfinit this assumes that all primes p gt primelimit have a certain property the equation order is p maximal This is never done by default and must be explicitly set by the user of such functions Nevertheless these functions now provide a more flexible interface and their use of the global default primelimit is deprecated 309 EMACS Deprecated feature factor N 0 was used to partially factor integers
259. dent gp will strive to only give you meaningful completions in a given context it will fail sometimes but only under rare and restricted conditions For instance shortly after a we expect a user name then a path to some file Directly after default has been typed we would expect one of the default keywords After whatnow we expect the name of an old function which may well have disappeared from this version After a we expect a member keyword And generally of course we expect any GP symbol which may be found in the hashing lists functions both yours and GP s and variables If at any time only one completion is meaningful gp will provide it together with e an ending comma if we are completing a default e a pair of parentheses if we are completing a function name In that case hitting lt TAB gt again will provide the argument list as given by the online help Otherwise hitting lt TAB gt once more will give you the list of possible completions Just ex periment with this mechanism as often as possible you will probably find it very convenient For instance you can obtain default seriesprecision 10 just by hitting def lt TAB gt se lt TAB gt 10 which saves 18 keystrokes out of 27 Hitting M h will give you the usual short online help concerning the word directly beneath the cursor M H will yield the extended help corresponding to the help default program usually opens a dvi previewer or runs a primitive
260. der q M is R furthermore mathouseholder q matid M Q the inverse of Q This function raises an error if the precision is too low or x is singular The library syntax is GEN matqr GEN M long flag long prec 241 3 8 39 matrank x Rank of the matrix zx The library syntax is long rank GEN x 3 8 40 matrix m n X Y expr 0 Creation of the m x n matrix whose coefficients are given by the expression expr There are two formal parameters in expr the first one X corresponding to the rows the second Y to the columns and X goes from 1 to m Y goes from 1 to n If one of the last 3 parameters is omitted fill the matrix with zeroes 3 8 41 matrixqz A p 0 A being an m x n matrix in M n Q let Im A resp Imz A the Q vector space resp the Z module spanned by the columns of A This function has varying behavior depending on the sign of p If p gt 0 A is assumed to have maximal rank n lt m The function returns a matrix B Mm n Z with IngB ImQqA such that the GCD of all its n x n minors is coprime to p in particular if p 0 default this GCD is 1 minors x vector x 1 i matdet x 7i A 8 1 7 5 3 7 7 5 7 minors A 11 4 7 8 7 4 7 determinants of all 2x2 minors B matrixqz A 42 3 1 5 2 7 3 minors 13 1 2 1 B integral with coprime minors If p 1 returns the HNF basis of the lattice Z N Imz A If p 2 returns the
261. des u v to check whether ux vy is close to d The library syntax is GEN ggcdO GEN x GEN y NULL Also available are GEN ggcd GEN x GEN y if y is not NULL and GEN content GEN x if y NULL 3 4 35 gedext x y Returns u v d such that d is the gcd of x y x x u y xwv ecd x y and u and v minimal in a natural sense The arguments must be integers or polynomials u v dl gcdext 32 102 1 16 5 2 d 12 2 gcdext x 2 x x 2 x 2 13 1 2 1 2 x 1 If x y are polynomials in the same variable and inexact coefficients then compute u v d such that xz x u y xwv d where d approximately divides both and x and y in particular we do not obtain gcd x y which is defined to be a scalar in this case a x 0 0 gcd a a 41 1 gcdext a a 12 O 1 x 0 E 28 gcdext x Pi 6 x 2 zeta 2 13 6 x 18 8495559 1 57 5726923 For inexact inputs the output is thus not well defined mathematically but you obtain explicit polynomials to check whether the approximation is close enough for your needs The library syntax is GEN gcdextO GEN x GEN y 3 4 36 hilbert z y p Hilbert symbol of x and y modulo the prime p p 0 meaning the place at infinity the result is undefined if p 4 0 is not prime It is possible to omit p in which case we take p 0 if both x and y are rational or one of them is a real number And take p q if one of x y is a t_INTMOD modulo q or a q a
262. diagonal U V block diagonal AT 1200 0 3 40 0 0 o 0 1 2 3 0 0 4 5 6 0 07 8 9 The library syntax is GEN matconcat GEN v 3 8 15 matdet z flag 0 Determinant of the square matrix x If flag 0 uses an appropriate algorithm depending on the coefficients e integer entries modular method due to Dixon Pernet and Stein e real or p adic entries classical Gaussian elimination using maximal pivot e intmod entries classical Gaussian elimination using first non zero pivot e other cases Gauss Bareiss If flag 1 uses classical Gaussian elimination with appropriate pivoting strategy maximal pivot for real or p adic coefficients This is usually worse than the default The library syntax is GEN detO GEN x long flag Also available are GEN det GEN x flag 0 GEN det2 GEN x flag 1 and GEN ZM_det GEN x for integer entries 3 8 16 matdetint B Let B be an m x n matrix with integer coefficients The determinant D of the lattice generated by the columns of B is the square root of det BT B if B has maximal rank m and 0 otherwise This function uses the Gauss Bareiss algorithm to compute a positive multiple of D When B is square the function actually returns D det B This function is useful in conjunction with mathnfmod which needs to know such a multiple If the rank is maximal and the matrix non square you can obtain D exactly using matdet mathnfmod B matdetint B Note that as so
263. dic Incompatible types will raise an error The library syntax is long hilbert GEN x GEN y GEN p NULL 3 4 37 isfundamental x True 1 if x is equal to 1 or to the discriminant of a quadratic field false 0 otherwise The library syntax is long isfundamental GEN x 3 4 38 ispolygonal z s amp N True 1 if the integer x is an s gonal number false 0 if not The parameter s gt 2 must be a t_INT If N is given set it to n if x is the n th s gonal number ispolygonal 36 3 amp N 1 1 N The library syntax is long ispolygonal GEN x GEN s GEN N NULL 114 3 4 39 ispower z k amp n If k is given returns true 1 if x is a k th power false 0 if not If k is omitted only integers and fractions are allowed for x and the function returns the maximal k gt 2 such that x n is a perfect power or 0 if no such k exist in particular ispower 1 ispower 0 and ispower 1 all return 0 If a third argument amp n is given and x is indeed a k th power sets n to a k th root of x For a t_FFELT x instead of omitting k which is not allowed for this type it may be natural to set k x p poldegree x pol 1 fforder x The library syntax is long ispower GEN x GEN k NULL GEN n NULL Also available is long gisanypower GEN x GEN pty k omitted 3 4 40 ispowerful x True 1 if x is a powerful integer false 0 if not an integer is powerful if and only if its valuation at a
264. dicative the curve coefficients are checked for compatibility possibly changing D for instance if D 1 and an t_INTMOD is found If inconsistencies are detected an error is raised ellinit 1 0 5 1 0 7 140 at top level ellinit 1 0 5 1 0 OK O xxx ellinit inconsistent moduli in ellinit 7 5 If the curve coefficients are too general to fit any of the above domain categories only basic operations such as point addition will be supported later If the curve seen over the domain D is singular fail and return an empty vector E ellinit 0 0 0 0 1 y 2 x 3 1 over Q E ellinit 0 1 the same curve short form E ellinit 36a1 sill the same curve Cremona s notations E ellinit 0 1 2 over F2 singular curve 14 7 E ellinit a4 a6 Mod 1 5 AN over F_5 a4 a6 basic support The result of ellinit is an ell structure It contains at least the following information in its components 41 02 03 04 46 ba ba be bg C4 C6 A j All are accessible via member functions In particular the discriminant is E disc and the j invariant is F j E ellinit a4 a6 E disc 42 64 a4 3 432 a6 2 E j 13 6912 ra473 4 ra473 27 a672 Further components contain domain specific data which are in general dynamic only com puted when needed and then cached in the structure E ellinit 2 3 10 60 7 E over F_p p la
265. difference between strings and keywords see Section 2 9 the latter don t get expanded at all and you can type them without any enclosing quotes The former are dynamic objects where everything outside quotes gets immediately expanded 3 12 1 Strprintf fmt x Returns a string built from the remaining arguments according to the format fmt The format consists of ordinary characters not printed unchanged and conversions specifications See printf 3 12 2 addhelp sym str Changes the help message for the symbol sym The string str is expanded on the spot and stored as the online help for sym It is recommended to document global variables and user functions in this way although gp will not protest if you don t You can attach a help text to an alias but it will never be shown aliases are expanded by the help operator and we get the help of the symbol the alias points to Nothing prevents you from modifying the help of built in PARI functions But if you do we would like to hear why you needed it Without addhelp the standard help for user functions consists of its name and definition gp gt f x x72 gp gt f f x gt x 2 Once addhelp is applied to f the function code is no longer included It can still be consulted by typing the function name gp gt addhelp f Square gp gt f Square gp gt f 12 x gt x 2 The library syntax is void addhelp const char sym const char str 287 3 12 3
266. different from the above floating point accuracy and depends on the type of z For exact types no change For x a vector or a matrix the operation is done componentwise For real x n is the number of desired significant decimal digits If n is smaller than the precision of x x is truncated otherwise x is extended with zeros For x a p adic or a power series n is the desired number of significant p adic or X adic digits where X is the main variable of x Note yes this is inconsistent Note that the precision is a 83 priori distinct from the exponent k appearing in O it is indeed equal to k if and only if z is a p adic or X adic unit precision 1 O x 10 1 1 O x710 precision x 2 O x710 3 2 x72 0 x 5 precision 772 0 7710 3 913 772 0 775 For the last two examples note that 1 0 2 2 14 0 23 indeed has 3 significant coefficients The library syntax is GEN precision0 GEN x long n Also available are GEN gprec GEN x long n and long precision GEN x In both the accuracy is expressed in words 32 bit or 64 bit depending on the architecture 3 2 48 random N 2 Returns a random element in various natural sets depending on the argument NV e t_INT returns an integer uniformly distributed between 0 and N 1 Omitting the argument is equivalent to random 2731 e t_REAL returns a real number in 0 1 with the same accuracy as N whose mantissa has the same numb
267. discreduced f Reduced discriminant vector of the integral monic polynomial f This is the vector of elementary divisors of Z a f a Zla where a is a root of the polynomial f The components of the result are all positive and their product is equal to the absolute value of the discriminant of f The library syntax is GEN reduceddiscsmith GEN f 220 3 7 17 polgraeffe f Returns the Graeffe transform g of f such that g x f x f x The library syntax is GEN polgraeffe GEN f 3 7 18 polhensellift A B p e Given a prime p an integral polynomial A whose leading co efficient is a p unit a vector B of integral polynomials that are monic and pairwise relatively prime modulo p and whose product is congruent to A Ic A modulo p lift the elements of B to polynomials whose product is congruent to A modulo p More generally if T is an integral polynomial irreducible mod p and B is a factorization of A over the finite field F t T you can lift it to Z t T p by replacing the p argument with lp T 7PAT t13 23p 7 A x 2 t 1 B x 34t72 t 1 x 4xt72 6xt 6 r polhensellift A B p T 6 1 x 20191 t 2 50604 t 75783 x 97458xt 2 67045 t 41866 liftall r 1 r 2 Mod Mod 1 p 6 T 42 x 2 t 1 The library syntax is GEN polhensellift GEN A GEN B GEN p long e 3 7 19 polhermite n a xP n Hermite polynomial H evaluated at a x by default i
268. e 298 gp_readvec_stream 298 Gracie uma hee Boe oe he eS 220 graphcolormap 273 306 307 graphcolors 307 roal 2 25 a be eee ee eee ee RS 85 GRH e bos 156 157 159 160 210 227 gEDOGOL pe dose aa a ad a 85 Ground ss eano w ee Ra Se ee 85 GYOUP exis bee EE ee eR eS 132 GShitt 344 0446 o ooo os 69 BSIBMG cs cc 00 eS 69 ESIN ne a Gs ge wR eg a ee Se a 5 96 ESTOD Es pas er a eek Ah as 96 gsizebyte sa se 54 Hie bee ena 86 Bsizeword gt ra cees 565 8 eee a es 86 EST sia bac ee aa ee be eo 65 96 BSGI coso coros oros eos 96 BSQrtn A ee ee a eS ols 97 SUD epa e ios Bee dk Sa ao eta a Se ae 4 65 ESUDSE csi sb ea Se ee ee 225 gsubstpol 2 4 66 b eee ewe eee Gs 225 GSUDStVEO e sep Ga ae ee ed ees 226 ELM see toe ea eee ae a ee a 97 Ptanh o 2 ose a Se a ee as OF BCOCOL swe das Baldo e Saw alan 71 ELOCOLO cura ee ee ee Oe 71 ptocolrev 2 422405 64 246 oa as 71 gtocolrevO ss scm ase ee cis 71 Stolist o e 2 sa cei ae eee ees FL Stomat ico ok ek a he ee a de gtopoly e ba ocs dcce ee ede ee ns 13 StopolyreV a i aaa ina amp i td 74 ETOSET Lisa a a 75 BOSE Sha rl rs a e OP Bas 75 LUOVES de a Gh ae Bon Kia oe da 76 StOVECO iia awe en aw ea aes 76 gLOVECTOV fos aae k bk a te we 76 gtovecrev0 rios 76 gtovecsmall 2 64 24556 84 be be 76 gtovecsmallO 76 B TACO ociosa a ek wes 253 Streams dm a a ee ke a eee a 244 tCTUNG messi Ta Ee Se 86 evaluation a acs e
269. e n a d a N n H 1 e dare The library syntax is GEN polhermite_eval long n GEN a NULL The variant GEN pol hermite long n long v returns the n th Hermite polynomial in variable v 3 7 20 polinterpolate X Y x amp e Given the data vectors X and Y of the same length n X containing the x coordinates and Y the corresponding y coordinates this function finds the interpolating polynomial passing through these points and evaluates it at x If Y is omitted return the polynomial interpolating the i X 1 Tf present e will contain an error estimate on the returned value The library syntax is GEN polint GEN X GEN Y NULL GEN x NULL GEN e NULL 3 7 21 poliscyclo f Returns 0 if f is not a cyclotomic polynomial and n gt 0 if f the n th cyclotomic polynomial poliscyclo x 4 x 2 1 1 12 polcyclo 12 42 x4 x2 1 poliscyclo x 4 x 2 1 43 0 The library syntax is long poliscyclo GEN f 221 3 7 22 poliscycloprod f Returns 1 if f is a product of cyclotomic polynomial and 0 otherwise f x76 x75 x73 xt1 poliscycloprod f L2 1 factor f 13 x 2 x 1 1 x 4 x72 1 1 poliscyclo T T lt 1 4 3 12 polcyclo 3 polcyclo 12 5 x 6 x b x 3 x 1 The library syntax is long poliscycloprod GEN f 3 7 23 polisirreducible pol pol being a polynomial univariate in the present version 2 7 0 returns 1 if p
270. e 1 1 Depth of scalar product combination to use e 1 2 Maximum level of Bacher polynomials to use Since this function computes the minimal vectors it can become very lengthy as the dimension of G grows The library syntax is GEN qfisominitO GEN G GEN fl NULL Also available is GEN qfi sominit GEN F GEN fl where F is a vector of zm 3 8 57 qfjacobi A Apply Jacobi s eigenvalue algorithm to the real symmetric matrix A This returns L V where e L is the vector of real eigenvalues of A sorted in increasing order e V is the corresponding orthogonal matrix of eigenvectors of A p19 7 A 1 2 2 1 mateigen A 1 1 1 1 1 L H qfjacobi A L 3 1 000000000000000000 3 000000000000000000 H 74 247 0 7071067811865475245 0 7071067811865475244 0 7071067811865475244 0 7071067811865475245 norml2 A L 1 H 1 approximate eigenvector 15 9 403954806578300064 E 38 norml2 H H 1 6 2 350988701644575016 E 38 close to orthogonal The library syntax is GEN jacobi GEN A long prec 3 8 58 qfill x flag 0 LLL algorithm applied to the columns of the matrix x The columns of x may be linearly dependent The result is a unimodular transformation matrix T such that x T is an LLL reduced basis of the lattice generated by the column vectors of x Note that if x is not of maximal rank T will not be square The LLL parameters are 0 51 0 99 meaning that
271. e function of i vectorv is similar for column vectors vector 10 i i72 1 hi 2 5 10 17 26 37 50 65 82 101 The functions Vec and Col convert objects to row and column vectors respectively as well as Vecrev and Colrev which revert the indexing T poltchebi 5 5 th Chebyshev polynomial 1 16 x75 20 x73 5 x Vec T 12 16 0 20 0 5 0 coefficients of T Vecrev T 3 0 5 0 20 O 16 in reverse order Remark For v a t_VEC t_COL t_LIST or t_MAT the alternative set notations lg x lt v f x x x lt v f x g x x lt v are available as shortcuts for apply g select f Vec v select f Vec v apply g Vec v respectively and may serve as t_VEC constructors p p lt primes 10 isprime p 2 2 3 5 11 17 29 returns the primes p among the first 10 primes such that p p 2 is a twin pair p 2 p lt primes 10 p 4 1 1 25 169 289 841 returns the squares of the primes congruent to 1 modulo 4 where p runs among the first 10 primes 24 2 3 15 Matrices t_MAT To enter a matrix type the components row by row the components being separated by commas the rows by semicolons and everything enclosed in brackets and J eg x y z t u v yields an empty 0 x 0 matrix The function Mat transforms any object into a matrix and matrix creates matrices whose i j th en
272. e 174 1dealapprO 2 siran raea aa eee om 174 idealchinese 174 idealcoprime 174 PACALGIV os ees ar e Bs GK ew 174 Pdeal day surs aoe iw eo a oe ii 174 idealdivexact 174 idealfactor 165 174 182 idealfactorback 174 175 idealfrobenius 175 idealhnft rasa sed 176 177 idealhnfO aa roa e e a IIT idealintersect 177 240 idealiny sss oe wb Roe oe ee 177 193 id allist ss 1040 04 L77 178 ideallistO 0 178 ideallistarch 178 ideallog 152 178 179 183 idealmin sose poo rr de i 179 idealm l olaa a ed 179 idealmulO 0 179 idealmulred 179 idealnotm era we Heer i eek ee Hs 179 idealnumden 2 2 5 5 2 4 8 bee Ge as 179 idealpow 179 180 182 idealpowO ooco bee ees 180 idealpowred o ooo 180 idealpoWS secs bee eee 0 03 180 idealprimedec 180 190 idealprincipalunits 180 181 idealramgroups 181 idealred 152 175 181 idealredO 24 40 mal eel 182 idealstar 165 180 182 Idealstar roscado de 183 IdealstarO o 23 2 4 443 d404 amp 5 183 idealtwoelt 183 idealtwoeltO 183 idealtwoelt2 ecu be ee bees 183 idealval gt lt oa e bee eee RG raad 183 Ot aes ak ele Gee da as 283 iferr 4 24 44 4 27 50 76 279 283 299 IMAG gs ooo 2 ES Sh ik
273. e GEN Idealstar GEN nf GEN ideal long flag where flag is an or ed combination of nf_GEN include generators and nf_INIT return a full bid not a group possibly 0 This offers one more combination gen but no init 3 6 70 idealtwoelt nf x a Computes a two element representation of the ideal x in the number field nf combining a random search and an approximation theorem x is an ideal in any form possibly an extended ideal whose principal part is ignored e When called as idealtwoelt nf x the result is a row vector a a with two components such that x aZk aZx and a is chosen to be the positive generator of x N Z unless x was given as a principal ideal in which case we may choose a 0 The algorithm uses a fast lazy factorization of x N Z and runs in randomized polynomial time e When called as idealtwoelt nf x a with an explicit non zero a supplied as third argument the function assumes that a x and returns a x such that z aZ kx 0Z xk Note that we must factor a in this case and the algorithm is generally much slower than the default variant The library syntax is GEN idealtwoeltO GEN nf GEN x GEN a NULL Also available are GEN idealtwoelt GEN nf GEN x and GEN idealtwoelt2 GEN nf GEN x GEN a 3 6 71 idealval nf x pr Gives the valuation of the ideal x at the prime ideal pr in the number field nf where pr is in idealprimedec format The library syntax is long idealval GEN nf GEN x GEN pr 3
274. e Pee nO 148 ellwpseries 148 ellzeta 131 148 149 ellztopoint ci 149 MACS mai eo Go es Hi ok e e E 62 328 Engel expansion 104 environment expansion 75 environment expansion 55 environment variable 75 ET C is awk A A A Bee A ed 92 e fio ess 2 ba ba ee ee eS ek 291 error recovery 2 ee ee 49 error trapping lt e s ceses esasa 50 LLO er ee E G 47 50 286 291 error E vic mr a da ere ds 283 Ota orcas 92 131 266 taO gt rta oe OSES EES BESO 93 Euclides ido Gt SR 113 Euclidean quotient 65 66 Euclidean remainder 66 Euler product 105 121 266 Euler totient function 99 106 Buller 262 tee a Bee ae a we aes 89 Euler Maclaurin 98 SULSEPOL sex ox as Gwe SS ne te ie 106 Val ie eda hae aie he ee ae 34 36 47 216 OXP sonoras is eee eS 93 EXPM 2s pes foe o Bs 93 expression sequence 15 EXPTESSION ss 4 as e goed W Ga ee ie o 15 extended gcd 0204 114 extern egies ara ea 47 291 311 external prettyprint 2 eee 308 CXtEINStr o se soe ee ee ee me a 291 OxtractO atari ees e 254 F facteantor sia sipe a a ae ES vos 109 factor coss eee eee aan 105 106 108 LActoRO s 24 e a Be eS 108 factorback s wa sa foe 4 a sad A 108 109 factorback2 iuris ss 109 factorcantor 109 factorff esere srana 107 109 110 factorial 2 hee o o
275. e available consecutive powers of 2 63 Mnemonics for flags Numeric flags as mentioned above are obscure error prone and quite rigid should the authors want to adopt a new flag numbering scheme for instance when noticing flags with the same meaning but different numeric values across a set of routines it would break backward compatibility The only advantage of explicit numeric values is that they are fast to type so their use is only advised when using the calculator gp As an alternative one can replace a numeric flag by a character string containing symbolic identifiers For a generic flag the mnemonic corresponding to the numeric identifier is given after it as in fun x flag 0 If flag is equal to 1 AGM use an agm formula which means that one can use indifferently fun x 1 or fun z AGM For a binary flag mnemonics corresponding to the various toggles are given after each of them They can be negated by prepending no_ to the mnemonic or by removing such a prefix These toggles are grouped together using any punctuation character such as or For instance taken from description of ploth X a b expr flag 0 n 0 Binary digits of flags mean 1 Parametric 2 Recursive so that instead of 1 one could use the mnemonic Parametric no_Recursive or simply Para metric since Recursive is unset by default default value of flag is 0 i e everything unset People used to th
276. e bit or notation in languages like C may also use the form Parametric no_Recursive Pointers Ifa parameter in the function prototype is prefixed with a amp sign as in foo z amp e it means that besides the normal return value the function may assign a value to e as a side effect When passing the argument the amp sign has to be typed in explicitly As of version 2 7 0 this pointer argument is optional for all documented functions hence the amp will always appear between brackets as in Z_issquare z amp e About library programming The library function foo as defined at the beginning of this section is seen to have two mandatory arguments x and flag no function seen in the present chapter has been implemented so as to accept a variable number of arguments so all arguments are mandatory when programming with the library usually variants are provided corresponding to the various flag values We include an default value token in the prototype to signal how a missing argument should be encoded Most of the time it will be a NULL pointer or 1 for a variable number Refer to the User s Guide to the PARI library for general background and details 64 3 1 Standard monadic or dyadic operators 3 1 1 The expressions x and x refer to monadic operators the first does nothing the second negates 2 The library syntax is GEN gneg GEN x for 2 3 1 2 The expression x y is the sum of x and y Addition b
277. e curve names as t_STR 3 5 42 ellsigma L z x flag 0 Computes the value at z of the Weierstrass function attached to the lattice L as given by ellperiods 1 including quasi periods is useful otherwise there are recomputed from scratch for each new z 2 olz L z II 1 ew tak w wEL It is also possible to directly input L w wa or an elliptic curve E as given by ellinit L E omega w ellperiods 1 I 1 ellsigma w 1 2 h2 0 47494937998792065033250463632798296855 E ellinit 1 0 ellsigma E at x implicitly at default seriesprecision 4 x 1 60 xx 5 1 10080 x 9 23 259459200 x 13 0 x 17 If flag 1 computes an arbitrary determination of log a z The library syntax is GEN ellsigma GEN L GEN z NULL long flag long prec 3 5 43 ellsub F 21 22 Difference of the points z1 and z2 on the elliptic curve corresponding to E The library syntax is GEN ellsub GEN E GEN z1 GEN z2 3 5 44 elltaniyama E d seriesprecision Computes the modular parametrization of the el liptic curve E Q where E is an ell structure as output by ellinit This returns a two component vector u v of power series given to d significant terms seriesprecision by default charac terized by the following two properties First the point u v satisfies the equation of the elliptic curve Second let N be the conductor of E and Xo N gt E be a modular parametr
278. e is set to width x and height y omitting either x or y means we use the full size of the device in that direction If flag 0 x and y represent pixel units Otherwise x and y are understood as fractions of the size of the current output device hence must be between 0 and 1 and internally converted to pixels The plotting device imposes an upper bound for x and y for instance the number of pixels for screen output These bounds are available through the plothsizes function The following sequence initializes in a portable way i e independent of the output device a window of maximal size accessed through coordinates in the 0 1000 x 0 1000 range s plothsizes plotinit 0 s 1 1 s 2 1 plotscale 0 0 1000 0 1000 3 10 16 plotkill w Erase rectwindow w and free the corresponding memory Note that if you want to use the rectwindow w again you have to use plotinit first to specify the new size So it s better in this case to use plotinit directly as this throws away any previous work in the given rectwindow 3 10 17 plotlines w X Y flag 0 Draw on the rectwindow w the polygon such that the x y coordinates of the vertices are in the vectors of equal length X and Y For simplicity the whole polygon is drawn not only the part of the polygon which is inside the rectwindow If flag is non zero close the polygon In any case the virtual cursor does not move X and Y are allowed to be scalars in this case both hav
279. e is zero and D applies componentwise to a vector or matrix When applied to a t_POLMOD if no value is provided for the variable of the modulus such value is derived using the implicit function theorem Some examples This function can be used to differentiate formal expressions If E exp X then we have E 2 X x E We can derivate X exp X as follow diffop E X X E 1 2 X E 1 2xX 2 1 x E Let Sin and Cos be two function such that Sin Cos 1 and Cos Sin We can differentiate Sin Cos as follow PARI inferring the value of Sin from the equation diffop Mod Sin Cos Sin 2 Cos 2 1 Cos Sin 1 Mod 1 Cos 2 Sin 2 Cos 2 1 Compute the Bell polynomials both complete and partial via the Faa di Bruno formula Bell k n 1 my var i eval Str X i my x v dv v vector k i if i 1 E var i 1 dv vector k i if i 1 X var 1 E var i x diffop E v dv k E if n lt 0O subst x X 1 polcoeff x n X The library syntax is GEN diffopO GEN x GEN v GEN d long n For n 1 the function GEN diffop GEN x GEN v GEN d is also available 216 3 7 5 eval 1 Replaces in x the formal variables by the values that have been assigned to them after the creation of x This is mainly useful in GP and not in library mode Do not confuse this with substitution see subst If x is a character string eval x executes xz as a
280. e library syntax is GEN ellconvertname GEN name 3 5 14 elldivpol E n v xP n division polynomial f for the curve E in the variable v In standard notation for any affine point P X Y on the curve we have n P n P n P wn P Yn PJ for some polynomials n wn Yn in Z a az a3 a4 as X Y We have f X Yn X for n odd and fn X Yn X Y 2Y a X az for n even We have fi 1 f2 4X b X 2b4X b6 fg 3X bX 3b4X 3b6X b8 fa fo 2X b2 X5 5b4 X4 10b6X 10bs X bobg babe X bsba b2 For n gt 2 the roots of fn are the X coordinates of points in Efn The library syntax is GEN elldivpol GEN E long n long v 1 where v is a variable number 136 3 5 15 elleisnum w k flag 0 k being an even positive integer computes the numerical value of the Eisenstein series of weight k at the lattice w as given by ellperiods namely Zir Jwa 1 2 1 k Ong 1 9 n gt 0 where q exp 2i7T and T w w 2 belongs to the complex upper half plane It is also possible to directly input w w1 w2 or an elliptic curve E as given by ellinit w ellperiods 1 1 elleisnum w 4 2 2268 8726415508062275167367584190557607 elleisnum w 6 3 3 977978632282564763 E 33 E ellinit 1 0 elleisnum E 4 1 5 47 999999999999999999999999999999999998 When flag is non zero and k 4 or 6 returns the elliptic inva
281. e nt column of x Using of this function requires detailed knowledge of the structure of the different PARI types and thus it should almost never be used directly Some useful exceptions 2x 3 0 3 75 component x 2 12 81 p p adic accuracy component x 1 13 3 p q Q b 1 2 3 component q 1 5 1 The library syntax is GEN compo GEN x long n 3 2 29 conj x Conjugate of x The meaning of this is clear except that for real quadratic numbers it means conjugation in the real quadratic field This function has no effect on integers reals intmods fractions or p adics The only forbidden type is polmod see conjvec for this The library syntax is GEN gconj GEN x 3 2 30 conjvec z Conjugate vector representation of z If z is a polmod equal to Mod a T this gives a vector of length degree T containing e the complex embeddings of z if T has rational coefficients i e the a r i where r polroots T e the conjugates of z if T has some intmod coefficients if z is a finite field element the result is the vector of conjugates z z PP D zr where n degree T If z is an integer or a rational number the result is z If z is a row or column vector the result is a matrix whose columns are the conjugate vectors of the individual elements of z The library syntax is GEN conjvec GEN z long prec 3 2 31 denominator x Denominator of x The meaning of this is clear when
282. e numbers 143 3 5 33 n is in if not 7 2 7 43 74 6 ellmul E z n Computes n z where z is a point on the elliptic curve E The exponent Z or may be a complex quadratic integer if the curve E has complex multiplication by n an error message is issued Ei ellinit 1 0 z 0 0 ellmul Ei z 10 0 unsurprising z has order 2 ellmul Ei z I 0 0 Ei has complex multiplication by Z i ellmul Ei z quadgen 4 0 0 an alternative syntax for the same query Ej ellinit 0 11 z 1 0 ellmul Ej z I kk at top level ellmul Ej z I Hook ellmul not a complex multiplication in ellmul ellmul Ej z 1 quadgen 3 1 w 0 The simple minded algorithm for the CM case assumes that we are in characteristic 0 and that the quadratic order to which n belongs has small discriminant The library syntax is GEN ellmul GEN E GEN z GEN n 3 5 34 ellneg E z Opposite of the point z on elliptic curve E The library syntax is GEN ellneg GEN E GEN z 3 5 35 ellorder E z o Gives the order of the point z on the elliptic curve E defined over Q or a finite field If the curve is defined over Q return the impossible value zero if the point has infinite order 7 E ellinit 157 2 0 the 157 is congruent curve P 2 2 ellorder E P 2 2 P ellheegner E ellorder E P infinite order 43 0
283. e oe oe E 80 IMAGE ses ca a eae a a e a a ew A 239 imagecompl seere eae pa ee a ae 240 INCLAM e wee wide a Bee Se 94 INCEMO oka a eos Sheed A eres 94 NCYAMC o Geb hee ter enigs 94 inclusive or 2 0 ee ee es 70 index sos aor aicn ap e a i adaa a a 155 Index sed 06 ecs gintan a 2 ee a 155 indexrank 444 ea ave ese a aor a a 240 infinite product 267 infinite sum 004 268 i 00 00001 se e sa a da e e e 266 IDIBZOLA oras a eeo bka 215 inline sie nee e A ee eae 292 INDUC ds BO Ge date Bw a aM A 292 install oe voii 47 53 292 311 ID CIEC np meee phia SS 257 258 INLO 4 6 on e A A eS 218 A Pe eR ae eae SS 1 05 17 integral basis 185 integral pseudo matriz 44 153 internal longword format 58 internal representation 58 interpolating polynomial 221 Intersect e e cag aoe he ee 4000 Bb ew 240 INELOEMAL eso a ere 218 intfouriercos 258 intfourierexp 21 462 eee eae 258 intfouriersine se cs rone e a eg 258 i tf ncinit s oe samo ere wee 258 intlaplaceiny lt lt 0 258 259 intmellininy s ente E ea eot 259 260 intmellininvshort 260 INTO a ei a Be a e a ia 7 MO escri 6246684 8 18 intnum 257 260 265 269 INCNUMINIT e s s s eee oes amumi 261 265 intnuminitgen 265 intnumromb 265 266 intnumstep 261 266 ING biG ss ene le aS eG e e a E 78
284. e rational or Gaussian primes are in fact pseudoprimes see ispseudoprime a priori not rigorously proven primes In fact any factor which is lt 101 whose norm is lt 10 for an irrational Gaussian prime is a genuine prime Use isprime to prove primality of other factors as in fa factor 2 2 7 1 1 59649589127497217 1 5704689200685129054721 1 isprime fa 1l 42 1 1 MAN both entries are proven primes Another possibility is to set the global default factor_proven which will perform a rigorous primality proof for each pseudoprime factor A t_INT argument lim can be added meaning that we look only for prime factors p lt lim The limit lim must be non negative In this case all but the last factor are proven primes but the remaining factor may actually be a proven composite If the remaining factor is less than lim then it is prime factor 2 2 7 1 1075 13 340282366920938463463374607431768211457 1 106 Deprecated feature Setting lim 0 is the same as setting it to primelimit 1 Don t use this 1t is unwise to rely on global variables when you can specify an explicit argument This routine uses trial division and perfect power tests and should not be used for huge values of lim at most 10 say factorint 1 8 will in general be faster The latter does not guarantee that all small prime factors are found but it also finds larger factors and in a much more efficient way
285. e to There a single segment will be drawn between the virtual cursor current position and the point X Y And only the part thereof which actually lies within the boundary of w Then move the virtual cursor to X Y even if it is outside the window If you want to draw a line from x1 y1 to 12 y2 where 1 yl is not necessarily the position of the virtual cursor use plotmove w x1 y1 before using this function 3 10 18 plotlinetype w type Change the type of lines subsequently plotted in rectwindow w type 2 corresponds to frames 1 to axes larger values may correspond to something else w 1 changes highlevel plotting This is only taken into account by the gnuplot interface 3 10 19 plotmove w x y Move the virtual cursor of the rectwindow w to position x y 3 10 20 plotpoints w X Y Draw on the rectwindow w the points whose x y coordinates are in the vectors of equal length X and Y and which are inside w The virtual cursor does not move This is basically the same function as plothraw but either with no scaling factor or with a scale chosen using the function plotscale As was the case with the plotlines function X and Y are allowed to be simultaneously scalar In this case draw the single point X Y on the rectwindow w if it is actually inside w and in any case move the virtual cursor to position x y 3 10 21 plotpointsize w size Changes the size of following points in rectwindow w If w
286. e values 0 1 2 3 of flag have a binary meaning analogous to the one in matsnf in this case binary digits of flag mean e 1 complete output if set outputs H U where H is the Hermite normal form of M and U is a transformation matrix such that MU 0 H The matrix U belongs to GL R When M has a large kernel the entries of U are in general huge e 2 generic input Deprecated If set assume that R K X is a polynomial ring otherwise assume that R Z This flag is now useless since the routine always checks whether the matrix has integral entries For these 4 values we use a naive algorithm which behaves well in small dimension only Larger values correspond to different algorithms are restricted to integer matrices and all output the unimodular matrix U From now on all matrices have integral entries e flag 4 returns H U as in complete output above using a variant of LLL reduction along the way The matrix U is provably small in the L sense and in general close to optimal but the reduction is in general slow although provably polynomial time If flag 5 uses Batut s algorithm and output H U P such that H and U are as before and P is a permutation of the rows such that P applied to MU gives H This is in general faster than flag 4 but the matrix U is usually worse it is heuristically smaller than with the default algorithm When the matrix is dense and the dimension is large bigger than 100 sa
287. eads the last file that was fed into gp The return value is the result of the last expression evaluated If a GP binary file is read using this command see Section 3 12 44 the file is loaded and the last object in the file is returned In case the file you read in contains an allocatemem statement to be generally avoided you should leave read instructions by themselves and not part of larger instruction sequences 3 12 31 readstr filename Reads in the file filename and return a vector of GP strings each component containing one line from the file If filename is omitted re reads the last file that was fed into gp 3 12 32 readvec filename Reads in the file filename subject to string expansion If filename is omitted re reads the last file that was fed into gp The return value is a vector whose components are the evaluation of all sequences of instructions contained in the file For instance if file contains 1 3 then we will get ra fi 1 12 2 43 3 read a 14 3 readvec a 5 1 2 3 In general a sequence is just a single line but as usual braces and may be used to enter multiline sequences The library syntax is GEN gp_readvec_file const char filename The underlying li brary function GEN gp_readvec_stream FILE f is usually more flexible 3 12 33 select f A flag 0 We first describe the default behaviour when flag is 0 or omitted Given a vector or list A and a t_CLOS
288. eaks in polynomials to enhance the probability of a good line break The default value is O 3 14 2 breakloop If true enables the break loop debugging mode see Section 2 10 3 The default value is 1 if we are running an interactive gp session and O otherwise 3 14 3 colors This default is only usable if gp is running within certain color capable terminals For instance rxvt color_xterm and modern versions of xterm under X Windows or standard Linux DOS text consoles It causes gp to use a small palette of colors for its output With xterms the colormap used corresponds to the resources Xterm colorn where n ranges from 0 to 15 see the file misc color dft for an example Accepted values for this default are strings a1 ag where k lt 7 and each a is either e the keyword no use the default color usually black on transparent background e an integer between 0 and 15 corresponding to the aforementioned colormap e a triple co c1 c2 where co stands for foreground color c for background color and cz for attributes 0 is default 1 is bold 4 is underline The output objects thus affected are respectively error messages history numbers prompt input line output help messages timer that s seven of them Tf k lt 7 the remaining a are assumed to be no For instance default colors 9 5 no no 4 typesets error messages in color 9 history numbers in color 5 output in color 4 and does not affect the res
289. ed it is much more efficient to initialize this data once and for all and pass it to the relevant functions as in p nextprime 10740 v p 1 factor p 1 data for discrete log amp order computations znorder Mod 2 p v 3 500000000000000000000000000028 g znprimroot p znlog 2 g v 15 543038070904014908801878611374 3 4 3 addprimes x Adds the integers contained in the vector x or the single integer x to a special table of user defined primes and returns that table Whenever factor is subsequently called it will trial divide by the elements in this table If x is empty or omitted just returns the current list of extra primes The entries in x must be primes there is no internal check even if the factor_proven default is set To remove primes from the list use removeprimes The library syntax is GEN addprimes GEN x NULL 3 4 4 bestappr z B Using variants of the extended Euclidean algorithm returns a rational approximation a b to x whose denominator is limited by B if present If B is omitted return the best approximation affordable given the input accuracy if you are looking for true rational numbers presumably approximated to sufficient accuracy you should first try that option Otherwise B must be a positive real scalar impose 0 lt b lt B e If x is a t_REAL or a t_FRAC this function uses continued fractions bestappr Pi 100 1 22 7 bestappr 0 142
290. een powers 1 s 5 of the series s with polynomial coefficients of degree lt r In case no relation is found return 0 s 1 10 y 46xy 2 460xy 3 5658xy 4 77740xy 5 O y 6 seralgdep s 2 2 12 x72 8xy 2 20xy 1 subst x s 13 0 y76 seralgdep s 1 3 14 77 y72 20xy 1 x 310 y73 231xy72 30 y 1 seralgdep s 1 2 45 0 The series main variable must not be x so as to be able to express the result as a polynomial in x The library syntax is GEN seralgdep GEN s long p long r 251 3 8 66 setbinop f X Y The set whose elements are the f x y where x y run through X Y respectively If Y is omitted assume that X Y and that f is symmetric f x y f y x for all x y in X X 1 2 3 Y 2 3 4 setbinop x y gt x ty X Y set X Y 2 3 4 5 6 7 setbinop x y gt x y X Y set X Y 13 3 2 1 0 1 setbinop x y gt xt ty X Al set 2X X X 12 2 3 4 5 6 The library syntax is GEN setbinop GEN f GEN X GEN Y NULL 3 8 67 setintersect z y Intersection of the two sets x and y see setisset If x or y is not a set the result is undefined The library syntax is GEN setintersect GEN x GEN y 3 8 68 setisset x Returns true 1 if x is a set false 0 if not In PARI a set is a row vector whose entries are strictly increasing with respect to a somewhat arbitray universal comparison functio
291. en by a vector of polynomials modulo rnf po1 The reason why we do not return the customary HNF in terms of a fixed Z basis for Zr is precisely that no such basis has been explicitly specified On the other hand if you already computed an absolute nf structure Labs associated to L then xabs rnfidealup L x xLabs mathnf matalgtobasis Labs xabs computes a traditional HNF xLabs for x in terms of the fixed Z basis Labs zk The library syntax is GEN rnfidealup GEN rnf GEN x 3 6 149 rnfinit nf pol nf being a number field in nfinit format considered as base field and pol a polynomial defining a relative extension over nf this computes data to work in the relative extension The main variable of pol must be of higher priority see Section 2 5 3 than that of nf and the coefficients of pol must be in nf The result is a row vector whose components are technical In the following description we let K be the base field defined by nf and L K the large field associated to the rnf Furthermore we let m K Q the degree of the base field n L K the relative degree r and ra the number of real and complex places of K Acces to this information via member functions is preferred since the specific data organization specified below will change in the future rnf 1 rnf pol contains the relative polynomial pol rnf 2 contains the integer basis A d of K as integral elements of L Q More precisely A is a vect
292. en rethrowing whatever we do not like we advise to only trap errors of a specific kind as above Of course sometimes one just want to trap everything because we do not know what to expect The following function check whether install works correctly in your gp broken_install can we install iferr install addii GG ERR return 0S can we use the installed function iferr if addii 1 1 2 return BROKEN ERR return USE return 0 The function returns OS if the operating system does not support install USE if using an installed function triggers an error BROKEN if the installed function did not behave as expected and O if everything works The ERR formal parameter contains more useful data than just the error name which we recovered using errname ERR In fact a t_ERROR object usually has extra components which can be accessed as component ERR 1 component ERR 2 and so on Or globally by casting the error to a t_VEC Vec ERR returns the vector of all components at once See Section 3 11 17 for the list of all exception types and the corresponding contents of ERR 53 2 11 Interfacing GP with other languages The PARI library was meant to be interfaced with C programs This specific use is dealt with extensively in the User s guide to the PARI library Of course gp itself provides a convenient interpreter to execute rather intricate scripts see Section 3 11 Scripts when properly wri
293. ent is converted in the style d ddd e dd where there is one digit before the decimal point and the number of digits after it is equal to the precision if the precision is missing use the current realprecision for the total number of printed digits If the precision is explicitly 0 no decimal point character appears An E conversion uses the letter E rather than e to introduce the exponent e f F The real argument is converted in the style ddd ddd where the number of digits after the decimal point is equal to the precision if the precision is missing use the current realprecision for the total number of printed digits If the precision is explicitly 0 no decimal point character appears If a decimal point appears at least one digit appears before it e g G The real argument is converted in style e or f or E or F for G conversions ddd ddd where the total number of digits printed is equal to the precision if the precision is missing use the current realprecision If the precision is explicitly 0 it is treated as 1 Style e is used when the decimal exponent is lt 4 to print 0 or when the integer part cannot be decided given the known significant digits and the f format otherwise e c The integer argument is converted to an unsigned char and the resulting character is written e s Convert to a character string If a precision is given no more than the specified number of characters are written e p
294. ently power series of power series i e power series in several variables However power series of polynomials which are power series in several variables of a special type are OK This is a difficult design problem the mathematical problem itself contains some amount of imprecision and it is not easy to design an intuitive generic interface for such beasts 1 3 6 Strings These contain objects just as they would be printed by the gp calculator 1 3 7 Zero What is zero This is a crucial question in all computer systems The answer we give in PARI is the following For exact types all zeros are equivalent and are exact and thus are usually represented as an integer zero The problem becomes non trivial for imprecise types there are infinitely many distinct zeros of each of these types For p adics and power series the answer is as follows every such object including 0 has an exponent e This p adic or X adic zero is understood to be equal to O p or O X respectively Real numbers also have exponents and a real zero is in fact O 2 where e is now usually a negative binary exponent This of course is printed as usual for a floating point number 0 00 or 0 Exx depending on the output format and not with a O symbol as with p adics or power series With respect to the natural ordering on the reals we make the following convention whatever its exponent a real zero is smaller than any positive number and any two real zeroes are equal
295. equence whose value is set by the exception handler to help the recovery code decide what to do about the error For instance one can define a fault tolerant inversion function as follows inv x iferr 1 x ERR oo ERR is unused for i 1 1 print inv i 1 oo 1 Protected codes can be nested without adverse effect Let s now see how ERR can be used as written inv is too tolerant inv blah 42 go Let s improve it by checking that we caught a division by 0 exception and not an unrelated one like the type error 1 blah inv2 x iferr 1 x ERR if errname ERR e_INV error ERR oo inv2 0 13 oo as before inv2 blah at top level inv2 blah kkk o xxx in function inv2 f errname ERR e_INV error ERR oo0 kk ER A SEN 52 error forbidden division t_INT t_STR In the inv2 blah example the error type was not expected so we rethrow the exception error ERR triggers the original error that we mistakenly trapped Since the recovery code should always check whether the error is the one expected this construction is very common and can be simplified to inv3 x iferr 1 x ERR oo errname ERR e_INV More generally iferr statements ERR recovery predicate only catches the exception if predicate allowed to check various things about ERR not only its name is non zero Rather than trapping everything th
296. er nf but only if it is squarefree If it is reducible but squarefree the result will be the absolute equation of the tale algebra defined by pol If pol is not squarefree raise an e_DOMAIN exception rnfequation y 2 1 x72 y 1 x 4 1 T y73 2 rnfequation nfinit T x73 2 x Mod y T A2 x 6 108 Galois closure of Q 27 1 3 If flag is non zero outputs a 3 component row vector z a k where e z is the absolute equation of L over Q as in the default behavior e a expresses as a t_POLMOD modulo z a root a of the polynomial defining the base field nf k is a small integer such that 6 8 ka is a root of z where is a root of pol T y 3 2 pol x 2 x y y 2 z a k rnfequation T pol 1 14 x76 108 206 subst T y a 15 0 alpha Mod y T beta Mod x Mod 1 T pol subst z x beta k alpha 8 0 The library syntax is GEN rnfequation0 GEN nf GEN pol long flag Also available are GEN rnfequation GEN nf GEN pol flag 0 and GEN rnfequation2 GEN nf GEN pol flag 1 3 6 139 rnfhnfbasis bnf x Given bnf as output by bnfinit and either a polynomial x with coefficients in bnf defining a relative extension L of bnf or a pseudo basis x of such an extension gives either a true bnf basis of L in upper triangular Hermite normal form if it exists and returns 0 otherwise The library syntax is GEN rnfhnfbasis GEN bnf GEN x 3 6 140 rnfidealabs
297. er of significant words e t_INTMOD returns a random intmod for the same modulus e t_FFELT returns a random element in the same finite field e t_VEC of length 2 N a b returns an integer uniformly distributed between a and b e t_VEC generated by ellinit over a finite field k coefficients are t_INTMODs modulo a prime or t_FFELTs returns a random k rational affine point on the curve More precisely if the curve has a single point at infinity we return it otherwise we return an affine point by drawing an abscissa uniformly at random until el lordinate succeeds Note that this is definitely not a uniform distribution over E k but it should be good enough for applications e t_POL return a random polynomial of degree at most the degree of N The coefficients are drawn by applying random to the leading coefficient of N random 10 11 9 random Mod 0 7 12 Mod 1 7 a ffgen ffinit 3 7 a random a 3 a6 2xa75 a4 a 3 a2 2xa E ellinit 3 7 Mod 1 109 random E 4 Mod 103 109 Mod 10 109 E ellinit 1 7 ra 0 random E 5 a6 a75 2xa74 2xa72 2xa76 2xa 4 2xa73 a72 2xa random Mod 1 7 x 4 76 Mod 5 7 x 4 Mod 6 7 xx 3 Mod 2 7 x 2 Mod 2 7 x Mod 5 7 These variants all depend on a single internal generator and are independent from your oper ating system s random number generators A random seed may be obtained via getrand and re
298. ereas a single would just yield a not too helpful amp amp unknown identifier message Also you can ask for extended help on section number n in Chapter 3 just by typing n where n would yield merely a list of functions Finally a few key concepts in gp are documented in this way metacommands e g defaults e g psfile and type names e g t_INT or integer as well as various miscellaneous keywords such as edit short summary of line editor commands operator member user defined nf ell Last but not least without argument will open a dvi previewer xdvi by default GPXDVI if it is defined in your environment containing the full user s manual tutorial and refcard do the same with the tutorial and reference card respectively 56 Technical note This functionality is provided by an external perl script that you are free to use outside any gp session and modify to your liking if you are perl knowledgeable It is called gphelp lies in the doc subdirectory of your distribution just make sure you run Configure first see Appendix A and is really two programs in one The one which is used from within gp is gphelp which runs T X on a selected part of this manual then opens a previewer gphelp detex is a text mode equivalent which looks often nicer especially on a colour capable terminal see misc gprc dft for examples The default help selects which help program will be used from within gp You
299. eries E s A 1 E being an elliptic curve given by an arbitrary model over Q as output by ellinit this function computes the value of the L series of E at the complex point s This function uses an O N1 algorithm where N is the conductor The optional parameter A fixes a cutoff point for the integral and is best left omitted the result must be independent of A up to realprecision so this allows to check the function s accuracy The library syntax is GEN elllseries GEN E GEN s GEN A NULL long prec 142 3 5 31 ellminimalmodel amp v Return the standard minimal integral model of the rational elliptic curve E If present sets v to the corresponding change of variables which is a vector lu r s t with rational components The return value is identical to that of ellchangecurve E v The resulting model has integral coefficients is everywhere minimal a is 0 or 1 az is 0 1 or 1 and az is 0 or 1 Such a model is unique and the vector v is unique if we specify that u is positive which we do The library syntax is GEN ellminimalmodel GEN E GEN v NULL 3 5 32 ellmodularegn N x y Return a vector eqn t where eqn is a modular equation of level N i e a bivariate polynomial with integer coefficients t indicates the type of this equation either canonical t 0 or Atkin t 1 This function currently requires the package seadata to be installed and is limited to N lt 500 N prime Let j be t
300. eriods 7 and na such that m 2 w 2 where is the Weierstrass zeta function associated to the period lattice see ellzeta In particular the Legendre relation holds mwi mw 27 Warning As for the orientation of the basis of the period lattice beware that many sources use the inverse convention where w2 w has positive imaginary part and our wa is the negative of theirs Our convention T w1 wa ensures that the action of PSLa is the natural one la b c d T ar b cr d au bw2 cw1 dw instead of a twisted one Our tau is 1 7 in the above inverse convention 3 5 1 3 Curves over Q We advise to input a model defined over Q for such curves In any case if you input an approximate model with t_PADIC coefficients it will be replaced by a lift to Q an exact model close to the one that was input and all quantities will then be computed in terms of this lifted model For the time being only curves with multiplicative reduction split or non split i e Up Jj lt 0 are supported by non trivial functions In this case the curve is analytically isomorphic to Q qo E Q for some p adic integer q the Tate period In particular we have j q j e p is the residual characteristic e roots is a vector with a single component equal to the p adic root es of the right hand side g x of the associated b model Y g x The point e 0 corresponds to 1 Q q under the Tate parametrizatio
301. ernative inputs e g ellinit 11a1 Functions using this package need to load chunks of a large database in memory and require at least 2MB stack to avoid stack overflows e gen returns the generators of E Q if known from John Cremona s database 3 5 2 ellL1 e r Returns the value at s 1 of the derivative of order r of the L function of the elliptic curve e assuming that r is at most the order of vanishing of the L function at s 1 The result is wrong if r is strictly larger than the order of vanishing at 1 e ellinit 11a1 order of vanishing is 0 ellLi e 0 12 0 2538418608559106843377589233 e ellinit 389a1 order of vanishing is 2 ellLi e 0 4 5 384067311837218089235032414 E 29 ellLi e 1 45 0 ellLi e 2 6 1 518633000576853540460385214 The main use of this function after computing at low accuracy the order of vanishing using el lanalyticrank is to compute the leading term at high accuracy to check or use the Birch and Swinnerton Dyer conjecture Xp18 realprecision 18 significant digits 132 ellanalyticrank ellinit 0 0 1 7 6 time 32 ms 1 3 10 3910994007158041 p200 realprecision 202 significant digits 200 digits displayed ellLi e 3 time 23 113 ms 3 10 3910994007158041387518505103609170697263563756570092797 The library syntax is GEN e11L1 GEN e long r long prec 3 5 3 elladd E z1 22 Sum of the points z1 and 22
302. erse deg minpoly z lt 4 Break loop type break to go back to GP prompt break gt Vec dbg_err ask for more info e_DOMAIN modreverse deg minpoly z lt 4 Mod x73 9 x x74 10 x72 1 break gt minpoly u x 2 8 The library syntax is GEN modreverse GEN z 3 6 75 newtonpoly z p Gives the vector of the slopes of the Newton polygon of the polynomial x with respect to the prime number p The n components of the vector are in decreasing order where n is equal to the degree of x Vertical slopes occur iff the constant coefficient of x is zero and are denoted by LONG_MAX the biggest single precision integer representable on the machine 2 1 resp 263 1 on 32 bit resp 64 bit machines see Section 3 2 55 The library syntax is GEN newtonpoly GEN x GEN p 184 3 6 76 nfalgtobasis nf x Given an algebraic number x in the number field nf transforms it to a column vector on the integral basis nf zk nf nfinit y 2 4 nf zk 12 1 1 2 y nfalgtobasis nf 1 1 43 1 1 nfalgtobasis nf y 4 0 2 nfalgtobasis nf Mod y y 2 4 4 0 2 This is the inverse function of nfbasistoalg The library syntax is GEN algtobasis GEN nf GEN x 3 6 77 nfbasis T Let T X be an irreducible polynomial with integral coefficients This function returns an integral basis of the number field defined by T that is a Z basis of its maximal order The basi
303. erstand them as permutations and compose them The library syntax is GEN gmul GEN x GEN y for x y Also available is GEN gsqr GEN x for x x 3 1 5 The expression x y is the quotient of x and y In addition to the impossibilities for multiplication note that if the divisor is a matrix it must be an invertible square matrix and in that case the result is y Furthermore note that the result is as exact as possible in particular division of two integers always gives a rational number which may be an integer if the quotient is exact and not the Euclidean quotient see x y for that and similarly the quotient of two polynomials is a rational function in general To obtain the approximate real value of the quotient of two integers add 0 to the result to obtain the approximate p adic value of the quotient of two integers add O p k to the result finally to obtain the Taylor series expansion of the quotient of two polynomials add 0 X k to the result or use the taylor function see Section 3 7 45 The library syntax is GEN gdiv GEN x GEN y for x y 65 3 1 6 The expression x y is the Euclidean quotient of x and y If y is a real scalar this is defined as floor x y if y gt 0 and ceil a y if y lt 0 and the division is not exact Hence the remainder x x y y is in 0 y Note that when y is an integer and x a polynomial y is first promoted to a polynomial of degree 0 When zx is a vector or matrix
304. es a oA A A 5 LIDIA e aura pe ey ae eS GUA a 110 Liber Be Hom we ew dodo i 78 81 PTE GO Fama NN 81 liftall eee eee EL 82 Liftint e e s sce eee ee aw Rows 81 82 TATED OM eoe eh eect hate he We Bond We a 81 82 LMC orar 43 o ecr ass bak he ew i 228 231 232 lind6pO rcar 6246 2 6 2 capea t 232 lind p2 25 2 624645 262 erto a 232 line editor o 60 linear dependence 231 TINGS ce Gb ee bh ok bh ee Be eek 307 lijenrap vee ae snag ae 307 DiS Daet ig sata 8 es die we eRe Se G 54 LSC a occ a we we HE a a 7 26 Literae wee ala neh Be es So a 71 Tistereat ssa ous wile Ne SK a ek T1 292 ITSTINASO Ducado do we E A 232 listkRill Loira sea e 202 LLISTDOD oe oe a ro o ee iio 232 listp t 260 oe at tads pap da 232 TISESOFG 1 4654 s a aoa 2 264 230 202 CED oi ee eS 127 181 191 237 241 248 LL oia ran is a a a i 248 ILLETAN ooe ae aee te ae eh E ae S 249 IA A eor aoma e a aE a 249 lllgramkerim 249 DUT omic a a A Ges je a 248 A Ade Se He Hae 6 248 lngamMa 94 local seere be ranma ra eA 34 LOG 2 pee a KE ee SE 56 57 95 297 307 logfile oca Rik a Be Bee A 297 Logfile esa ee bea ewe Re eS e 308 LOB DE 225 ea Pe ee crasas 118 LOGINCO w see Wide es ok gea arie AS a 118 LONG MAX ouvir creara 83 87 184 LUQUE ee he rta Ea Se eo 28 30 IVALUE i ipaa ar y we Ue a ada A 28 M Mat 2 bea eee eh ees 24 26 71 230 238 Matadjoint 229 233 M
305. es low precision floating point computations for maximal speed which gives incorrect result when x has large entries The condition is checked in the code and the routine raises an error if large rounding errors occur A more robust but much slower implementation is chosen if the following flag is used If flag 2 x can have non integral real entries In this case if b is omitted the minimal vectors only have approximately the same norm If bis omitted mis an upper bound for the number of vectors that will be stored and returned but all minimal vectors are nevertheless enumerated If m is omitted all vectors found are stored and returned note that this may be a huge vector x matid 2 qfminim x 4 minimal vectors of norm 1 0 1 1 0 12 4 1 0 1 1 0 x 4 2 0 0 0 2 0 0 0 0 0 O 1 1 0 0 O 1 0 1 0 O 0 2 2 4 2 2 0 2 0 0 0 O O O 0 1 O O O O 0 1 O 1 1 1 0 2 4 0 2 0 0 0 O O O 0 1 1 0 O 1 O O 1 1 1 O O 0 2 0 4 0 0 0 0 0 O O O 1 1 O O O 1 1 O 1 1 1 0 0 0 2 0 4 0 0 O 1 1 0 O 1 0 O 0 2 O 0 1 1 1 0 O 2y 240 0 0 4 2 0113 07 0 05 Os 05 0 05 05 0 1 1 0 1 1 1 0 0 0 0 0 2 4 2 0 0 0 0 O 1 O O 0 1 O O O 1 1 0 0 0 O O O 0 2 4 O O O 0 1 O O O O 0 1 1 1 O 1 0 0 0 0 O 1 1 O O 4 0 2 O 1 1 0 1 0 1 0 0 O O O
306. es the value at z of the Weierstrass function attached to the lattice w as given by ellperiods 1 including quasi periods is useful otherwise there are recomputed from scratch for each new z a 1 z L ITE gt w2 z w It is also possible to directly input w w wa or an elliptic curve E as given by ellinit w E omega The quasi periods of such that C z aw bw C z an bnz for integers a and b are obtained as 7 2 w 2 Or using directly elleta w ellperiods 1 1 1 ellzeta w 1 2 12 1 5707963267948966192313216916397514421 E ellinit 1 0 ellzeta E E omega 1 2 14 0 84721308479397908660649912348219163647 One can also compute the series expansion around z 0 the quasi periods are useless in this case E ellinit 0 11 ellzeta E Ml at x implicitly at default seriesprecision 14 x 1 1 35 x 5 1 7007 x 11 O x715 ellzeta E x 0 x720 explicit precision 15 x 1 1 35 x75 1 7007 x711 1 1440257 x717 0 x 18 The library syntax is GEN ellzeta GEN w GEN z NULL long prec 3 5 50 ellztopoint E z E being an ell as output by ellinit computes the coordinates x y on the curve E corresponding to the complex number z Hence this is the inverse function of ellpointtoz In other words if the curve is put in Weierstrass form y 4x gox gs x y represents the Weierstrass g function and its derivative More precisely we
307. ese two take place in Q y x x Mod 1 y 13 Mod 1 y x M n Q y yQ y z Qlz Mod x y 74 0 In the last example the division by y takes place in Q y x hence the Mod object is a coset in Q y z yQ y z which is the null ring since y is invertible So be very wary of variable ordering when your computations involve implicit divisions and many variables This also affects functions like numerator denominator or content denominator x y hi 1 denominator y x 42 x content x y 3 1 y content y x 74 y content 2 x 15 2 Can you see why Hint x y 1 y x x is in Q y r and denominator is taken with respect to Q y x y x y x zx is in Q y z so y is invertible in the coefficient ring On the other hand 2 x involves a single variable and the coefficient ring is simply Z These problems arise because the variable ordering defines an implicit variable with respect to which division takes place This is the price to pay to allow and operators on polynomials instead of requiring a more cumbersome divrem z y var which also exists Unfortunately in some functions like content and denominator there is no way to set explicitly a main variable like in divrem and remove the dependence on implicit orderings This will hopefully be corrected in future versions 33 2 5 4 Multivariate power series Just like multivariate polynomials power series are funda mentall
308. etting this option also set enable tls see below This option requires the pthread library For benchmarking it is often useful to set time ftime so that GP report wall clock instead of the sum of the time spent by each threads e mpi use the MPI interface to parallelism This allows to take advantage of clusters using MPI This option requires a MPI library It is usually necessary to set the environment variable CC to mpicc enable tls build the thread safe version of the library Implied by mt pthread This tends to slow down the shared library libpari so by about 15 so you probably want to use the static library libpari a instead 2 3 Compilation To compile the gp binary and build the documentation type make all To only compile the gp binary type make gp in the toplevel directory If your make program supports parallel make you can speed up the process by going to the build directory that Configure created and doing a parallel make here for instance make j4 with GNU make It should even work from the toplevel directory 2 4 Basic tests To test the binary type make bench This runs a quick series of tests for a few seconds on modern machines In many cases this will also build a different binary named gp sta or gp dyn linked in a slightly different way and run the tests with both In exotic configurations one may pass all the tests while the other fails and we want to check for this To test only the defa
309. eturns the ideal of K below x i e the intersection of x with K The library syntax is GEN rnfidealdown GEN rnf GEN x 3 6 142 rnfidealhnf rnf x rnf being a relative number field extension L K as output by rn finit and z being a relative ideal which can be as in the absolute case of many different types including of course elements computes the HNF pseudo matrix associated to x viewed as a Zg module The library syntax is GEN rnfidealhnf GEN rnf GEN x 3 6 143 rnfidealmul rnf x y rnf being a relative number field extension L K as output by rnfinit and z and y being ideals of the relative extension L K given by pseudo matrices outputs the ideal product again as a relative ideal The library syntax is GEN rnfidealmul GEN rnf GEN x GEN y 3 6 144 rnfidealnormabs rnf x Let rnf be a relative number field extension L K as output by rnfinit and let x be a relative ideal which can be as in the absolute case of many different types including of course elements This function computes the norm of the x considered as an ideal of the absolute extension L Q This is identical to idealnorm rnf rnfidealnormrel rnf x but faster The library syntax is GEN rnfidealnormabs GEN rnf GEN x 3 6 145 rnfidealnormrel rnf x Let rnf be a relative number field extension L K as output by rnfinit and let x be a relative ideal which can be as in the absolute case of many different types including of course elements This fun
310. etween a scalar type x and a t_COL or t_MAT y returns respectively y 1 x y 2 and y xId Other additions between a scalar type and a vector or a matrix or between vector matrices of incompatible sizes are forbidden The library syntax is GEN gadd GEN x GEN y 3 1 3 The expression x y is the difference of x and y Subtraction between a scalar type x and a t_COL or t_MAT y returns respectively y 1 x y 2 and y xId Other subtractions between a scalar type and a vector or a matrix or between vector matrices of incompatible sizes are forbidden The library syntax is GEN gsub GEN x GEN y for x y 3 1 4 The expression x y is the product of x and y Among the prominent impossibilities are multiplication between vector matrices of incompatible sizes between a t_INTMOD or t_PADIC Re stricted to scalars is commutative because of vector and matrix operations it is not commutative in general Multiplication between two t_VECs or two t_COLs is not allowed to take the scalar product of two vectors of the same length transpose one of the vectors using the operator or the function mattranspose see Section 3 8 and multiply a line vector by a column vector a 1 2 3 Para eK at top level a a kK o x _ _ forbidden multiplication t_VEC t_VEC ak a 12 14 If x y are binary quadratic forms compose them see also qfbnucomp and qfbnupow If x y are t_VECSMALL of the same length und
311. f a prime power q one may input directly the polynomial P monic ir reducible with t_INTMOD coefficients and the function returns the generator g X mod P X inferring p from the coefficients of P If v is given the variable name is used to display g else the variable of the polynomial P is used If P is not irreducible we create an invalid object and behaviour of functions dealing with the resulting t_FFELT is undefined in fact it is much more costly to test P for irreducibility than it would be to produce it via ffinit The library syntax is GEN ffgen GEN q long v 1 where v is a variable number To create a generator for a prime finite field the function GEN p_to_GEN GEN p long v returns 1 ffgen x Mod 1 p v 3 4 28 ffinit p n v x Computes a monic polynomial of degree n which is irreducible over Fp where p is assumed to be prime This function uses a fast variant of Adleman and Lenstra s algorithm It is useful in conjunction with ffgen for instance if P ffinit 3 2 you can represent elements in F32 in term of g ffgen P t This can be abbreviated as g ffgen 3 2 t where the defining polynomial P can be later recovered as g mod The library syntax is GEN ffinit GEN p long n long v 1 where vis a variable num ber 3 4 29 fllog x g o Discrete logarithm of the finite field element x in base g i e ane in Z such that gf o If present o represents the multiplicative order of g see Se
312. factorization difficulties In the latter case if you must use a partial discriminant factorization as allowed by both nfdisc or nfbasis to build a partially correct nf structure always input nf pol to nffactor and not your makeshift nf otherwise factors could be missed The library syntax is GEN nffactor GEN nf GEN T 3 6 98 nffactorback nf f e Gives back the nf element corresponding to a factorization The integer 1 corresponds to the empty factorization If e is present e and f must be vectors of the same length e being integral and the corre sponding factorization is the product of the ffi If not and f is vector it is understood as in the preceding case with e a vector of 1s we return the product of the f i Finally f can be a regular factorization matrix nf nfinit y 2 1 nffactorback nf 3 y 1 1 21 1 1 2 3 2 12 66 3 1 1 72 1 2 1 3 43 12 66 I The library syntax is GEN nffactorback GEN nf GEN f GEN e NULL 3 6 99 nffactormod nf Q pr Factors the univariate polynomial Q modulo the prime ideal pr in the number field nf The coefficients of Q belong to the number field scalar polmod polynomial even column vector and the main variable of nf must be of lower priority than that of Q see Section 2 5 3 The prime ideal pr is either in idealprimedec or preferred modprinit format The coefficients of the polynomial factors are lifted to elements of nf K nfi
313. fficient explosion unless the base field is finite or R If flag 3 uses Berkowitz s division free algorithm valid over any ring commutative with unit Uses O n scalar operations If flag 4 x must be integral Uses a modular algorithm Hessenberg form for various small primes then Chinese remainders If flag 5 default uses the best method given x This means we use Berkowitz unless the base ring is Z use flag 4 or a field where coefficient explosion does not occur e g a finite field or the reals use flag 2 The library syntax is GEN charpoly0 GEN A long v 1 long flag where v is a vari able number Also available are GEN charpoly GEN x long v flag 5 GEN caract GEN A 229 long v flag 1 GEN carhess GEN A long v flag 2 GEN carberkowitz GEN A long v flag 3 and GEN caradj GEN A long v GEN pt In this last case if pt is not NULL pt receives the address of the adjoint matrix of A see matadjoint so both can be obtained at once 3 8 3 concat z y Concatenation of x and y If x or y is not a vector or matrix it is considered as a one dimensional vector All types are allowed for x and y but the sizes must be compatible Note that matrices are concatenated horizontally i e the number of rows stays the same Using transpositions one can concatenate them vertically but it is often simpler to use matconcat x matid 2 y 2 matid 2 concat x y 2 1 0 2
314. g a dash is an alias for a fully qualified kernel name An alias stands for name none but gmp stands for auto gmp 3 5 Problems related to readline Configure does not try very hard to find the readline library and include files If they are not in a standard place it will not find them You can invoke Configure with one of the following arguments with readline prefix to 1ib libreadline rz and include readline h with readline lib path to libreadline zzx with readline include path to readline h 319 Known problems e on Linux Linux distributions have separate readline and readline devel packages You need both of them installed to compile gp with readline support If only readline is installed Configure will complain Configure may also complain about a missing libncurses so in which case you have to install the ncurses devel package some distributions let you install readline devel without ncurses devel which is a bug in their package dependency handling e on OS X 4 or higher these systems comes equipped with a fake readline which is not sufficient for our purpose As a result gp is built without readline support Since readline is not trivial to install in this environment a step by step solution can be found in the PARI FAQ see http pari math u bordeaux fr 3 6 Testing 3 6 1 Known problems if BUG shows up in make bench e program the GP function install may not be available on your platform t
315. g n long fl long v GEN polchebyshevi long n long v and GEN polchebyshev2 long n long v for T and U respectively 3 7 11 polcoeff z n v Coefficient of degree n of the polynomial x with respect to the main variable if v is omitted with respect to v otherwise If n is greater than the degree the result is Zero Naturally applies to scalars polynomial of degree 0 as well as to rational functions whose denominator is a monomial It also applies to power series if n is less than the valuation the result is zero If it is greater than the largest significant degree then an error message is issued For greater flexibility x can be a vector or matrix type and the function then returns compo nent x n The library syntax is GEN polcoeff0 GEN x long n long v 1 where v is a variable number 219 3 7 12 polcyclo n a xP n th cyclotomic polynomial evaluated at a x by default The integer n must be positive Algorithm used reduce to the case where n is squarefree to compute the cyclotomic polyno mial use D 1 x x to compute it evaluated use D 1 Taj 2 1 4 4 In the evaluated case the algorithm assumes that at 1 is either 0 or invertible for all d n If this is not the case the base ring has zero divisors use subst polcyclo n x a The library syntax is GEN polcyclo_eval long n GEN a NULL The variant GEN polcy clo long n long v returns the n th cyclotom
316. ge s E has one component 1 t_STR the package name s 284 Syntax errors type errors e e_DIM The dimensions of arguments x and y submitted to function s does not match up E g multiplying matrices of inconsistent dimension adding vectors of different lengths E has three component 1 t_STR the function name s 2 the argument x 3 the argument y e e FLAG A flag argument is out of bounds in function s E has one component 1 t_STR the function name s e e NOTFUNC Generated by the PARI evaluator tried to use a GEN x which is not a t_CLOSURE in a function call syntax as in f 1 2 E has one component 1 the offending GEN z e e OP Impossible operation between two objects than cannot be typecast to a sensible common domain for deeper reasons than a type mismatch usually for arithmetic reasons As in 0 2 0 3 it is valid to add two t_PADICs provided the underlying prime is the same so the addition is not forbidden a priori for type reasons it only becomes so when inspecting the objects and trying to perform the operation E has three components 1 t_STR the operator name op 2 first argument 3 second argument e e_TYPE An argument x of function s had an unexpected type As in factor blah E has two components 1 t_STR the function name s 2 the offending argument z e e_TYPE2 Forbidden operation between two objects than cannot be typecast to a sensible common domain because
317. general better to use intmellininvshort p 105 intmellininv s 2 4 gamma s 3 time 1 190 ms reasonable p 308 intmellininv s 2 4 gamma s 3 time 51 300 ms slow because of I s The library syntax is intmellininv void E GEN eval void GEN GEN sig GEN z GEN tab long prec 3 9 9 intmellininvshort sig z tab Numerical integration of 2i7 7 s X z with respect to X on the line R X sig In other words inverse Mellin transform of s X at the value z Here s X is implicitly contained in tab in intfuncinit format typically tab intfuncinit T 1 1 s sig I T or similar commands Take the example of the inverse Mellin transform of TP s given in int mellininv Ap 105 oo 1 for clarity 7 A intmellininv s 2 4 gamma s 3 time 2 500 ms not too fast because of T s function of real type decreasing as exp 37 2 t tab intfuncinit t 00 3 Pi 2 o0 3 Pi 2 gamma 2 I t 73 1 time 1 370 ms intmellininvshort 2 4 tab A time 50 ms 14 1 26 3 25 E 109 I 50 times faster than A and perfect tab2 intfuncinit t 00 oo gamma 2 1 t 73 1 intmellininvshort 2 4 tab2 76 1 2 E 42 3 2 E 109 I 63 digits lost In the computation of tab it was not essential to include the exact exponential decrease of T 2 it But as the last example shows a rough indication must be given otherwise slow decrease is
318. global functions which have access to the lexical variables private to init The following would work in exactly the same way init5 my x 5 add y x y mul y x y 2 7 7 Closures as Objects Contrary to what you might think after the preceding examples GP s closures may not be used to simulate true objects with private and public parts and methods to access and manipulate them In fact closures indeed incorporate an existing context they may access lexical variables that existed at the time of their definition but then may not change it More precisely they access a copy which they are welcome to change but a further function call still accesses the original context as it existed at the time the function was defined initO my count 0 inc count dec count inc fyi 1 inc 12 1 inc 3 1 2 8 Member functions Member functions use the dot notation to retrieve information from complicated structures The built in structures are bid ell galois ff nf bnf bnr and prid which will be described at length in Chapter 3 The syntax structure member is taken to mean retrieve member from structure e g E j returns the j invariant of the elliptic curve E or outputs an error message if E is not a proper ell structure To define your own member functions use the syntax var member seq where the formal variable var is scoped to the function body seq This is
319. gpre In older versions gphelp was hidden in pari lib directory and was not meant to be used from the shell prompt but not anymore If gp complains it cannot find gphelp check whether your gprc or the system wide gprc does contain explicit paths If so correct them according to the current misc gprc dft 5 Getting Started 5 1 Printable Documentation Building gp with make all also builds its documentation You can also type directly make doc In any case you need a working plain T X installation After that the doc directory contains various dvi files libpari dvi manual for the PARI library users dvi manual for the gp calculator tutorial dvi a tutorial and refcard dvi a reference card for GP You can send these files to your favourite printer in the usual way probably via dvips The reference card is also provided as a PostScript document which may be easier to print than its dvi equivalent it is in Landscape orientation and assumes A4 paper size If pdftex is part of your TRX setup you can produce these documents in PDF format which may be more convenient for online browsing the manual is complete with hyperlinks type make docpdf All these documents are available online from PARI home page see the last section 5 2 C programming Once all libraries and include files are installed you can link your C programs to the PARI library A sample makefile examples Makefile is provided to illustrate the use of
320. gt sig or sig 1 Warning This function uses the intnum routines and is orders of magnitude slower than sumalt It is only given for completeness and should not be used in practice 270 Warning 2 The expression expr must not include the 1 coefficient Thus sumalt n a 1 f n is approximately equal to sumnumalt n a sig f n sig is coded as in sumnum However for slowly decreasing functions where sig is coded as o a with a lt 1 it is not really important to indicate a In fact as for sumalt the program will often give meaningful results usually analytic continuations even for divergent series On the other hand the exponential decrease must be indicated tab is as in intnum but if used must be initialized with sumnuminit If flag is nonzero assumes that the function f to be summed is of real type i e satisfies f z f Z and then twice faster when tab is precomputed Xp 308 tab sumnuminit 2 omitted 1 abscissa o 2 alternating sums time 1 620 ms slow but done once and for all a sumnumalt n 1 2 1 n 3 n 1 tab 1 time 230 ms similar speed to sumnum b sumalt n 1 1 7n n 3 n 1 time 0 ms infinitely faster a b time O ms 1 1 66 E 308 perfect The library syntax is sumnumalt void E GEN eval void GEN GEN a GEN sig GEN tab long flag long prec 3 9 26 sumnuminit sig m 0 sgn 1 Initialize table
321. h has integral coefficients e Finally MDI is a two element representation for faster ideal product of d A times the codifferent ideal nf discx nf codiff which is an integral ideal MDI is only used in idealinv nf 6 is the vector containing the r1 r2 roots nf roots of nf 1 corresponding to the rl r2 embeddings of the number field into C the first r1 components are real the next r2 have positive imaginary part nf 7 is an integral basis for Zg nf zk expressed on the powers of 6 Its first element is guaranteed to be 1 This basis is LLL reduced with respect to To strictly speaking it is a permutation of such a basis due to the condition that the first element be 1 nf 8 is the n x n integral matrix expressing the power basis in terms of the integral basis and finally nf 9 is the n x n matrix giving the multiplication table of the integral basis If a non monic polynomial is input nfinit will transform it into a monic one then reduce it see flag 3 It is allowed though not very useful given the existence of nfnewprec to input a nf or a bnf instead of a polynomial nf nfinit x 3 12 initialize number field Q X X73 12 nf pol defining polynomial 12 x73 12 193 nf disc field discriminant 13 972 nf index index of power basis order in maximal order 14 2 nf zk integer basis lifted to Q X 5 1 x 1 2 x 2 nf sign signature 6 1 1
322. h level plotting functions all the functions starting with ploth in which the user has little to do but explain what type of plot he wants and whose syntax is similar to the one used in the preceding section 3 10 2 Low level plotting functions called rectplot functions sharing the prefix plot where every drawing primitive point line box etc is specified by the user These low level functions work as follows You have at your disposal 16 virtual windows which are filled independently and can then be physically ORed on a single window at user defined positions These windows are numbered from 0 to 15 and must be initialized before being used by the function plotinit which specifies the height and width of the virtual window called a rectwindow in the sequel At all times a virtual cursor initialized at 0 0 is associated to the window and its current value can be obtained using the function plotcursor A number of primitive graphic objects called rect objects can then be drawn in these win dows using a default color associated to that window which can be changed using the plotcolor function and only the part of the object which is inside the window will be drawn with the ex ception of polygons and strings which are drawn entirely The ones sharing the prefix plotr draw relatively to the current position of the virtual cursor the others use absolute coordinates Those having the prefix plotrecth put in the rectwindow a large
323. h returns 1 when it calls f O the binding stack for the x identifier contains two items the global binding to 0 and the binding to 1 introduced in h which is still present on the stack since the control flow has not left h yet 2 6 1 Scoping rules Named parameters in a function definition as well as all loop indices have lexical scope within the function body and the loop body respectively p 0 forprime p 2 11 print p p prints O at the end x 0 f x x f 1 returns 2 and leave global x unaffected 0 If you exit the loop prematurely e g using the break statement you must save the loop index in another variable since its value prior the loop will be restored upon exit For instance for i 1 n if ok i break if i gt n return failure The names are borrowed from the Perl scripting language More generally in all iterative constructs which use a variable name for prod sum vector matrix plot etc the given variable is lexically scoped to the construct s body 35 is incorrect since the value of i tested by the i gt n is quite unrelated to the loop index One ugly workaround is for i 1 n if ok i isave i break if isave gt n return failure But it is usually more natural to wrap the loop in a user function and use return instead of break try for i 1 n if ok i return i J3 0 failure A list of variables can be
324. h root of unity K nfinit polcyclo 11 nfrootsof1 K he 22 0 0 O O O 1 O 0 O 015 z nfbasistoalg K 2 MA in algebraic form 43 Mod x75 x710 x 9 x 8 x 7 x76 x75 x74 4 x 3 x 2 x 1 lift z711 lift 272 proves that the order of z is 22 4 1 x 9 x78 x 7 x76 x 5 x 4 x 3 x 2 x 1 This function guesses the number w as the gcd of the k v for unramified v above odd primes then computes the roots in nf of the w th cyclotomic polynomial the algorithm is polynomial time with respect to the field degree and the bitsize of the multiplication table in nf both of them polynomially bounded in terms of the size of the discriminant Fields of degree up to 100 or so should require less than one minute The library syntax is GEN rootsof1 GEN nf Also available is GEN rootsof1_kannan GEN nf that computes all algebraic integers of T norm equal to the field degree all roots of 1 by Kronecker s theorem This is in general a little faster than the default when there are roots of 1 in the field say twice faster but can be much slower say days slower since the algorithm is a priori exponential in the field degree 3 6 114 nfsnf nf x Given a Zkx module x associated to the integral pseudo matrix A I J returns an ideal list d d which is the Smith normal form of x In other words x is isomorphic to ZK d ZK d and d divides d _ for i
325. h x but with its content content 2 4 matid 3 11 2 The library syntax is GEN content GEN x 3 4 11 contfrac x b nmaz Returns the row vector whose components are the partial quo tients of the continued fraction expansion of x In other words a result ag a means that x ay 1 a 1 a The output is normalized so that a 4 1 unless we also have n 0 The number of partial quotients n 1 is limited by nmax If nmax is omitted the expansion stops at the last significant partial quotient p19 realprecision 19 significant digits contfrac Pi 1 3 7 15 1 292 1 1 contfrac Pi 3 Mn 2 42 3 7 15 1525 0 Bs 1 14 Dy As sty 12021 x can also be a rational function or a power series If a vector b is supplied the numerators are equal to the coefficients of b instead of all equal to 1 as above more precisely x 1 bp a0 b1 a1 b a for a numerical continued fraction x real the a are integers as large as possible if x is a rational function they are polynomials with deg a degb 1 The length of the result is then equal to the length of b unless the next partial quotient cannot be reliably computed in which case the expansion stops This happens when a partial remainder is equal to zero or too small compared to the available significant digits for x a t_REAL A direct implementation of the numerical continued fraction contfrac x b described above wou
326. have z p z b2 12 y z az as 2 If z is in the lattice defining E over C the result is the point at infinity 0 The library syntax is GEN pointell GEN E GEN z long prec 3 5 51 genus2red Q P p Let Q P be polynomials with integer coefficients Determines the reduction at p gt 2 of the proper smooth genus 2 curve C Q defined by the hyperelliptic equation y Qy P The special fiber X of the minimal regular model X of C over Z If p is omitted determines the reduction type for all odd prime divisors of the discriminant This function rewritten from an implementation of Liu s algorithm by Cohen and Liu 1994 genus2reduction 0 3 see http www math u bordeaux fr liu G2R 149 CAVEAT The function interface may change for the time being it returns N FaN T V where N is either the local conductor at p or the global conductor FaN is its factorization y T defines a minimal model over Z 1 2 and V describes the reduction type at the various considered p Unfortunately the program is not complete for p 2 and we may return the odd part of the conductor only this is the case if the factorization includes the impossible term 271 if the factorization contains another power of 2 then this is the exact local conductor at 2 and N is the global conductor default debuglevel 1 genus2red 0 x 6 3 x 3 63 3 potential stable reduction 1 reduction at p III 9 page 184
327. hcolors Entries in the graphcolormap that will be used to plot multi curves The successive curves are drawn in colors graphcolormap graphcolors 1 graphcolormap graphcolors 2 cycling when the graphcolors list is exhausted The default value is 4 5 3 14 15 help Name of the external help program to use from within gp when extended help is invoked usually through a or request see Section 2 13 1 or M H under readline see Section 2 15 The default value is the path to the gphelp script we install 3 14 16 histfile Name of a file where gp will keep a history of all input commands results are omitted If this file exists when the value of histfile changes it is read in and becomes part of the session history Thus setting this default in your gprc saves your readline history between sessions Setting this default to the empty string changes it to lt undefined gt The default value is lt undefined gt no history file 3 14 17 histsize gp keeps a history of the last histsize results computed so far which you can recover using the notation see Section 2 13 4 When this number is exceeded the oldest values are erased Tampering with this default is the only way to get rid of the ones you do not need anymore The default value is 5000 3 14 18 lines If set to a positive value gp prints at most that many lines from each result terminating the last line shown with if further material has been supp
328. he variables to more restricted blocks than the whole function body Arguments are passed by value not as variables modifying a function s argument in the function body is allowed but does not modify its value in the calling scope In fact a copy of the actual parameter is assigned to the formal parameter when the function is called Formal parameters are lexically scoped to the function body It is not allowed to use the same variable name for different parameters of your function f x x 1 xx variable declared twice f x x 1 CK ata Finishing touch You can add a specific help message for your function using addhelp but the online help system already handles it By default name will print the definition of the function name the list of arguments as well as their default values the text of seq as you input it Just as c prints the list of all built in commands u outputs the list of all user defined functions Backward compatibility lexical scope Lexically scoped variables were introduced in ver sion 2 4 2 Before that the formal parameters were dynamically scoped If your script depends on this behavior you may use the following trick replace the initial f x by f x_orig local x x_orig 38 Backward compatibility disjoint namespaces Before version 2 4 2 variables and functions lived in disjoint namespaces and it was not possible to have a variable and a function share the same name Hence the need for
329. he cardinality of the multiplicative group of the underlying finite field t ffgen ffinit nextprime 1078 5 g ffprimroot t amp o o will be useful fforder g 1000000 o time O ms 5 5000001750000245000017150000600250008403 fforder g 1000000 time 16 ms noticeably slower same result of course 16 5000001750000245000017150000600250008403 The library syntax is GEN fforder GEN x GEN o NULL 3 4 32 ffprimroot z amp o Return a primitive root of the multiplicative group of the definition field of the finite field element x not necessarily the same as the field generated by x If present o is set to a vector ord fa where ord is the order of the group and fa its factorisation factor ord This last parameter is useful in fflog and fforder see Section 3 4 2 t ffgen ffinit nextprime 1077 5 g ffprimroot t ko o 1 13 100000950003610006859006516052476098 o 2 LA 2 1 112 7 2 31 1 41 1 67 1 1523 1 10498781 1 15992881 1 46858913131 1 fflog g 1000000 g o time 1 312 ms 5 1000000 The library syntax is GEN ffprimroot GEN x GEN o NULL 3 4 33 fibonacci z zt Fibonacci number The library syntax is GEN fibo long x 3 4 34 gcd z y Creates the greatest common divisor of x and y If you also need the u and v such that zx u yx v gcd z y use the bezout function x and y can have rather quite general types
330. he inner structure of A complete if n is omitted up to level n otherwise This is useful for debugging This is similar to x but does not require A to be an history entry In particular it can be used in the break loop 3 11 7 for X a b seq Evaluates seq where the formal variable X goes from a to b Nothing is done if a gt b a and b must be in R 3 11 8 forcomposite n a b seq Evaluates seg where the formal variable n ranges over the composite numbers between the non negative real numbers a to b including a and b if they are composite Nothing is done if a gt b forcomposite n 0 10 print n 4 6 8 9 10 Omitting b means we will run through all composites gt a starting an infinite loop it is expected that the user will break out of the loop himself at some point using break or return Note that the value of n cannot be modified within seq forcomposite n 2 10 n kk at top level forcomposite n 2 10 n OK xxx index read only was changed to 279 3 11 9 fordiv n X seq Evaluates seq where the formal variable X ranges through the divisors of n see divisors which is used as a subroutine It is assumed that factor can handle n without negative exponents Instead of n it is possible to input a factorization matrix i e the output of factor n This routine uses divisors as a subroutine then loops over the divisors In particular if n is an integer divisors are sorted b
331. he j invariant function The polynomial eqn satisfies the following functional equa tion which allows to compute the values of the classical modular polynomial y of prime level N such that y j 7 j N7 0 while being much smaller than the latter e for canonical type P f 7 j 7 P N f r N7 0 where s 12 gcd 12 N 1 e for Atkin type P f r j 7 P f 7 9 N7T 0 In both cases f is a suitable modular function see below The following GP function returns values of the classical modular polynomial by eliminating f T in the above two equations for N lt 31 or N 41 47 59 71 classicaleqn N X X Y Y my E ellmodulareqn N P E 1 t E 2 Q d if poldegree P y gt 2 error level unavailable in classicaleqn if t 0 my s 12 gcd 12 N 1 Q x N 1 substvec P x y N s x Y d N s 2 N 1 1 7 N 1 Q subst P y Y d X Y N 1 polresultant subst P y X Q d More precisely let Wy T 51 be the Atkin Lehner involution we have j Wy 7 j N7 and the function f also satisfies e for canonical type F Wy 7 N f 7 e for Atkin type F Wy 7 f r Furthermore for an equation of canonical type f is the standard n quotient 2s f r N NT nr where 7 is Dedekind s eta function which is invariant under Po N The library syntax is GEN ellmodularegn long N long x 1 long y 1 where x y are variabl
332. he message x bnrL1 overflow in zeta_get_NO need too many primes if the approximate functional equation requires us to sum too many terms if the discriminant of K is too large The library syntax is GEN bnrL1 GEN bnr GEN H NULL long flag long prec 3 6 21 bnrclassno A B C Let A B C define a class field L over a ground field K of type bnr bnr subgroup or bnf modulus or bnf modulus subgroup Section 3 6 5 this function returns the relative degree L K In particular if A is a bnf with units and B a modulus this function returns the correspond ing ray class number modulo B One can input the associated bid with generators if the subgroup C is non trivial for B instead of the module itself saving some time This function is faster than bnrinit and should be used if only the ray class number is desired See bnrclassnolist if you need ray class numbers for all moduli less than some bound The library syntax is GEN bnrclassno0 GEN A GEN B NULL GEN C NULL Also avail able is GEN bnrclassno GEN bnf GEN f to compute the ray class number modulo f 163 3 6 22 bnrclassnolist bnf list bnf being as output by bnfinit and list being a list of moduli with units as output by ideallist or ideallistarch outputs the list of the class numbers of the corresponding ray class groups To compute a single class number bnrclassno is more efficient bnf bnfinit x 2 2 L ideallist bnf 100
333. he name of a function that was present in GP version 1 39 15 or lower outputs the new function name and syntax if it changed at all 387 out of 560 did 301 3 12 42 write filename str Writes appends to filename the remaining arguments and appends a newline same output as print 3 12 43 writel filename str Writes appends to filename the remaining arguments without a trailing newline same output as print1 3 12 44 writebin filename x Writes appends to filename the object x in binary format This format is not human readable but contains the exact internal structure of x and is much faster to save load than a string expression as would be produced by write The binary file format includes a magic number so that such a file can be recognized and correctly input by the regular read or r function If saved objects refer to polynomial variables that are not defined in the new session they will be displayed in a funny way see Section 3 12 22 Installed functions and history objects can not be saved via this function If x is omitted saves all user variables from the session together with their names Reading such a named object back in a gp session will set the corresponding user variable to the saved value E g after x 1 writebin log reading log into a clean session will set x to 1 The relative variables priorities see Section 2 5 3 of new variables set in this way remain the same preset v
334. he output is a vector x q where a Norm x x q The algorithm looks for a solution x which is an S integer with S a list of places of K containing at least the ramified primes the generators of the class group of L as well as those primes dividing a If L K is Galois then this is enough otherwise flag is used to add more primes to S all the places above the primes p lt flag resp p flag if flag gt 0 resp flag lt 0 The answer is guaranteed i e a is a norm iff q 1 if the field is Galois or under GRH if S contains all primes less than 12log disc M where M is the normal closure of L K If rnfisnorminit has determined or was told that L K is Galois and flag 4 0 a Warning is issued so that you can set flag 1 to check whether L K is known to be Galois according to T Example 210 bnf bnfinit y 3 y72 2 y 1 p x 2 Mod y 2 2 y 1 bnf pol T rnfisnorminit bnf p rnfisnorm T 17 checks whether 17 is a norm in the Galois extension Q 8 Q a where a a 2a 1 0 and B a 2a 1 0 it is The library syntax is GEN rnfisnorm GEN T GEN a long flag 3 6 153 rnfisnorminit pol polrel flag 2 Let K be defined by a root of pol and L K the extension defined by the polynomial polrel As usual pol can in fact be an nf or bnf etc if pol has degree 1 the base field is Q polrel is also allowed to be an nf etc Computes technical data needed by rnfisnorm to solve n
335. he required accuracy The library syntax is GEN roots GEN x long prec 3 7 30 polrootsmod pol p flag 0 Row vector of roots modulo p of the polynomial pol Multiple roots are not repeated polrootsmod x 2 1 2 If p is very small you may set flag 1 which uses a naive search The library syntax is GEN rootmod0 GEN pol GEN p long flag 3 7 31 polrootspadic z p r Vector of p adic roots of the polynomial pol given to p adic preci sion r p is assumed to be a prime Multiple roots are not repeated Note that this is not the same as the roots in Z p Z rather it gives approximations in Z p Z of the true roots living in Qp polrootspadic x 3 x 2 64 2 5 1 273 00275 273 2 74 0 275 1 0 275 If pol has inexact t_PADIC coefficients this is not always well defined in this case the poly nomial is first made integral by dividing out the p adic content then lifted to Z using truncate coefficientwise Hence the roots given are approximations of the roots of an exact polynomial which is p adically close to the input To avoid pitfalls we advise to only factor polynomials with eact rational coefficients The library syntax is GEN rootpadic GEN x GEN p long r 223 3 7 32 polsturm pol a bj Number of real roots of the real squarefree polynomial pol in the interval Ja b using Sturm s algorithm a resp b is taken to be oo resp 00 if omitted The library syntax is long sturmpart
336. he version of PARI tested there was 1 39 which succeeded reliably from precision 265 on in about 200 as much time as the current version The library syntax is GEN algdepO GEN x long k long flag Also available is GEN al gdep GEN x long k flag 0 3 8 2 charpoly A v x flag 5 characteristic polynomial of A with respect to the variable v i e determinant of v J A if A is a square matrix charpoly 1 2 3 4 fi x72 5ex 2 charpoly 1 2 3 4 t 12 472 Set 2 If A is not a square matrix the function returns the characteristic polynomial of the map multi plication by A if A is a scalar charpoly Mod x 2 x 3 2 1 x73 6 x72 12 x 10 charpoly I 42 x2 1 charpoly quadgen 5 13 x 2 x 1 charpoly ffgen ffinit 2 4 14 Mod 1 2 x 4 Mod 1 2 x73 Mod 1 2 x 2 Mod 1 2 x Mod 1 2 The value of flag is only significant for matrices and we advise to stick to the default value Let n be the dimension of A If flag 0 same method Le Verrier s as for computing the adjoint matrix i e using the traces of the powers of A Assumes that n is invertible uses O n scalar operations If flag 1 uses Lagrange interpolation which is usually the slowest method Assumes that n is invertible uses O n scalar operations If flag 2 uses the Hessenberg form Assumes that the base ring is a field Uses O n scalar operations but suffers from coe
337. his is not to be confused with the history of your commands maintained by readline The gp history contains the results they produced in sequence The successive elements of the history array are called 1 2 As a shortcut the latest computed expression can also be called the previous one the one before that and so on When you suppress the printing of the result with a semicolon it is still stored in the history but its history number will not appear either It is a better idea to assign it to a variable for later use than to mentally recompute what its number is Of course on the next line you may just use The time used to compute that history entry is also stored as part of the entry and can be recovered using the 4 operator 1 2 tt by itself returns the time needed to compute the last result the one returned by Remark The history array is in fact better thought of as a queue its size is limited to 5000 entries by default after which gp starts forgetting the initial entries So 41 becomes unavailable as gp prints 5001 You can modify the history size using histsize 2 2 3 Special editing characters A GP program can of course have more than one line Since your commands are executed as soon as you have finished typing them there must be a way to tell gp to wait for the next line or lines of input before doing anything There are three ways of doing this The first one is to use the
338. hnical parameters 159 3 6 12 bnfisintnorm bnf x Computes a complete system of solutions modulo units of positive norm of the absolute norm equation Norm a x where a is an integer in bnf If bnf has not been certified the correctness of the result depends on the validity of GRH See also bnfisnorm The library syntax is GEN bnfisintnorm GEN bnf GEN x The function GEN bnfisint normabs GEN bnf GEN a returns a complete system of solutions modulo units of the abso lute norm equation Norm z a As fast as bnfisintnorm but solves the two equations Norm x a simultaneously 3 6 13 bnfisnorm bnf x flag 1 Tries to tell whether the rational number g is the norm of some element y in bnf Returns a vector a b where x Norm a xb Looks for a solution which is an S unit with S a certain set of prime ideals containing among others all primes dividing z If bnf is known to be Galois set flag 0 in this case x is a norm iff b 1 If flag is non zero the program adds to S the following prime ideals depending on the sign of flag If flag gt 0 the ideals of norm less than flag And if flag lt 0 the ideals dividing flag Assuming GRH the answer is guaranteed i e x is a norm iff b 1 if S contains all primes less than 12log disc Bnf where Bnf is the Galois closure of bnf See also bnfisintnorm The library syntax is GEN bnfisnorm GEN bnf GEN x long flag 3 6 14 bnfisprincipal bnf x flag
339. iable values Strexpand HOME doc 1 home pari doc The individual arguments are read in string context see Section 2 9 3 2 14 Strtex z Translates its arguments to TeX format and concatenates the results into a single character string type t_STR the empty string if x is omitted The individual arguments are read in string context see Section 2 9 3 2 15 Vec z n Transforms the object x into a row vector The dimension of the resulting vector can be optionally specified via the extra parameter n If n is omitted or 0 the dimension depends on the type of x the vector has a single component except when x is e a vector or a quadratic form in which case the resulting vector is simply the initial object considered as a row vector e a polynomial or a power series In the case of a polynomial the coefficients of the vector start with the leading coefficient of the polynomial while for power series only the significant coefficients are taken into account but this time by increasing order of degree In this last case Vec is the reciprocal function of Pol and Ser respectively e a matrix return the vector of columns comprising the matrix e a character string return the vector of individual characters e an error context t_ERROR return the error components see iferr In the last three cases matrix character string error n is meaningless and must be omitted or an error is raised Otherwise if n is given O
340. ibrary syntax is intfuncinit void E GEN eval void GEN GEN a GEN b long m long flag long prec Note that the order of m and flag are reversed compared to the GP syntax 258 3 9 7 intlaplaceinv X sig z expr tab Numerical integration of 2im lexpr X e with respect to X on the line R X sig In other words inverse Laplace transform of the function corresponding to expr at the value z sig is coded as follows Either it is a real number equal to the abscissa of integration and then the integrand is assumed to be slowly decreasing when the imaginary part of the variable tends to too Or it is a two component vector o a where is as before and either a 0 for slowly decreasing functions or a gt 0 for functions decreasing like exp at Note that it is not necessary to choose the exact value of a tab is as in intnum It is often a good idea to use this function with a value of m one or two higher than the one chosen by default which can be viewed thanks to the function intnumstep or to increase the abscissa of integration o For example Ap 105 intlaplaceinv x 2 1 1 x 1 time 350 ms 1 7 37 E 55 1 72 E 54 I not so good m intnumstep 12 7 intlaplaceinv x 2 1 1 x m 1 1 time 700 ms 13 3 95 E 97 4 76 E 98 I better intlaplaceinv x 2 1 1 x m 2 1 time 1400 ms 14 0 E 105 0 E 106 I perfect but slow intlaplaceinv x 5 1
341. ic polynomial in variable v 3 7 13 polcyclofactors f Returns a vector of polynomials whose product is the product of distinct cyclotomic polynomials dividing f f x710 5 x78 x77 8 x7 6 4 x 7 5 8 x7 4 3 x7 34 7 X7 243 v polcyclofactors f A2 x72 1 x 2 x 1 x 4 x73 x72 x 1 apply poliscycloprod v 3 1 1 1 apply poliscyclo v 4 4 3 10 In general the poynomials are products of cyclotomic polynomials and not themselves irreducible g x78 2x x 7 6x xx 6 9 xx 5 12 x 4 11 x 3 10x x 2 6x x 3 polcyclofactors g 2 x 6 24x75 3 x 4 3 x 3 3 x 2 2 x 1 factor 1 3 x 2 x 11 x74 x 3 x 2 x 1 1 The library syntax is GEN polcyclofactors GEN f 3 7 14 poldegree z v Degree of the polynomial x in the main variable if v is omitted in the variable v otherwise The degree of 0 is a fixed negative number whose exact value should not be used The degree of a non zero scalar is 0 Finally when x is a non zero polynomial or rational function returns the ordinary degree of x Raise an error otherwise The library syntax is long poldegree GEN x long v 1 where v is a variable number 3 7 15 poldisc pol v Discriminant of the polynomial pol in the main variable if v is omitted in v otherwise The algorithm used is the subresultant algorithm The library syntax is GEN poldiscO GEN pol long v 1 where v is a variable number 3 7 16 pol
342. ich should be useless since Q 1 P is a priori a better representation for the number field and its elements The library syntax is GEN polredbest GEN T long flag 3 6 122 polredord x Finds polynomials with reasonably small coefficients and of the same degree as that of x defining suborders of the order defined by x One of the polynomials always defines Q hence is equal to x 1 where n is the degree and another always defines the same order as x if x is irreducible Useless function try polredbest The library syntax is GEN polredord GEN x 3 6 123 poltschirnhaus x Applies a random Tschirnhausen transformation to the polynomial x which is assumed to be non constant and separable so as to obtain a new equation for the tale algebra defined by x This is for instance useful when computing resolvents hence is used by the polgalois function The library syntax is GEN tschirnhaus GEN x 3 6 124 rnfalgtobasis rnf x Expresses x on the relative integral basis Here rnf is a relative number field extension L K as output by rnfinit and x an element of L in absolute form i e expressed as a polynomial or polmod with polmod coefficients not on the relative integral basis The library syntax is GEN rnfalgtobasis GEN rnf GEN x 3 6 125 rnfbasis bnf M Let K the field represented by bnf as output by bnfinit Misa projective Zx module of rank n M amp K is an n dimensional K vector space given by a pseudo basis
343. icients The library syntax is GEN qflllgramO GEN G long flag Also available are GEN 111 gram GEN G flag 0 GEN 111gramint GEN G flag 1 and GEN 111gramkerim GEN G flag 4 3 8 60 qfminim z b m flag 0 x being a square and symmetric matrix representing a positive definite quadratic form this function deals with the vectors of x whose norm is less than or equal to b enumerated using the Fincke Pohst algorithm storing at most m vectors no limit if m is omitted The function searches for the minimal non zero vectors if b is omitted The behavior is undefined if x is not positive definite a precision too low error is most likely although more precise error messages are possible The precise behavior depends on flag If flag 0 default seeks at most 2m vectors The result is a three component vector the first component being the number of vectors found the second being the maximum norm found and the last vector is a matrix whose columns are the vectors found only one being given for each pair v at most m such pairs unless m was omitted The vectors are returned in no particular order If flag 1 ignores m and returns N v where v is a non zero vector of length N lt b or if no non zero vector has length lt b If no explicit b is provided return a vector of smallish norm smallest vector in an LLL reduced basis In these two cases x must have integral entries The implementation us
344. ile building a large user community linked by helpful mailing lists and a tradition of great user support from the developers And of course PARI GP is Free Software covered by the GNU General Public License either version 2 of the License or at your option any later version PARI is used in three different ways 1 as a library libpari which can be called from an upper level language application for instance written in ANSI C or C 2 as a sophisticated programmable calculator named gp whose language GP contains most of the control instructions of a standard language like C 3 the compiler gp2c translates GP code to C and loads it into the gp interpreter A typical script compiled by gp2c runs 3 to 10 times faster The generated C code can be edited and optimized by hand It may also be used as a tutorial to libpari programming The present Chapter 1 gives an overview of the PARI GP system gp2c is distributed separately and comes with its own manual Chapter 2 describes the GP programming language and the gp calculator Chapter 3 describes all routines available in the calculator Programming in library mode is explained in Chapters 4 and 5 in a separate booklet User s Guide to the PARI library libpari dvi A tutorial for gp is provided in the standard distribution A tutorial for PARI GP tuto rial dvi and you should read this first You can then start over and read the more boring stuff which lies ahead You can have a qui
345. imation to the curve Otherwise add the middle point to the reference points This is fast and usually more precise than usual plot Compare the results of ploth X 1 1 sin 1 X Recursive ploth X 1 1 sin 1 X for instance But beware that if you are extremely unlucky or choose too few reference points you may draw some nice polygon bearing little resemblance to the original curve For instance you should never plot recursively an odd function in a symmetric interval around 0 Try ploth x 20 20 sin x Recursive to see why Hence it s usually a good idea to try and plot the same curve with slightly different parameters 274 The other values toggle various display options e 4 no Rescale do not rescale plot according to the computed extrema This is used in conjunction with plotscale when graphing multiple functions on a rectwindow as a plotrecth call s plothsizes plotinit 0 s 2 1 s 2 1 plotscale 0 1 1 1 1 plotrecth 0 t 0 2 Pi cos t sin t Parametric no_Rescale plotdraw 0 1 1 This way we get a proper circle instead of the distorted ellipse produced by ploth t 0 2 Pi cos t sin t Parametric e 8 no_X_axis do not print the z axis e 16 no_Y_axis do not print the y axis e 32 no_Frame do not print frame e 64 no Lines only plot reference points do not join them e 128 Points_too plot both lines and points e 256 Splines use splines to interpol
346. imple metacommands are meant as shortcuts and should not be used in GP scripts see Sec tion 3 11 Beware that these as all of gp input are case sensitive For example Q is not identical to q In the following list braces are used to denote optional arguments with their default values when applicable e g n 0 means that if n is not there it is assumed to be 0 Whitespace or spaces between the metacommand and its arguments and within arguments is optional This can cause problems only with Mw when you insist on having a file name whose first character is a digit and with r or w if the file name itself contains a space In such cases just use the underlying read or write function see Section 3 12 42 55 2 13 1 command The gp on line help interface If you type n where n is a number from 1 to 11 you will get the list of functions in Section 3 n of the manual the list of sections being obtained by simply typing These names are in general not informative enough More details can be obtained by typing function which gives a short explanation of the function s calling convention and effects Of course to have complete information read Chapter 3 of this manual the source code is at your disposal as well though a trifle less readable If the line before the copyright message indicates that extended help is available this means perl is present on your system and the PARI distribution was correctly installed yo
347. in the unknown x print Hello Hello void return value f x x72 3 x gt x 2 AN a user function In the third example Hello is printed as a side effect but is not the return value The print command is a procedure which conceptually returns nothing But in fact procedures return a special void object meant to be ignored but which evaluates to 0 in a numeric context and stored as 0 in the history or results The final example assigns to the variable f the function gt 2 the alternative form f x gt x 2 achieving the same effect the return value of a function definition is unsurprisingly a function object of type t_CLOSURE Several expressions are combined on a single line by separating them with semicolons Such an expression sequence will be called a seg A seg also has a value which is the value of the last expression in the sequence Under gp the value of the seg and only this last value becomes an history entry The values of the other expressions in the seg are discarded after the execution of the seg is complete except of course if they were assigned into variables In addition the value of the seq is printed if the line does not end with a semicolon Not exactly since not all their arguments need be evaluated For instance it would be stupid to evaluate both branches of an if statement since only one will apply only this one is evaluated 15 2 2 2 The gp history of results T
348. inant are included Note that nfinit has built in support for such a check K nfinit T listP nfcertify K we computed an actual maximal order 2 D The first line initializes a number field structure incorporating nfbasis T listP in place of a proven integral basis The second line certifies that the resulting structure is correct This allows 185 to create an nf structure associated to the number field K Q X T when the discriminant of T cannot be factored completely whereas the prime divisors of discK are known Of course if listP contains a single prime number p the function returns a local integral basis for Zp X T nfbasis x 2 x 1001 1 1 1 3 x 1 3 nfbasis x 2 x 1001 2 2 1 x The Buchmann Lenstra algorithm We now complicate the picture it is in fact allowed to include composite numbers instead of primes in listP Vector or Matrix case provided they are pairwise coprime The result will still be a correct integral basis if the field discriminant factors completely over the actual primes in the list Adding a composite C such that C divides D may help because when we consider C as a prime and run the algorithm two good things can happen either we succed in proving that no prime dividing C can divide the index without actually needing to find those primes or the computation exhibits a non trivial zero divisor thereby factoring C and we go on with the refined facto
349. inear relation with coefficients in the field lindep x y x72 y x 2 y x y 2 1 4 ly y 1 y 2 For better control it is preferable to use t_POL rather than t_SER in the input otherwise one gets a linear combination which is t adically small but not necessarily 0 Indeed power series are first converted to the minimal absolute accuracy occuring among the entries of v which can cause some coefficients to be ignored then truncated to polynomials v t72 0 t74 1 0 t72 L lindep v 1 1 0 v L 12 t72 0 t 4 small but not 0 The library syntax is GEN lindepO GEN v long flag Also available are GEN lindep GEN v real complex entries flag 0 GEN lindep2 GEN v long flag real complex entries GEN padic_lindep GEN v p adic entries and GEN Xadic_lindep GEN v polynomial entries Fi nally GEN deplin GEN v returns a non zero kernel vector for a t_MAT input 3 8 6 listcreate Creates an empty list This routine used to have a mandatory argument which is now ignored for backward compatibility In fact this function has become redundant and obsolete it will disappear in future versions of PARI just use List 3 8 7 listinsert L x n Inserts the object x at position n in L which must be of type t_LIST This has complexity O F L n 1 all the remaining elements of list from position n 1 onwards are shifted to the right The library syntax is GEN listinsert GEN L GEN x long n
350. ing points If tab is e a positive integer m we use 2 sampling points hopefully increasing accuracy But note that the running time is roughly proportional to 2 One may try consecutive values of m until they give the same value up to an accepted error If tab is omitted the algorithm guesses a reasonable value for m depending on the current precision only which should be sufficient for regular functions That value may be obtained from intnumstep and increased in case of difficulties e a set of integration tables as output by intnuminit they are used directly This is useful if several integrations of the same type are performed on the same kind of interval and func tions for a given accuracy in particular for multivariate integrals since we then skip expensive precomputations Specifying the behavior at endpoints This is done as follows An endpoint a is either given as such a scalar real or complex or 1 for 00 or as a two component vector a a to indicate the behavior of the integrand in a neighborhood of a If a is finite the code a a means the function has a singularity of the form x a up to logarithms If a gt 0 we only assume the function is regular which is the default assumption If a wrong singularity exponent is used the result will lose a catastrophic number of decimals intnum x 0 1 x 1 2 assume is regular at 0 1 1 999999999999999999990291881
351. ing the time for printing the results If the time is negligible lt 1 ms nothing is printed in particular no timing should be printed when defining a user function or an alias or installing a symbol from the library The default value is O off 313 314 Appendix A Installation Guide for the UNIX Versions 1 Required tools Compiling PARI requires an ANSI C or a C compiler If you do not have one we suggest that you obtain the gcc g compiler As for all GNU software mentioned afterwards you can find the most convenient site to fetch gcc at the address http www gnu org order ftp html On Mac OS X this is also provided in the Xcode tool suite You can certainly compile PARI with a different compiler but the PARI kernel takes advantage of optimizations provided by gcc This results in at least 20 speedup on most architectures Optional libraries and programs The following programs and libraries are useful in conjunction with gp but not mandatory In any case get them before proceeding if you want the functionalities they provide All of them are free The download page on our website http pari math u bordeaux fr download html contains pointers on how to get these e GNU MP library This provides an alternative multiprecision kernel which is faster than PARI s native one but unfortunately binary incompatible so the resulting PARI library SONAME is libpari gmp e GNU readline library This provides line
352. intnum x 0 1 2 1 x 1 2 no it s not 2 2 000000000000000000000000000 intnum x 0 1 10 1 x 1 2 13 1 999999999999999999999946438 using a wrong exponent is bad 261 If a is 00 which is coded as 1 the situation is more complicated and 1 a means e a 0 or no a at all ie simply 1 assumes that the integrand tends to zero but not exponentially fast and not oscillating such as sin x x e a gt 0 assumes that the function tends to zero exponentially fast approximately as exp az This includes oscillating but quickly decreasing functions such as exp 2 sin x oo 1 intnum x 0 00 exp 2 x kk at top level intnum x 0 00 exp WER Fa nnn kk exp exponent expo overflow intnum x 0 oo 2 exp 2 x 1 0 5000000000000000000000000000 OK intnum x 0 oo 4 exp 2 x 12 0 4999999999999999999961990984 wrong exponent gt imprecise result intnum x 0 oo 20 exp 2 x 2 0 4999524997739071283804510227 disaster e a lt 1 assumes that the function tends to 0 slowly like 1 Here it is essential to give the correct a if possible but on the other hand a lt 2 is equivalent to a 0 in other words to no a at all The last two codes are reserved for oscillating functions Let k gt 0 real and g x a non oscillating function tending slowly to 0 e g like a negative power of x then e a k x I assumes that the
353. iptic curves Ej and E gt intersecting transversally at one point vecj contains their modular invariants j and j2 which may live in a quadratic extension of F are need not be distinct The Jacobian J C has potentially good reduction isomorphic to the product of the reductions of E and E gt e 6 the union of an elliptic curve E and a projective line which has an ordinary double point and these two components intersect transversally at one point vecj contains 7 mod p the modular invariant of E The potential semi abelian reduction of J C is the extension of an elliptic curve with modular invariant j mod p by a torus 150 e 7 asin type 6 but the two components are both singular The Jacobian J C has potentially multiplicative reduction The component red NUtype neron contains two data concerning the reduction at p without any ramified field extension The NUtype is a t_STR describing the reduction at p of C following Namikawa Ueno The complete classification of fibers in pencils of curves of genus two Manuscripta Math vol 9 1973 pages 143 186 The reduction symbol is followed by the corresponding page number in this article The second datum neron is the group of connected components over an algebraic closure of F of the N ron model of J C given as a finite abelian group vector of elementary divisors If p 2 the red component may be omitted altogether and replaced by in the case where the progr
354. is GEN znstar GEN n 3 5 Functions related to elliptic curves 3 5 1 Elliptic curve structures An elliptic curve is given by a Weierstrass model y axy azy aya 04 06 whose discriminant is non zero Affine points on E are represented as two component vectors x y the point at infinity i e the identity element of the group law is represented by the one component vector 0 Given a vector of coefficients a1 az a3 a4 a the function ellinit initializes and returns an ell structure An additional optional argument allows to specify the base field in case it cannot be inferred from the curve coefficients This structure contains data needed by elliptic curve related functions and is generally passed as a first argument Expensive data are skipped on initialization they will be dynamically computed when and if needed and then inserted in the structure The precise layout of the ell structure is left undefined and should never be used directly The following member functions are available depending on the underlying domain 3 5 1 1 All domains e a1 a2 a3 a4 a6 coefficients of the elliptic curve e b2 b4 b6 b8 b invariants of the curve in characteristic 4 2 for Y 2y a x a3 the curve equation becomes Y 4r box 2bax bg g x e c4 c6 c invariants of the curve in characteristic 4 2 3 for X x b2 12 and Y 2y a x a3 the curve equation becomes Y 4X c4 12 X
355. is is the ground field K e mod is a modulus f as described above e subgroup a subgroup of the ray class group modulo f of K As described above this is input as a square matrix expressing generators of a subgroup of the ray class group bnr clgp on the given generators The corresponding bnr is the subfield of the ray class field of K modulo f fixed by the given subgroup K bnfinit y 2 1 bnr bnrinit K 13 clgp 13 36 12 3 bnrdisc bnr discriminant of the full ray class field bnrdisc bnr 3 1 0 1 discriminant of cyclic cubic extension of K We could have written directly bnrdisc K 13 bnrdisc K 13 3 1 0 1 avoiding one bnrinit but this would actually be slower since the bnrinit is called internally anyway And now twice 154 3 6 6 General use All the functions which are specific to relative extensions number fields Buchmann s number fields Buchmann s number rays share the prefix rnf nf bnf bnr respectively They take as first argument a number field of that precise type respectively output by rnfinit nfinit bnfinit and bnrinit However and even though it may not be specified in the descriptions of the functions below 1t is permissible if the function expects a nf to use a bnf instead which contains much more information On the other hand if the function requires a bnf it will not launch bnfinit for you which is a costly operation Instead it will give
356. isc 0 1 returns 0 1 Note that quaddisc n returns the same value as coredisc n and also works with rational inputs n Q The library syntax is GEN corediscO GEN n long flag Also available are GEN core disc GEN n flag 0 and GEN coredisc2 GEN n flag 1 3 4 15 dirdiv x y x and y being vectors of perhaps different lengths but with y 1 4 0 considered as Dirichlet series computes the quotient of by y again as a vector The library syntax is GEN dirdiv GEN x GEN y 3 4 16 direuler p a b expr c Computes the Dirichlet series associated to the Euler product of expression expr as p ranges through the primes from a to b expr must be a polynomial or rational function in another variable than p say X and expr X is understood as the local factor expr p The series is output as a vector of coefficients If cis present output only the first c coefficients in the series The following command computes the sigma function associated to s s 1 direuler p 2 10 1 1 X 1 p X 1 1 3 4 7 6 12 8 15 13 18 The library syntax is direuler void E GEN eval void GEN GEN a GEN b 3 4 17 dirmul x y x and y being vectors of perhaps different lengths representing the Dirichlet series gt nn and gt gt Ynn computes the product of x by y again as a vector dirmul vector 10 n 1 vector 10 n moebius n 1 1 O O O O 0 O O O 0 The product length is the
357. istinguishable from 0 Note on output formats A zero real number is printed in e format as 0 Exx where xz is the usually negative decimal exponent of the number cf Section 1 3 7 This allows the user to check the accuracy of that particular zero When the integer part of a real number x is not known exactly because the exponent of x is greater than the internal precision the real number is printed in e format 2 3 3 Intmods t_INTMOD To create the image of the integer a in Z bZ for some non zero integer b type Mod a b not a b Internally all operations are done on integer representatives belonging to 0 b 1 Note that this type is available for convenience not for speed each elementary operation involves a reduction modulo b If x is a t_INTMOD Mod a b the following member function is defined x mod return the modulus b 2 3 4 Rational numbers t_FRAC All fractions are automatically reduced to lowest terms so it is impossible to work with reducible fractions To enter n m just type it as written As explained in Section 3 1 5 floating point division is not performed only reduction to lowest terms Note that rational computation are almost never the fastest method to proceed in the PARI implementation each elementary operation involves computing a gcd It is generally a little more efficient to cancel denominators and work with integers only P Pol vector 1073 i 1 i big polynomial with small rati
358. istoalg nf 1 1 3 Mod 1 2 y 1 y72 4 nfbasistoalg nf y 74 Mod y y 2 4 nfbasistoalg nf Mod y y 2 4 4 Mod y y 2 4 This is the inverse function of nfalgtobasis The library syntax is GEN basistoalg GEN nf GEN x 3 6 79 nfcertify nf nf being as output by nfinit checks whether the integer basis is known un conditionally This is in particular useful when the argument to nfinit was of the form T 1istP specifying a finite list of primes when p maximality had to be proven The function returns a vector of composite integers If this vector is empty then nf zk and nf disc are correct Otherwise the result is dubious In order to obtain a certified result one must completely factor each of the given integers then addprime each of them then check whether nfdisc nf pol is equal to nf disc The library syntax is GEN nfcertify GEN nf 3 6 80 nfdetint nf x Given a pseudo matrix x computes a non zero ideal contained in i e mul tiple of the determinant of x This is particularly useful in conjunction with nfhnfmod The library syntax is GEN nfdetint GEN nf GEN x 3 6 81 nfdisc T field discriminant of the number field defined by the integral preferably monic irreducible polynomial T X Returns the discriminant of the number field Q X T using the Round 4 algorithm Local discriminants valuations at certain primes As in nfbasis the argument T can be replaced by T list
359. istsort L 1 L fi List 1 2 3 5 j setsearch L 4 1 4 should have been inserted at index j 42 4 listinsert L 4 j L 3 List 1 2 3 4 51 The library syntax is long setsearch GEN S GEN x long flag 3 8 71 setunion x y Union of the two sets x and y see setisset If x or y is not a set the result is undefined The library syntax is GEN setunion GEN x GEN y 3 8 72 trace x This applies to quite general x If x is not a matrix it is equal to the sum of x and its conjugate except for polmods where it is the trace as an algebraic number For x a square matrix it is the ordinary trace If x is a non square matrix but not a vector an error occurs The library syntax is GEN gtrace GEN x 3 8 73 vecextract z y 12 Extraction of components of the vector or matrix x according to y In case x is a matrix its components are the columns of x The parameter y is a component specifier which is either an integer a string describing a range or a vector If y is an integer it is considered as a mask the binary bits of y are read from right to left but correspond to taking the components from left to right For example if y 13 1101 2 then the components 1 3 and 4 are extracted If y is a vector t_VEC t_COL or t_VECSMALL which must have integer entries these entries correspond to the component numbers to be extracted in the order specified If y is a string 1t can be e a single non
360. ively 2 12 Defaults There are many internal variables in gp defining how the system will behave in certain situations unless a specific override has been given Most of them are a matter of basic customization colors prompt and will be set once and for all in your preferences file see Section 2 14 but some of them are useful interactively set timer on increase precision etc The function used to manipulate these values is called default which is described in Sec tion 3 12 7 The basic syntax is default def value which sets the default def to value In interactive use most of these can be abbreviated using gp metacommands mostly starting with which we shall describe in the next section Here we will only describe the available defaults and how they are used Just be aware that typing default by itself will list all of them as well as their current values see Xd Just after the default name we give between parentheses the initial value when gp starts assuming you did not tamper with factory settings using command line switches or a gprc see http sagemath org see http clisp cons org 54 Note The suffixes k M or G can be appended to a value which is a numeric argument with the effect of multiplying it by 10 10 and 10 respectively Case is not taken into account there so for instance 30k and 30K both stand for 30000 This is mostly useful to modify or set the default parisize which
361. ivisors or U V D as would normally be output with the cleanup flag unset The library syntax is GEN matsnfO GEN X long flag 3 8 44 matsolve M B M being an invertible matrix and B a column vector finds the solution X of MX B using Dixon p adic lifting method if M and B are integral and Gaussian elimination otherwise This has the same effect as but is faster than M7 x B The library syntax is GEN gauss GEN M GEN B For integral input the function GEN ZM_gauss GEN M GEN B is also available 3 8 45 matsolvemod M D B flag 0 M being any integral matrix D a column vector of non negative integer moduli and B an integral column vector gives a small integer solution to the system of congruences iMi jtj bi mod d if one exists otherwise returns zero Shorthand notation B resp D can be given as a single integer in which case all the b resp d above are taken to be equal to B resp D M 1 2 3 4 matsolvemod M 3 4 1 2 12 2 0 matsolvemod M 3 1 MX 1 1 over F_3 13 1 1 matsolvemod M 3 0 1 2 x 2y 1 mod 3 3x 4y 2 in Z 7 4 6 4 If flag 1 all solutions are returned in the form of a two component row vector x u where x is a small integer solution to the system of congruences and wu is a matrix whose columns give a basis of the homogeneous system so that all solutions can be obtained by adding x to any linear combination of
362. ization the pullback by of the N ron differential du 2v au az is equal to 2inf z dz a holomor phic differential form The variable used in the power series for u and v is x which is implicitly understood to be equal to exp 2izz The algorithm assumes that E is a strong Weil curve and that the Manin constant is equal to 1 in fact f x Do ys ellan E n x The library syntax is GEN elltaniyama GEN E long precdl 147 3 5 45 elltatepairing E P Q m Computes the Tate pairing of the two points P and Q on the elliptic curve E The point P must be of m torsion The library syntax is GEN elltatepairing GEN E GEN P GEN Q GEN m 3 5 46 elltors E flag 0 If E is an elliptic curve defined over Q outputs the torsion subgroup of E as a 3 component vector t v1 v2 where t is the order of the torsion group v1 gives the structure of the torsion group as a product of cyclic groups sorted by decreasing order and v2 gives generators for these cyclic groups E must be an ell structure as output by ellinit defined over Q E ellinit 1 0 elltors E 41 4 2 2 C00 0 1 0 1 Here the torsion subgroup is isomorphic to Z 2Z x Z 2Z with generators 0 0 and 1 0 If flag 0 find rational roots of division polynomials If flag 1 use Lutz Nagell much slower If flag 2 use Doud s algorithm bound torsion by computing E F for small primes of good reduction then look for torsion points
363. k and provides simple diagnostics if it does not Only useful if make bench fails horribly e g things like 1 1 do not work make test all runs all available test suites Thorough but slow Some of the tests require extra packages elldata galdata etc to be available If you want to test such an extra package before make install which would install it to its final location where gp expects to find it run env GP_DATA_DIR PWD data make test all from the PARI toplevel directory otherwise the test will fail 320 4 Installation When everything looks fine type make install You may have to do this with superuser privileges depending on the target directories Tip for MacOS X beginners use sudo make install In this case it is advised to type make all first to avoid running unnecessary commands as root Caveat Install directories are created honouring your umask settings if your umask is too re strictive e g 077 the installed files will not be world readable Beware that running sudo may change your user umask This installs in the directories chosen at Configure time the default gp executable probably gp dyn under the name gp the default PARI library probably libpari so the necessary include files the manual pages the documentation and help scripts To save on disk space you can manually gzip some of the documentation files if you wish usersch tex and all dvi files assuming your xdvi knows how to deal
364. l 1 gives a real number The library syntax is GEN mpfactr long x long prec GEN mpfact long x returns zq as a t_INT 3 4 25 factorint z flag 0 Factors the integer n into a product of pseudoprimes see ispseu doprime using a combination of the Shanks SQUFOF and Pollard Rho method with modifications due to Brent Lenstra s ECM with modifications by Montgomery and MPQS the latter adapted from the LiDIA code with the kind permission of the LiDIA maintainers as well as a search for pure powers The output is a two column matrix as for factor the first column contains the prime divisors of n the second one contains the positive exponents By convention 0 is factored as 0 and 1 as the empty factorization also the divisors are by default not proven primes is they are larger than 2 they only failed the BPSW compositeness test see ispseudoprime Use isprime on the result if you want to guarantee primality or set the factor_proven default to 1 Entries of the private prime tables see addprimes are also included as is This gives direct access to the integer factoring engine called by most arithmetical functions flag is optional its binary digits mean 1 avoid MPQS 2 skip first stage ECM we may still fall back to it later 4 avoid Rho and SQUFOF 8 don t run final ECM as a result a huge composite may be declared to be prime Note that a strong probabilistic primality test is used thus composites might
365. l runs from 1 to n In particular vector n i expr is not equivalent to v vector n for i 1 n vlil expr as the following example shows n 3 v vector n vector n i i gt 2 3 4 vector n for i 1 n vli i gt 2 0 4 3 8 78 vectorsmall n X expr 0 Creates a row vector of small integers type t_VECSMALL with n components whose components are the expression expr evaluated at the integer points between 1 and n If one of the last two arguments is omitted fill the vector with zeroes 3 8 79 vectorv n X expr 0 As vector but returns a column vector type t_COL 256 3 9 Sums products integrals and similar functions Although the gp calculator is programmable it is useful to have a number of preprogrammed loops including sums products and a certain number of recursions Also a number of functions from numerical analysis like numerical integration and summation of series will be described here One of the parameters in these loops must be the control variable hence a simple variable name In the descriptions the letter X will always denote any simple variable name and represents the formal parameter used in the function The expression to be summed integrated etc is any legal PARI expression including of course expressions using loops Library mode Since it is easier to program directly the loops in library mode these functions are mainly useful for GP program
366. l s ss moe moose doa men Eo s 195 HELSINGI sunrise 195 MEISTSOM 2a 44d be eee aa e 195 DEKSEMO PE e wes eB Se he 195 DEMORA ko ea Se wee OC Oe e ee eo 188 nfmodprinit 4 4 2 ss as 188 189 195 Nimul isc ee eke ee we e eb we we a 188 nfmulmodpr gt io sek Ha hee es 188 minewprec co oo ooo eee ees 193 195 MENOTM s e eow ee ewe eee HE Oe we 188 NEPON A 189 NfpowmodpYr sss ee he eee ee 189 nfreduce so s s soe eta aor etu aoe mos 189 nfreducemodpr 189 HNEYOOUS be sab SSE waS Se BES 195 196 nfrootsofl ooo cmo bw eee Re es 196 nfroots 2 5 4 4 025 dsd bes 196 NESNE 2 ei gels bu Gace oe Be o s 196 nfsolvemodpr 197 HESON 250 a ha E a a E 189 nisubfield xose 44 eS eae e ass 169 nfsubfields 197 N TTACO se aa sa ee ee a 189 Nivel rpg sa Oe Ew aa 189 ME ADD AK e e ha aa la ds ei y a 201 DEZA ura so rs Oe E 201 HE FORCE ute e ice 159 161 DEGEN avs ar a we EK ee 161 183 MINIT eos Oe Sw SO g 183 Nf ORIG sa 2 baa eae Oe 195 201 nf_PARTIALFACT 195 201 309 BE RAW Saca See ee ee e me amp a a 201 nf RED soss nied Moe a Ke ee 194 195 nt _ROUND2 s ek how a a 195 TO de ee aoe are dices sw EE ae 131 155 DOT 26 Gee oe BS RS Se Pw eS Raw 82 Norm oscar ora eee owe 244 HOMID 2426 heehee alee he ees 244 NOU be 3 Se Get ok bh Bee oe ed 70 NUCOMP isa o Hie PS wee AS Soe 122 MUGUp ess Bed db ae ae 122 number field 20 numbpart co omm
367. l way always as 0 Eezp e in conversion style we switch to style e if the exponent is greater or equal to the precision e in conversion g and G we do not remove trailing zeros from the fractional part of the result nor a trailing decimal point 0 is printed in a special way always as 0 Eezp 3 12 26 printsep sep str Outputs its string arguments in raw format ending with a new line Successive entries are separated by sep printsep 1 2 3 4 1 2 3 4 3 12 27 printsep1 sep str Outputs its string arguments in raw format without ending with a newline Successive entries are separated by sep printsep1 1 2 3 4 print 1 2 3 4 3 12 28 printtex str Outputs its string arguments in TeX format This output can then be used in a TEX manuscript The printing is done on the standard output If you want to print it to a file you should use writetex see there Another possibility is to enable the log default see Section 2 12 You could for instance do default logfile new tex default log 1 printtex result 3 12 29 quit status 0 Exits gp and return to the system with exit status status a small integer A non zero exit status normally indicates abnormal termination Note the system actually sees only status mod 256 see your man pages for exit 3 or wait 2 297 3 12 30 read filename Reads in the file filename subject to string expansion If filename is omitted re r
368. lag 0 use standard Gauss pivot If flag 1 use matsupplement much slower keep the default flag The library syntax is GEN matimageO GEN x long flag Also available is GEN image GEN x flag 0 3 8 28 matimagecompl x Gives the vector of the column indices which are not extracted by the function matimage as a permutation t_VECSMALL Hence the number of components of matimagecompl x plus the number of columns of matimage x is equal to the number of columns of the matrix zx The library syntax is GEN imagecomp1 GEN x 3 8 29 matindexrank x x being a matrix of rank r returns a vector with two t_VECSMALL components y and z of length r giving a list of rows and columns respectively starting from 1 such that the extracted matrix obtained from these two vectors using vecextract z y z is invertible The library syntax is GEN indexrank GEN x 3 8 30 matintersect z y x and y being two matrices with the same number of rows each of whose columns are independent finds a basis of the Q vector space equal to the intersection of the spaces spanned by the columns of x and y respectively The faster function idealintersect can be used to intersect fractional ideals projective Zg modules of rank 1 the slower but much more general function nfhnf can be used to intersect general Z g modules The library syntax is GEN intersect GEN x GEN y 3 8 31 matinverseimage z y Given a matrix x and a column vector or matrix y retu
369. ld be greedy generalized continued fraction cf x b 1 my a vector b t x b 1 for i 1 b ali floor x t x ali if t i b break 103 x bli 1 t a There is some degree of freedom when choosing the a the program above can easily be modified to derive variants of the standard algorithm In the same vein although no builtin function implements the related Engel expansion a special kind of Egyptian fraction decomposition 1 a 1 a1a2 it can be obtained as follows n terms of the Engel expansion of x engel x n 10 my u x a vector n for k 1 n alk ceil 1 u u uxalk 1 if lu break a Obsolete hack don t use this If b is an integer nmaz is ignored and the command is understood as contfrac x b The library syntax is GEN contfracO GEN x GEN b NULL long nmax Also available are GEN gboundcf GEN x long nmax GEN gcf GEN x and GEN gcf2 GEN b GEN x 3 4 12 contfracpnqn z n 1 When z is a vector or a one row matrix x is considered as the list of partial quotients ao a1 an of a rational number and the result is the 2 by 2 matrix Pn Pn 1 dn dn 1 in the standard notation of continued fractions so Pn qn ao 1 a1 1 an If x is a matrix with two rows bo b1 bn and lao a1 an this is then considered as a generalized continued fraction and we have similarly pn qn
370. le is GEN anellsmal1 GEN e long n which returns a t_VECSMALL instead of a t_VEC saving on memory 133 3 5 6 ellanalyticrank e eps Returns the order of vanishing at s 1 of the L function of the elliptic curve e and the value of the first non zero derivative To determine this order it is assumed that any value less than eps is zero If no value of eps is given a value of half the current precision is used e ellinit 11a1 rank O ellanalyticrank e 12 O 0 2538418608559106843377589233 e ellinit 37a1 rank 1 ellanalyticrank e 4 1 0 3059997738340523018204836835 e ellinit 389a1 rank 2 ellanalyticrank e 6 2 1 518633000576853540460385214 e ellinit 5077a1 rank 3 ellanalyticrank e 18 3 10 39109940071580413875185035 The library syntax is GEN ellanalyticrank GEN e GEN eps NULL long prec 3 5 7 ellap E py Let E be an ell structure as output by ellinit defined over Q or a finite field F The argument p is best left omitted if the curve is defined over a finite field and must be a prime number otherwise This function computes the trace of Frobenius t for the elliptic curve E defined by the equation E F q 1 t If the curve is defined over Q p must be explicitly given and the function computes the trace of the reduction over F The trace of Frobenius is also the a coefficient in the curve L series L E s gt gt ann whence the functi
371. lgroup GEN E GEN p corresponding to flag 0 3 5 21 ellheegner E Let E be an elliptic curve over the rationals assumed to be of analytic rank 1 This returns a non torsion rational point on the curve whose canonical height is equal to the product of the elliptic regulator by the analytic Sha This uses the Heegner point method described in Cohen GTM 239 the complexity is propor tional to the product of the square root of the conductor and the height of the point thus it is preferable to apply it to strong Weil curves E ellinit 15772 01 u ellheegner E print u 1 n u 2 69648970982596494254458225 166136231668185267540804 538962435089604615078004307258785218335 67716816556077455999228495435742408 7 ellheegner ellinit 0 1 AV E has rank 0 eK at top level ellheegner E ellinit OK x ellheegner The curve has even analytic rank The library syntax is GEN ellheegner GEN E 3 5 22 ellheight E x flag 2 Global N ron Tate height of the point z on the elliptic curve E defined over Q using the normalization in Cremona s Algorithms for modular elliptic curves E must be an ell as output by ellinit it needs not be given by a minimal model although the computation will be faster if it is flag selects the algorithm used to compute the Archimedean local height If flag 0 we use sigma and theta functions and Silverman s trick Computing heights on elliptic curves Math
372. ll OK test m 3 time 120 ms 3 7 23 E 60 64 2 times faster lost 45 decimals The library syntax is intnum void E GEN eval void GEN GEN a GEN b GEN tab long prec where an omitted tab is coded as NULL 3 9 11 intnuminit a b m 0 Initialize tables for integration from a to b where a and b are coded as in intnum Only the compactness the possible existence of singularities the speed of decrease or the oscillations at infinity are taken into account and not the values For instance intnuminit 1 1 is equivalent to intnuminit 0 Pi and intnuminit 0 1 2 1 is equiv alent to intnuminit 1 1 1 2 If mis not given it is computed according to the current precision Otherwise the integration step is 1 2 Reasonable values of m are m 6 or m 7 for 100 decimal digits and m 9 for 1000 decimal digits The result is technical but in some cases it is useful to know the output Let x t be the change of variable which is used tab 1 contains the integer m as above either given by the user or computed from the default precision and can be recomputed directly using the function intnumstep tab 2 and tab 3 contain respectively the abscissa and weight corresponding to t 0 0 and 0 tab 4 and tab 5 contain the abscissas and weights corresponding to positive t nh for 1 lt n lt N and h 1 2 nh and nh Finally tab 6 and tab 7 contain either the abscissas and
373. ll primes is greater than 1 ispowerful 50 hi 0 ispowerful 100 42 1 ispowerful 573 10 1000 1 72 43 1 The library syntax is long ispowerful GEN x 3 4 41 isprime z flag 0 True 1 if x is a prime number false 0 otherwise A prime number is a positive integer having exactly two distinct divisors among the natural numbers namely 1 and itself This routine proves or disproves rigorously that a number is prime which can be very slow when x is indeed prime and has more than 1000 digits say Use ispseudoprime to quickly check for compositeness See also factor It accepts vector matrices arguments and is then applied componentwise If flag 0 use a combination of Baillie PSW pseudo primality test see ispseudoprime Selfridge p 1 test if x 1 is smooth enough and Adleman Pomerance Rumely Cohen Lenstra APRCL for general x If flag 1 use Selfridge Pocklington Lehmer p 1 test and output a primality certificate as follows return e 0 if x is composite e 1 if x is small enough that passing Baillie PSW test guarantees its primality currently x lt 2 as checked by Jan Feitsma e 2 if x is a large prime whose primality could only sensibly be proven given the algorithms implemented in PARI using the APRCL test e Otherwise x is large and x 1 is smooth output a three column matrix as a primality certificate The first column contains prime divisors p of x 1 s
374. lly gives you 19 digits 19 lt log 2 lt 20 The value returned when typing default realprecision is the internal number of significant digits not the number of printed digits default realprecision 2 realprecision 19 significant digits 2 digits displayed default realprecision 1 19 The default value is 38 resp 28 on a 64 bit resp 32 bit machine 3 14 34 recover This toggle is either 1 on or 0 off If you change this to 0 any error becomes fatal and causes the gp interpreter to exit immediately Can be useful in batch job scripts The default value is 1 3 14 35 secure This toggle is either 1 on or 0 off If on the system and extern command are disabled These two commands are potentially dangerous when you execute foreign scripts since they let gp execute arbitrary UNIX commands gp will ask for confirmation before letting you or a script unset this toggle The default value is 0 3 14 36 seriesprecision Number of significant terms when converting a polynomial or rational function to a power series see ps The default value is 16 3 14 37 simplify This toggle is either 1 on or 0 off When the PARI library computes something the type of the result is not always the simplest possible The only type conversions which the PARI library does automatically are rational numbers to integers when they are of type t_FRAC and equal to integers and similarly rational functions to polynomia
375. lmods For vectors and matrices the norm is taken componentwise and hence is not the L norm see norm12 Note that the norm of an element of R is its square so as to be compatible with the complex norm The library syntax is GEN gnorm GEN x 3 2 43 numerator x Numerator of x The meaning of this is clear when z is a rational number or function If x is an integer or a polynomial it is treated as a rational number or function respectively and the result is x itself For polynomials you probably want to use numerator content x instead In other cases numerator x is defined to be denominator x x This is the case when z is a vector or a matrix but also for t_COMPLEX or t_QUAD In particular since a t_PADIC or t_INTMOD has denominator 1 its numerator is itself 82 Warning Multivariate objects are created according to variable priorities with possibly surprising side effects x y is a polynomial but y x is a rational function See Section 2 5 3 The library syntax is GEN numer GEN x 3 2 44 numtoperm n k Generates the k th permutation as a row vector of length n of the numbers 1 to n The number k is taken modulo n i e inverse function of permtonum The numbering used is the standard lexicographic ordering starting at 0 The library syntax is GEN numtoperm long n GEN k 3 2 45 padicprec z p Absolute p adic precision of the object x This is the minimum precision of the components of x The result is
376. lotting devices few significant digits of the result will be meaningful Hence you should keep the current precision to a minimum e g 9 before calling this function n specifies the number of reference point on the graph where a value of 0 means we use the hardwired default values 1000 for general plot 1500 for parametric plot and 8 for recursive plot If no flag is given expr is either a scalar expression f X in which case the plane curve y f X will be drawn or a vector f X f X and then all the curves y f X will be drawn in the same window The binary digits of flag mean e 1 Parametric parametric plot Here expr must be a vector with an even number of components Successive pairs are then understood as the parametric coordinates of a plane curve Each of these are then drawn For instance ploth X 0 2 Pi sin X cos X Parametric ploth X 0 2 Pi sin X cos X ploth X 0 2 Pi X X sin X cos X Parametric draw successively a circle two entwined sinusoidal curves and a circle cut by the line y x e 2 Recursive recursive plot If this flag is set only one curve can be drawn at a time i e expr must be either a two component vector for a single parametric curve and the parametric flag has to be set or a scalar function The idea is to choose pairs of successive reference points and if their middle point is not too far away from the segment joining them draw this as a local approx
377. ls when they are of type t_RFRAC and equal to polynomials This feature is useful in many cases and saves time but can be annoying at times Hence you can disable this and whenever you feel like it use the function simplify see Chapter 3 which allows you to simplify objects to the simplest possible types recursively see y The default value is 1 311 3 14 38 sopath This is a list of directories separated by colons semicolons in the DOS world since colons are preempted for drive names When asked to install an external symbol from a shared library whose name is not given by an absolute path does not start with or gp will look for it in these directories in the order they were written in sopath Here as usual means the current directory and its immediate parent Environment expansion is performed The default value is corresponding to an empty list of directories install will use the library name as input and look in the current directory if the name is not an absolute path 3 14 39 strictargs This toggle is either 1 on or 0 off If on all arguments to new user functions are mandatory unless the function supplies an explicit default value Otherwise arguments have the default value 0 In this example fun a b 2 a b a is mandatory while b is optionnal If strictargs is on fun x at top level fun 4K gt Hook in function fun a b 2 KK Rees k missing
378. lt from the input 0 represents any sufficiently small real number But it was thought more useful to try and go on with an approximate computation than to raise an early exception If the series precision is insufficient errors may occur mostly division by 0 which could have been avoided by a better global understanding of the computation A 1 y 0 B 1 0 y 7 B denominator A 12 0 E 28 O y A B kk _ _ Warning normalizing a series with O leading term 3 1 000000000000000000000000000 y 1 0 1 AXB _ x_ Warning normalizing a series with O leading term 4 1 000000000000000000000000000 y 1 O 1 22 2 3 12 Rational functions t_RFRAC As for fractions all rational functions are automatically reduced to lowest terms All that was said about fractions in Section 2 3 4 remains valid here 2 3 13 Binary quadratic forms of positive or negative discriminant t_QFR and t_QF1 These are input using the function Qfb For example Qfb 1 2 3 creates the binary form x 21y 34 It is imaginary of internal type t_QFI since its discriminant 2 4x3 8 is negative Although imaginary forms could be positive or negative definite only positive definite forms are implemented In the case of forms with positive discriminant t_QFR you may add an optional fourth component related to the regulator more precisely to Shanks and Lenstra s distance which must be a real number See also the f
379. lt i lt 4 Finally a more useful example APPLY f v vector v i f v i applies the function f to every element in the vector v The built in function apply is more powerful since it also applies to lists and matrices 2 7 6 Defining functions within a function Defining a single function is easy init x add y gt x y Basically we are defining a global variable add whose value is the function y gt x y The parentheses were added for clarity and are not mandatory init 5 add 2 42 7 A more refined approach is to avoid global variables and return the function init x y gt x y add init 5 Then add 2 still returns 7 as expected Of course if add is in global scope there is no gain but we can lexically scope it to the place where it is useful my add init 5 How about multiple functions then We can use the last idea and return a vector of functions but if we insist on global variables The first idea 43 init x add y x y mul y x y does not work since in the construction seq the function body contains everything until the end of the expression Hence executing init defines the wrong function add itself defining a function mul The way out is to use parentheses for grouping so that enclosed subexpressions will be evaluated independently init x add y x y mul y x y init 5 add 2 13 7 mul 3 4 15 This defines two
380. many examples to get a feeling for what the various parameters achieve All examples below assume precision is set to 105 decimal digits We first type Ap 105 oo 1 MM for clarity 262 Apparent singularities Even if the function f x represented by expr has no singularities it may be important to define the function differently near special points For instance if f a 1 exp x 1 exp 2 x then J f x du y Euler s constant Euler But f x 1 exp x 1 exp x x intnum x O 00 1 f x Euler 1 6 00 E 67 thus only correct to 67 decimal digits This is because close to 0 the function f is computed with an enormous loss of accuracy A better solution is f x 1 exp x 1 exp x x F truncate f t 0 t77 expansion around t 0 g x if x gt 1e 18 f x subst F t x AN note that6 18 gt 105 intnum x 0 00 1 g x Euler 2 0 E 106 perfect It is up to the user to determine constants such as the 10718 and 7 used above True singularities With true singularities the result is worse For instance intnum x 0 1 1 sqrt x 2 1 1 92 E 59 only 59 correct decimals intnum x 0 1 2 1 1 sqrt x 2 2 0 E 105 better Oscillating functions intnum x 0 oo sin x x Pi 2 1 20 78 nonsense intnum x 0 00 1 sin x x Pi 2 2 0 004 bad intnum x O oo 1 sin x x Pi 2 3 0 E
381. matrices v v where s is half the number of minimal vectors and the v 1 lt i lt s are the minimal vectors Since this requires computing the minimal vectors the computations can become very lengthy as the dimension of x grows The library syntax is GEN perf GEN G 3 8 63 qfrep q B flag 0 q being a square and symmetric matrix with integer entries repre senting a positive definite quadratic form count the vectors representing successive integers e If flag 0 count all vectors Outputs the vector whose i th entry 1 lt i lt B is half the number of vectors v such that q v i e If flag 1 count vectors of even norm Outputs the vector whose i th entry 1 lt i lt Bis half the number of vectors such that q v 2i q 2 1 1 3 qfrep q 5 2 Vecsmal1 0O 1 2 0 OJ 1 vector of norm 2 2 of norm 3 etc qfrep q 5 1 3 Vecsmal1 1 0 O 1 OJ 1 vector of norm 2 O of norm 4 etc This routine uses a naive algorithm based on qfminim and will fail if any entry becomes larger than 2 or 28 The library syntax is GEN qfrepO GEN q GEN B long flag 3 8 64 qfsign x Returns p m the signature of the quadratic form represented by the symmetric matrix x Namely p resp m is the number of positive resp negative eigenvalues of x The result is computed using Gaussian reduction The library syntax is GEN qfsign GEN x 3 8 65 seralgdep s p r finds a linear relation betw
382. me Hence bnr gen would produce an error If flag 1 as the default except that generators are computed The library syntax is GEN bnrinitO GEN bnf GEN f long flag Instead the above hard coded numerical flags one should rather use GEN Buchray GEN bnf GEN module long flag where flag is an or ed combination of nf GEN include generators and nf_INIT if omitted return just the cardinal of the ray class group and its structure possibly 0 3 6 28 bnrisconductor A B C A B C represent an extension of the base field given by class field theory see Section 3 6 5 Outputs 1 if this modulus is the conductor and 0 otherwise This is slightly faster than bnrconductor The library syntax is long bnrisconductor0 GEN A GEN B NULL GEN C NULL 3 6 29 bnrisprincipal bnr x flag 1 bnr being the number field data which is output by bnrinit 1 and x being an ideal in any form outputs the components of x on the ray class group generators in a way similar to bnfisprincipal That is a 2 component vector v where v 1 is the vector of components of x on the ray class group generators v 2 gives on the integral basis an element a such that z a g If flag 0 outputs only v1 In that case bnr need not contain the ray class group generators i e it may be created with bnrinit 0 If x is not coprime to the modulus of bnr the result is undefined The library syntax is GEN bnrisprincipal GEN bnr GEN x long flag
383. me reason killing a user function which is an ordinary variable holding a t_CLOSURE does not remove its name from the list of variable names If the symbol is associated to a variable user functions being an important special case one may use the quote operator a a to reset variables to their starting values However this will not delete a help message associated to a and is also slightly slower than kill a x 1 addhelp x foo x 1 1 x x x same as kill except we don t delete help 12 x x foo On the other hand kill is the only way to remove aliases and installed functions alias fun sin ki11 fun install addii GG kill addii The library syntax is void kill0 const char sym 3 12 23 print str Outputs its string arguments in raw format ending with a newline 3 12 24 print1 str Outputs its string arguments in raw format without ending with a newline Note that you can still embed newlines within your strings using the n notation 294 3 12 25 printf fmt x This function is based on the C library command of the same name It prints its arguments according to the format fmt which specifies how subsequent arguments are converted for output The format is a character string composed of zero or more directives e ordinary characters not printed unchanged e conversions specifications followed by some characters which fetch o
384. med a factorization and is interrupted by an error or via Control C this lets you recover the prime factors already found The downside is that a huge addprimes table unrelated to the current computations will slow down arithmetic functions relying on integer factorization one should then empty the table using removeprimes The defaut value is 0 3 14 11 factor_proven This toggle is either 1 on or 0 off By default the factors output by the integer factorization machinery are only pseudo primes not proven primes If this toggle is set a primality proof is done for each factor and all results depending on integer factorization are fully proven This flag does not affect partial factorization when it is explicitly requested It also does not affect the private table managed by addprimes its entries are included as is in factorizations without being tested for primality The defaut value is 0 3 14 12 format Of the form x n where x conversion style is a letter in e f g and n precision is an integer this affects the way real numbers are printed e If the conversion style is e real numbers are printed in scientific format always with an explicit exponent e g 3 3 E 5 e In style f real numbers are generally printed in fixed floating point format without exponent e g 0 000033 A large real number whose integer part is not well defined not enough significant digits is printed in style e For instance 10 7100 known to
385. ming On the other hand numerical routines code a function to be integrated summed etc with two parameters named GEN eval voidx GEN void E AN context eval E x must evaluate your function at x see the Libpari manual for details Numerical integration Starting with version 2 2 9 the double exponential univariate integra tion method is implemented in intnum and its variants Romberg integration is still available under the name intnumromb but superseded It is possible to compute numerically integrals to thousands of decimal places in reasonable time as long as the integrand is regular It is also reasonable to compute numerically integrals in several variables although more than two becomes lengthy The integration domain may be non compact and the integrand may have reasonable singularities at endpoints To use intnum you must split the integral into a sum of subintegrals where the function has no singularities except at the endpoints Polynomials in logarithms are not considered singular and neglecting these logs singularities are assumed to be algebraic asymptotic to C x a for some a gt 1 when x is close to a or to correspond to simple discontinuities of some higher derivative of the function For instance the point 0 is a singularity of abs x See also the discrete summation methods below sharing the prefix sum 3 9 1 derivnum X a expr Numerical derivation of expr with respect to X at X a
386. mmands or keyboard shortcuts to be dealt with in Section 2 13 2 1 3 Input Just type in an instruction e g 1 1 or Pi No action is undertaken until you hit the lt Return gt key Then computation starts and a result is eventually printed To suppress printing of the result end the expression with a sign Note that many systems use to indicate end of input Not so in gp a final semicolon means the result should not be printed Which is certainly useful if it occupies several screens 2 1 4 Interrupt Quit Typing quit at the prompt ends the session and exits gp At any point you can type Ctr1 C that is press simultaneously the Control and C keys the current computation is interrupted and control given back to you at the gp prompt together with a message like kk at top level gcd a b xk gcd user interrupt after 236 ms telling you how much time elapsed since the last command was typed in and in which GP function the computation was aborted It does not mean that that much time was spent in the function only that the evaluator was busy processing that specific function when you stopped it 14 2 2 The general gp input line The gp calculator uses a purely interpreted language GP The structure of this language is reminiscent of LISP with a functional notation f x y rather than f x y all programming constructs such as if while etc are functions and the main loop does not really execute b
387. monic polynomial at this point gp issues a warning then transforms your polynomial so that it becomes monic The nfinit routine will return a different result in this case instead of res you get a vector res Mod a Q where Mod a Q Mod X P gives the change of variables In all other routines the variable change is simply lost The tech interface is obsolete and you should not tamper with these parameters Indeed from version 2 4 0 on e the results are always rigorous under GRH before that version they relied on a heuristic strengthening hence the need for overrides e the influence of these parameters on execution time and stack size is marginal They can be useful to fine tune and experiment with the bnfinit code but you will be better off modifying all tuning parameters in the C code there are many more than just those three We nevertheless describe it for completeness The numbers c lt cg are non negative real numbers By default they are chosen so that the result is correct under GRH For i 1 2 let B c log dx and denote by S B the set of maximal ideals of K whose norm is less than B We want S B to generate CI K and hope that S B2 can be proven to generate Cl K More precisely S B1 is a factorbase used to compute a tentative CI K by generators and relations We then check explicitly using essentially bnfisprincipal that the elements of S B2 belong to the span of S B Under the assumption that S
388. mpt Here the computation is aborted as soon as we try to evaluate 1 0 and 1 1 is never executed Exceptions can be trapped using iferr however we can evaluate some expression and either recover an ordinary result no error occurred or an exception an error did occur i Mod 6 12 iferr 1 i E print E 1 1 error impossible inverse modulo Mod 6 12 1 2 One can ignore the exception print it as above or extract non trivial information from the error context i Mod 6 12 iferr 1 i E print component E 1 Mod 6 12 We can also rethrow the exception error E 2 4 GP operators Loosely speaking an operator is a function usually associated to basic arithmetic operations whose name contains only non alphanumeric characters For instance or but also or or even the selection operator As all functions operators take arguments and return a value assignment operators also have side effects besides returning a value they change the value of some variable Each operator has a fixed and unchangeable priority which means that in a given expression the operations with the highest priority is performed first Unless mentioned otherwise opera tors at the same priority level are left associative performed from left to right unless they are assignments in which case they are right associative Anything enclosed between parenthesis is considered a complete subexpression and is resolved
389. ms see qfbnucomp in particular the final warning The library syntax is GEN nupow GEN x GEN n 3 4 66 qfbpowraw x n n th power of the binary quadratic form x computed without doing any reduction i e using qfbcompraw Here n must be non negative and n lt 231 The library syntax is GEN qfbpowraw GEN x long n 3 4 67 qfbprimeform z p Prime binary quadratic form of discriminant x whose first coefficient is p where p is a prime number By abuse of notation p 1 is also valid and returns the unit form Returns an error if x is not a quadratic residue mod p or if lt 0 and p lt 0 Negative definite t_QFI are not implemented In the case where x gt 0 the distance component of the form is set equal to zero according to the current precision The library syntax is GEN primeform GEN x GEN p long prec 3 4 68 qfbred z flag 0 d isd sd Reduces the binary quadratic form x updating Shanks s distance function if x is indefinite The binary digits of flag are toggles meaning 1 perform a single reduction step 2 don t update Shanks s distance The arguments d isd sd if present supply the values of the discriminant va and vd respectively no checking is done of these facts If d lt 0 these values are useless and all references to Shanks s distance are irrelevant The library syntax is GEN qfbredO GEN x long flag GEN d NULL GEN isd NULL GEN sd NULL Also available a
390. ms for s gt 0 and odd For s a p adic number Kubota Leopoldt zeta function at s that is the unique continuous p adic function on the p adic integers that interpolates the values of 1 p k at negative integers k such that k 1 mod p 1 resp k is odd if p is odd resp p 2 The library syntax is GEN gzeta GEN s long prec 98 3 4 Arithmetic functions These functions are by definition functions whose natural domain of definition is either Z or Zs The way these functions are used is completely different from transcendental functions in that there are no automatic type conversions in general only integers are accepted as arguments An integer argument N can be given in the following alternate formats e t_MAT its factorization fa factor N e t_VEC a pair V fa giving both the integer and its factorization This allows to compute different arithmetic functions at a given N while factoring the latter only once N 10 faN factor N eulerphi N 12 829440 eulerphi faN 13 829440 eulerphi S N faN 4 829440 sigma S 15 15334088 3 4 1 Arithmetic functions and the factoring engine All arithmetic functions in the narrow sense of the word Euler s totient function the Moebius function the sums over divisors or powers of divisors etc call after trial division by small primes the same versatile factoring machinery described under factorint It includes Shanks SQU
391. multaneously Decide whether you agree with what Configure printed on your screen in particular the ar chitecture compiler and optimization flags Look for messages prepended by which report genuine problems Look especially for the gmp readline and X11 libraries and the perl and gun zip or zcat binaries If anything should have been found and was not consider that Configure failed and follow the instructions in section 3 The Configure run creates a file config 1log in the build directory which contains debugging information in particular all messages from compilers that may help diagnose problems This file is erased and recreated from scratch each time Configure is run 2 2 Advanced configuration Configure accepts many other flags and you may use any number of them to build quite a complicated configuration command See Configure help for a complete list In particular there are sets of flags related to GNU MP with gmp and GNU readline library with readline Here we focus on the non obvious ones tune fine tunes the library for the host used for compilation This adjusts thresholds by running a large number of comparative tests and creates a file tune h in the build directory that will be used from now on overriding the ones in src kernel none and src kernel gmp It will take a while about 30 minutes Expect a small performance boost perhaps a 10 speed increase compared to default settings
392. n e tate returns u u q a b in the notation of Henniart Mestre CRAS t 308 p 391 395 1989 q is as above u Qp y c6 is such that dx 2y a x a3 udt t where Eg gt E is an isomorphism well defined up to sign and dt t is the canonical invariant differential on the Tate curve u Qp does not depend on Technicality if u Qp it is stored as a quadratic t_POLMOD Finally a b satisfy 4u2b agm ya b 1 1 as in Theorem 2 loc cit 131 3 5 1 4 Curves over F e pis the characteristic of F4 e no is E F e cyc gives the cycle structure of E F e gen returns the generators of E F e group returns no cyc gen i e E F as an abelian group structure 3 5 1 5 Curves over Q All functions should return a correct result whether the model is minimal or not but it is a good idea to stick to minimal models whenever gcd c4 ce is easy to factor minor speed up The construction E ellminimalmodel E0 amp v replaces the original model Eg by a minimal model Y and the variable change v allows to go between the two models ellchangepoint PO v ellchangepointinv P v respectively map the point Py on Ep to its image on E and the point P on E to its pre image on Eo A few routines namely ellgenerators ellidentify ellsearch forell require the optional package elldata John Cremona s database to be installed In that case the function ellinit will allow alt
393. n To convert any object into a set this is most useful for vectors of course use the function Set e Qj Ss de deg 2 setisset a 12 0 Set a 73 1 2 3 The library syntax is long setisset GEN x 3 8 69 setminus x y Difference of the two sets x and y see setisset i e set of elements of x which do not belong to y If x or y is not a set the result is undefined The library syntax is GEN setminus GEN x GEN y 3 8 70 setsearch S x flag 0 Determines whether x belongs to the set S see setisset We first describe the default behaviour when flag is zero or omitted If x belongs to the set S returns the index j such that S j x otherwise returns 0 T 7 2 3 5 S Set T setsearch S 2 12 1 setsearch S 4 not found 43 0 setsearch T 7 search in a randomly sorted vector 4 O AN WRONG If S is not a set we also allow sorted lists with respect to the cmp sorting function without repeated entries as per listsort L 1 otherwise the result is undefined L List 1 4 2 3 21 setsearch L 4 252 41 O WRONG listsort L 1 L sort L first 12 List 1 2 3 4 setsearch L 4 13 4 now correct If flag is non zero this function returns the index j where x should be inserted and 0 if it already belongs to S This is meant to be used for dynamically growing sorted lists in conjunction with listinsert L List 1 5 2 3 21 l
394. n and other system independent data under share_prefix pari documenta tion sample GP scripts and C code extra packages like elldata or galdata You can also set directly bindir executables libdir library includedir include files mandir manual pages datadir other architecture independent data and finally sysdatadir other architecture dependent data 3 2 Environment variables Configure lets the following environment variable override the defaults if set Cc C compiler DLLD Dynamic library linker LD Static linker For instance Configure may avoid bin cc on some architectures due to various problems which may have been fixed in your version of the compiler You can try env CC cc Configure and compare the benches Also if you insist on using a C compiler and run into trouble with a fussy g try to use g fpermissive The contents of the following variables are appended to the values computed by Configure CFLAGS Flags for CC CPPFLAGS Flags for CC preprocessor LDFLAGS Flags for LD The contents of the following variables are prepended to the values computed by Configure 318 C_INCLUDE PATH is prepended to the list of directories searched for include files Note that adding I flags to CFLAGS is not enough since Configure sometimes relies on finding the include files and parsing them and it does not parse CFLAGS at this time LIBRARY _PATH is prepended to the list of directories
395. n element of R X T X the variable X is a mute variable and the integral is taken with respect to the main variable used in the base ring R In particular it is meaningless to integrate with respect to the main variable of x mod intformal Mod 1 x 2 1 x intformal incorrect priority in intformal variable x x The library syntax is GEN integ GEN x long v 1 where v is a variable number 218 3 7 8 padicappr pol a Vector of p adic roots of the polynomial pol congruent to the p adic number a modulo p and with the same p adic precision as a The number a can be an ordinary p adic number type t_PADIC i e an element of Zp or can be an integral element of a finite extension of Qp given as a t_POLMOD at least one of whose coefficients is a t_PADIC In this case the result is the vector of roots belonging to the same extension of Q as a The library syntax is GEN padicappr GEN pol GEN a Also available is GEN Zp_appr GEN f GEN a when a is a t_PADIC 3 7 9 padicfields p N flag 0 Returns a vector of polynomials generating all the extensions of degree N of the field Q of p adic rational numbers N is allowed to be a 2 component vector n d in which case we return the extensions of degree n and discriminant pt The list is minimal in the sense that two different polynomials generate non isomorphic ex tensions in particular the number of polynomials is the number of classes of non isomorphic extensions
396. n subgrouplist is not an option due to the sheer size of the output For maximal speed the subgroups have been left as produced by the algorithm To print them in canonical form as left divisors of G in HNF form one can for instance use G matdiagonal 2 2 forsubgroup H G 2 print mathnf concat G H 2 1 0 1 1 0 O 2 2 0 O 1 1 0 O 1 Note that in this last representation the index G H is given by the determinant See galois subcyclo and galoisfixedfield for applications to Galois theory The library syntax is forsubgroup void data long cal1 void GEN GEN G GEN bound 282 3 11 15 forvec X v seq flag 0 Let v be an n component vector where n is arbitrary of two component vectors a b for 1 lt i lt n This routine evaluates seq where the formal variables X 1 X n go from a to b from an to bn ie X goes from a an to b1 bn with respect to the lexicographic ordering The formal variable with the highest index moves the fastest If flag 1 generate only nondecreasing vectors X and if flag 2 generate only strictly increasing vectors X The type of X is the same as the type of v t_VEC or t_COL 3 11 16 if a seq1 seq2 Evaluates the expression sequence seq1 if a is non zero otherwise the expression seg2 Of course seq1 or seq2 may be empty if a seq evaluates seq if a is not equal to zero you don t have to write the second comma
397. n zero some extra messages may be printed according to what is going on see g The default value is 0 no debugging messages 3 14 7 debugfiles File usage debugging level If it is non zero gp will print information on file descriptors in use from PARI s point of view see gf The default value is 0 no debugging messages 3 14 8 debugmem Memory debugging level If it is non zero gp will regularly print information on memory usage If it s greater than 2 it will indicate any important garbage collecting and the function it is taking place in see gm Important Note As it noticeably slows down the performance the first functionality memory usage is disabled if you re not running a version compiled for debugging see Appendix A The default value is 0 no debugging messages 305 3 14 9 echo This toggle is either 1 on or 0 off When echo mode is on each command is reprinted before being executed This can be useful when reading a file with the r or read commands For example it is turned on at the beginning of the test files used to check whether gp has been built correctly see Ne The default value is 0 no echo 3 14 10 factor_add_primes This toggle is either 1 on or 0 off If on the integer factorization machinery calls addprimes on primes factor that were difficult to find larger than 2 4 so they are automatically tried first in other factorizations If a routine is performing or has perfor
398. nd a are as follows e n nmaz resp n nmin nmaz restricts partitions to length less than nmaz resp length between nmin and nmax where the length is the number of nonzero entries 280 ea amaz resp a amin amaz restricts the parts to integers less than amaz resp between amin and amaz By default parts are positive and we remove zero entries unless amin lt 0 in which case X is of constant length nmaz at most 3 non zero parts all lt 4 forpart v 5 print Vec v 4 3 1 4 2 3 1 1 3 1 2 2 between 2 and 4 parts less than 5 fill with zeros forpart v 5 print Vec v 0 51 1 2 4 0 0 1 4 0 0 2 3 0 1 1 3 0 1 2 2 1 1 1 2 The behaviour is unspecified if X is modified inside the loop The library syntax is forpart void data long call void GEN long k GEN a GEN n 3 11 12 forprime p a b seq Evaluates seq where the formal variable p ranges over the prime numbers between the real numbers a to b including a and b if they are prime More precisely the value of p is incremented to nextprime p 1 the smallest prime strictly larger than p at the end of each iteration Nothing is done if a gt b forprime p 4 10 print p 5 7 Omitting b means we will run through all primes gt a starting an infinite loop it is expected that the user will break out of the loop himself at some point using break or return Note that the v
399. nd string horizontal if bit 16 is set vertical if bit 32 is set see the tutorial for an example 3 10 31 psdraw list flag 0 Same as plotdraw except that the output is a PostScript program appended to the psfile and flag 0 scales the plot from size of the current output device to the standard PostScript plotting size 277 3 10 32 psploth X a b expr flags 0 n 0 Same as ploth except that the output is a PostScript program appended to the psfile 3 10 33 psplothraw listz listy flag 0 Same as plothraw except that the output is a PostScript program appended to the psfile 3 11 Programming in GP control statements A number of control statements are available in GP They are simpler and have a syntax slightly different from their C counterparts but are quite powerful enough to write any kind of program Some of them are specific to GP since they are made for number theorists As usual X will denote any simple variable name and seq will always denote a sequence of expressions including the empty sequence Caveat In constructs like for X a b seq the variable X is lexically scoped to the loop leading to possibly unexpected behavior n 5 for n 1 10 if something_nice break at this point n is 5 If the sequence seq modifies the loop index then the loop is modified accordingly for n 1 10 n 2 print n 3 6 9 12 3 11 1 break n 1 Interrupts executi
400. nd the class field must be totally real When the base field is Q the vastly simpler galoissubcyclo is used instead Here is an example bnf bnfinit y 2 3 bnr bnrinit bnf 5 1 bnrstark bnr returns the ray class field of Q V3 modulo 5 Usually one wants to apply to the result one of rnfpolredabs bnf pol 16 compute a reduced relative polynomial rnfpolredabs bnf pol 16 2 compute a reduced absolute polynomial The routine uses Stark units and needs to find a suitable auxiliary conductor which may not exist when the class field is not cyclic over the base In this case bnrstark is allowed to return a vector of polynomials defining independent relative extensions whose compositum is the requested class field It was decided that it was more useful to keep the extra information thus made available hence the user has to take the compositum herself Even if it exists the auxiliary conductor may be so large that later computations become unfeasible And of course Stark s conjecture may simply be wrong In case of difficulties try rnfkummer bnr bnrinit bnfinit y 8 12 y 6 36 y 4 36 y 2t9 1 2 1 bnrstark bnr at top level bnrstark bnr xk xxx bnrstark need 3919350809720744 coefficients in initzeta Computation impossible 167 lift rnfkummer bnr time 24 ms Y2 x72 1 3 y 6 11 3 y74 8xy 2 5 The library syntax is GEN bnrstark GEN bnr GEN subgroup NUL
401. ne argument from the list and prints it according to the specification More precisely a conversion specification consists in a one or more optional flags among 0 an optional decimal digit string specifying a minimal field width an optional precision in the form of a period followed by a decimal digit string and the conversion specifier among d i o u x X p e E f g G s The flag characters The character is followed by zero or more of the following flags e The value is converted to an alternate form For o conversion octal a 0 is prefixed to the string For x and X conversions hexa respectively 0x and OX are prepended For other conversions the flag is ignored e 0 The value should be zero padded For d i o u x X e E f F g and G conversions the value is padded on the left with zeros rather than blanks If the 0 and flags both appear the 0 flag is ignored e The value is left adjusted on the field boundary The default is right justification The value is padded on the right with blanks rather than on the left with blanks or zeros A overrides a 0 if both are given e a space A blank is left before a positive number produced by a signed conversion e A sign or is placed before a number produced by a signed conversion A overrides a space if both are used The field width An optional decimal digit string whose first digit is non zer
402. nes whether x belongs to the sorted vector or list v return the positive index where x was found or 0 if it does not belong to v If the comparison function cmpf is omitted we assume that v is sorted in increasing order according to the standard comparison function lt thereby restricting the possible types for x and the elements of v integers fractions or reals If cmpf is present it is understood as a comparison function and we assume that v is sorted according to it see vecsort for how to encode comparison functions v 1 3 4 5 7 vecsearch v 3 2 2 vecsearch v 6 13 0 XX not in the list vecsearch 7 6 5 5 unsorted vector result undefined 14 0 By abuse of notation x is also allowed to be a matrix seen as a vector of its columns again by abuse of notation a t_VEC is considered as part of the matrix if its transpose is one of the matrix columns v vecsort 3 0 2 1 0 21 sort matrix columns according to lex order 1 0 2 3 254 o 2 1 vecsearch v 3 1 12 3 vecsearch v 3 1 can search for x or x 43 3 vecsearch v 1 2 4 0 XX not in the list The library syntax is long vecsearch GEN v GEN x GEN cmpf NULL 3 8 75 vecsort x cmpf flag 0 Sorts the vector x in ascending order using a mergesort method x must be a list vector or matrix seen as a vector of its columns Note that mergesort is stable hence the initial ordering of
403. nf Proc 27 2000 pp 29 34 proved that the formula used in sumalt with n terms computes 1 2 s with a relative error of the order of 3 8 I s 7 41 2 7 3 Beware scopes Be extra careful with the scopes of variables What is wrong with the following definition FirstPrimeDiv x my p forprime p 2 x if x p 0 break P FirstPrimeDiv 10 1 0 Hint The function body is equivalent to my newp 0 forprime p 2 x if x p 0 break newp Detailed explanation The index p in the forprime loop is lexically scoped to the loop and is not visible to the outside world Hence it will not survive the break statement More precisely at this point the loop index is restored to its preceding value The initial my p although well meant adds to the confusion it indeed scopes p to the function body with initial value 0 but the forprime loop introduces another variable unfortunately also called p scoped to the loop body which shadows the one we wanted So we always return 0 since the value of the p scoped to the function body never changes and is initially 0 To sum up the routine returns the p declared local to it not the one which was local to forprime and ran through consecutive prime numbers Here is a corrected version FirstPrimeDiv x forprime p 2 x if x p 0 return p 2 7 4 Recursive functions Recursive functions can easily be written as long as one pays proper
404. nf for short which contains enough information to recover a full bnf vector very rapidly but which is much smaller and hence easy to store and print Calling bnfinit on the result recovers a true bnf in general different from the original Note that an snbf is useless for almost all purposes besides storage and must be converted back to bnf form before use for instance no nf bnf or member function accepts them An sbnf is a 12 component vector v as follows Let bnf be the result of a full bnfinit complete with units Then v 1 is bnf pol v 2 is the number of real embeddings bnf sign 1 v 3 is bnf disc v 4 is bnf zk v 5 is the list of roots bnf roots v 7 is the matrix W bnf 1 v 8 is the matrix matalpha bnf 2 v 9 is the prime ideal factor base bnf 5 coded in a compact way and ordered according to the permutation bnf 6 v 10 is the 2 component vector giving the number of roots of unity and a generator expressed on the integral basis v 11 is the list of fundamental units expressed on the integral basis v 12 is a vector containing the algebraic numbers alpha corresponding to the columns of the matrix matalpha expressed on the integral basis All the components are exact integral or rational except for the roots in v 5 The library syntax is GEN bnfcompress GEN bnf 3 6 10 bnfdecodemodule nf m If m is a module as output in the first component of an extension given by bnrdisclist outputs the true module
405. nfidealtwoelt 208 rnfidealup lt i 4 42 46 86 24 208 209 TOTO oh ooh we Ae aa e 209 210 rnfisabelian exceso 210 INTIS FES ue ras RO a 210 IO TISDOTD estes 210 211 rnfisnorminit 210 211 ynfkummer 167 211 214 OTI Eramos se 2 wah as S 211 rnfnormgroup 211 rnfpolr d rim Ad Skea ws 211 212 IN polredabS 212 338 rnfpolredbest 211 212 215 InfpseudobasiS 213 rnfsimplifybasis 192 TH STSIDILZ o s s soeces wos a a So 213 A boa en ae a Ae E E aes 185 FOOTMOdO sei esua e m e i a 223 FOOUPAdLG ss a ee ds 223 TOOTS pai go Oe ees 130 131 155 223 GOOCSOL 2240 408 ee oS E 196 rootsofi_kannan 196 round A se seata eoe a a 185 218 TOUDO 2 si aeee oie ee S45 e ia 85 TOUD O o e eoma e a ee He aes 85 TOW VEC OE osis sosom s eh eo rod T23 S scalar product 65 scalar type eor e eii ee ee ew SS T SCHeCMZ e pus de S 125 Schonage s sess saans eoa ia gpd pei 225 scientific format 306 SEA ada eee Be 134 seadata usa Bas Skee poirie 134 S CUTE morrison ee ee ee wae 311 Select 24 44 4 4 we Beh Ss 6 ek As 298 GOY a et dae ie e DT Ae Soe 22 74 226 Sseralgdep e murio he aide a alk 251 serconvol 224 seriesprecision 57 89 147 311 serlaplace 224 serreverse 224 225 SOL a e Gem Bos Oh ook OO ee ee bs 75 SO BDIDOD 224 25
406. ng the defaults In particular file input is delayed until the gprc has been fully loaded This is the right place to input files containing alias commands or your favorite macros For instance you could set your prompt in the following portable way self modifying prompt looking like 18 03 gp gt prompt H M eL imgp elm gt readline wants non printing characters to be braced between A B pairs if READL prompt H M ANe lim Bgp Alel m B gt escape sequences not supported under emacs if EMACS prompt H M gp gt Note that any of the last two lines could be broken in the following way if EMACS 59 prompt H M gp gt since the preprocessor directive applies to the next line if the current one is empty A sample gprc file called misc gprc dft is provided in the standard distribution It is a good idea to have a look at it and customize it to your needs Since this file does not use multiline constructs here is one note the terminating to separate the expressions if VERSION gt 2 2 3 read my_scripts syntax errors in older versions new_galois_format 1 default introduced in 2 2 4 if EMACS colors 9 5 no no 4 1 2 help gphelp detex ch 4 cb 0 cu 2 2 14 2 The gprc location When gp is started it looks for a customization file or gprc in the following places in this order only the first one found will be loaded e gp che
407. nit y 2 1 P idealprimedec K 3 1 nffactormod K x 2 y x 18xy 1 P 13 x 2 y 1 1 x Qty 2 1 P nfmodprinit K P convert to nfmodprinit format nffactormod K x 2 y x 18 y 1 x 2xy 1 1 x 2 y 2 1 Same result of course here about 10 faster due to the precomputation The library syntax is GEN nffactormod GEN nf GEN Q GEN pr 190 3 6 100 nfgaloisapply nf aut 1 Let nf be a number field as output by nfinit and let aut be a Galois automorphism of nf expressed by its image on the field generator such automorphisms can be found using nfgaloisconj The function computes the action of the automorphism aut on the object x in the number field x can be a number field element or an ideal possibly extended Because of possible confusion with elements and ideals other vector or matrix arguments are forbidden nf nfinit x 2 1 L nfgaloisconj nf 42 x x aut L 1 the non trivial automorphism nfgaloisapply nf aut x 4 Mod x x72 1 P idealprimedec nf 5 prime ideals above 5 nfgaloisapply nf aut P 2 P 1 17 O 1 idealval nf nfgaloisapply nf aut P 21 P 1 28 1 The surprising failure of the equality test 7 is due to the fact that although the corresponding prime ideals are equal their representations are not A prime ideal is specificed by a uniformizer and there is no guarantee that a
408. not be detected although no example is known You are invited to play with the flag settings and watch the internals at work by using gp s debug default parameter level 3 shows just the outline 4 turns on time keeping 5 and above show an increasing amount of internal details The library syntax is GEN factorint GEN x long flag 3 4 26 factormod z p flag 0 Factors the polynomial x modulo the prime integer p using Berlekamp The coefficients of x must be operation compatible with Z pZ The result is a two column matrix the first column being the irreducible polynomials dividing x and the second the exponents If flag is non zero outputs only the degrees of the irreducible polynomials for example for computing an L function A different algorithm for computing the mod p factorization is factorcantor which is sometimes faster The library syntax is GEN factormod0 GEN x GEN p long flag 110 3 4 27 ffgen q v Return a t_FFELT generator for the finite field with q elements q p must be a prime power This functions computes an irreducible monic polynomial P F X of degree f via ffinit and returns g X mod P X If v is given the variable name is used to display g else the variable x is used g ffgen 8 t g mod 12 t 3 t 2 1 g p 3 2 g f h4 3 f gen 6 kk at top level ffgen 6 OK ffgen not a prime number in ffgen 6 Alternative syntax instead o
409. ns 223 polsubcyclo ox dene ee een 224 polsylvestermatrix 224 LE ses psa Pees ye a ee Rie 4 224 poltchebi gt se aires eee kee en 224 poltschirnhaus 202 polylog srca ressar bbe a es 95 POLVLOSO ios ee ee 96 polynomial eee al polzag euros phe ae eee Pad 224 polzagier esa reata ee eas 224 PostScript ses ib e tn ae od we 272 power series ooo a T 8 22 powering 66 89 DIECISION 2 5 2 va 4 fee ba ee oe Bes 88 Precision p m s sews Hace Bes 83 84 precisionO ss esas be a 84 PTECprime s ss dei HE is 120 preferences file 13 54 58 prettymatriz format 4 308 prettyprinter 308 309 Prid ne ek a PRESS 44 180 PLING eo e lc ie Fk 120 Primeform sa ses eakad baaie 123 primelimit sa sasis a pore els 199 309 PEIMOPI z ee ot Pe Bow aa bos 120 121 PRIMES spoe se sd dupe da o amp amp 121 Primes ceea ne Sy ea ee ew v s 121 principal ideal 160 182 PEINE era a a 46 47 294 PRIMtL 2 2 Gia aw ras de ha ws 294 printf cia a iw caw tae s 294 297 306 PYINUSEP sosoo aosa ee eee ee ee a 297 Printsepl a sroti eae dade ed A 297 PEI Hee kek med Yee Ge oo 297 PLOT sica a a 28 Prod ecra oi sa ES SS 266 prodeuler saose toemon er 266 267 Prodin se s e sa a fieito a de ets 267 Prodidtl sts ge ae wks g a Ha es 267 Produ a wala ke He eae aa 65 PrOdUlt Leccion 266 programming 278 projective module
410. nvert the polynomials to vectors of the same length first P xt1 Q x72 2 xt1 chinese Px Mod 1 2 Q Mod 1 3 4 Mod 1 3 x 2 Mod 5 6 x Mod 3 6 chinese Vec P 3 Mod 1 2 Vec Q 3 Mod 1 3 5 Mod 1 6 Mod 5 6 Mod 4 6 Pol 76 Mod 1 6 x 2 Mod 5 6 x Mod 4 6 If y is omitted and x is a vector chinese is applied recursively to the components of zx yielding a residue belonging to the same class as all components of x Finally chinese x x x regardless of the type of x this allows vector arguments to contain other data so long as they are identical in both vectors The library syntax is GEN chinese GEN x GEN y NULL GEN chinese1 GEN x is also available 102 3 4 10 content x Computes the gcd of all the coefficients of x when this gcd makes sense This is the natural definition if x is a polynomial and by extension a power series or a vector matrix This is in general a weaker notion than the ideal generated by the coefficients content 2 xty 1 1 ged 2 y over Qly l If x is a scalar this simply returns the absolute value of x if x is rational t_INT or t_FRAC and either 1 inexact input or x exact input otherwise the result should be identical to gcd x 0 The content of a rational function is the ratio of the contents of the numerator and the de nominator In recursive structures if a matrix or vector coefficient x appears the gcd is taken not wit
411. o specifying a minimum field width If the value has fewer characters than the field width it is padded with spaces on the left or right if the left adjustment flag has been given In no case does a small field width cause truncation of a field if the value is wider than the field width the field is expanded to contain the conversion result Instead of a decimal digit string one may write to specify that the field width is given in the next argument The precision An optional precision in the form of a period followed by a decimal digit string This gives the number of digits to appear after the radix character for e E f and F conversions the maximum number of significant digits for g and G conversions and the maximum number of characters to be printed from an s conversion Instead of a decimal digit string one may write to specify that the field width is given in the next argument The length modifier This is ignored under gp but necessary for libpari programming De scription given here for completeness e 1 argument is a long integer e P argument is a GEN 295 The conversion specifier A character that specifies the type of conversion to be applied e d i A signed integer eo u x X An unsigned integer converted to unsigned octal o decimal u or hexadecimal x or X notation The letters abcdef are used for x conversions the letters ABCDEF are used for X conversions e e E The real argum
412. o the elliptic curve E E is either e a 5 component vector a1 az a3 a4 a6 defining the elliptic curve with Weierstrass equation Y Em XY aY X a2X at a6 e a 2 component vector a4 ag defining the elliptic curve with short Weierstrass equation Y X 4 aX 06 e a character string in Cremona s notation e g 11a1 in which case the curve is retrieved from the elldata database if available The optional argument D describes the domain over which the curve is defined e the t_INT 1 default the field of rational numbers Q e a t_INT p where p is a prime number the prime finite field F e an t_INTMOD Mod a p where pis a prime number the prime finite field F e a t_FFELT as returned by ffgen the corresponding finite field Fy e a t_PADIC O p the field Qp where p adic quantities will be computed to a relative accuracy of n digits We advise to input a model defined over Q for such curves In any case if you input an approximate model with t_PADIC coefficients it will be replaced by a lift to Q an exact model close to the one that was input and all quantities will then be computed in terms of this lifted model at the given accuracy e a t_REAL zx the field C of complex numbers where floating point quantities are by default computed to a relative accuracy of precision x If no such argument is given the value of realprecision at the time ellinit is called will be used This argument D is in
413. od coefficients computes x as an element of K as a polmod assuming x is in K otherwise a domain error occurs K nfinit y 2 1 L rnfinit K x 2 y L pol 12 x 4 1 rnfeltdown L Mod x 2 L pol 43 Mod y y 2 1 rnfeltdown L Mod y x 2 y 4 Mod y y 2 1 rnfeltdown L Mod y K pol 45 Mod y y 2 1 rnfeltdown L Mod x L pol Hook at top level rnfeltdown L Mod x x Ax O xxx rnfeltdown domain error in rnfeltdown element not in the base field The library syntax is GEN rnfeltdown GEN rnf GEN x 3 6 134 rnfeltnorm rnf x rnf being a relative number field extension L K as output by rn finit and z being an element of L returns the relative norm N x x as an element of K K nfinit y 2 1 L rnfinit K x 2 y rnfeltnorm L Mod x L pol 12 Mod x x72 Mod y y 2 1 rnfeltnorm L 2 13 4 rnfeltnorm L Mod x x 2 y The library syntax is GEN rnfeltnorm GEN rnf GEN x 3 6 135 rnfeltreltoabs rnf x rnf being a relative number field extension L K as output by rnfinit and x being an element of L expressed as a polynomial or polmod with polmod coefficients computes x as an element of the absolute extension L Q as a polynomial modulo the absolute equation rnf pol K nfinit y 2 1 L rnfinit K x 2 y L pol 12 x 4 1 rnfeltreltoabs L Mod x L pol 13 Mod x x 4 1 rnfeltreltoabs L Mod y x 2 y 14 Mod x 2 x 4
414. of course reminiscent of a user function with a single formal variable var For instance the current implementation of the ell type is a vector the j invariant being the thirteenth component It could be implemented as 44 Xi if type x t_VEC tx lt 14 error not an elliptic curve x x 13 As for user functions you can redefine your member functions simply by typing new definitions On the other hand as a safety measure you cannot redefine the built in member functions so attempting to redefine x j as above would in fact produce an error you would have to call it e g x myj in order for gp to accept it Rationale In most cases member functions are simple accessors of the form x a x 1 x b x 2 x c x 3 where x is a vector containing relevant data There are at least three alternative approaches to the above member functions 1 hardcode x 1 etc in the program text 2 define constant global variables AINDEX 1 BINDEX 2 and hardcode x AINDEX 3 user functions a x x 1 and so on Even if 2 improves on 1 these solutions are neither elegant nor flexible and they scale badly 3 is a genuine possibility but the main advantage of member functions is that their namespace is independent from the variables and functions namespace hence we can use very short identifiers without risk The j invariant is a good example it would clearly not be a good idea to define j E E 13 because cl
415. of size n The routine returns either a true Zx basis of M of size n if it exists or an n l element generating set of M if not It is allowed to use an irreducible polynomial P in K X instead of M in which case M is defined as the ring of integers of K X P viewed as a Zx module The library syntax is GEN rnfbasis GEN bnf GEN M 3 6 126 rnfbasistoalg rnf x Computes the representation of x as a polmod with polmods coefficients Here rnf is a relative number field extension L K as output by rnfinit and x an element of L expressed on the relative integral basis The library syntax is GEN rnfbasistoalg GEN rnf GEN x 202 3 6 127 rnfcharpoly nf T a var xj Characteristic polynomial of a over nf where a be longs to the algebra defined by T over nf i e nf X T Returns a polynomial in variable v x by default nf nfinit y 2 1 rnfcharpoly nf x 2 y x 1 x y 12 x72 Mod y y 2 1 x 1 The library syntax is GEN rnfcharpoly GEN nf GEN T GEN a long var 1 where var is a variable number 3 6 128 rnfconductor bnf pol Given bnf as output by bnfinit and pol a relative polynomial defining an Abelian extension computes the class field theory conductor of this Abelian extension The result is a 3 component vector conductor rayclgp subgroup where conductor is the conductor of the extension given as a 2 component row vector fo foo rayclgp is the full ray class group corresponding to the c
416. ol is non constant and irreducible O otherwise Irreducibility is checked over the smallest base field over which pol seems to be defined The library syntax is long isirreducible GEN pol 3 7 24 pollead x v Leading coefficient of the polynomial or power series x This is computed with respect to the main variable of x if v is omitted with respect to the variable v otherwise The library syntax is GEN pollead GEN x long v 1 where v is a variable number 3 7 25 pollegendre n a xP n Legendre polynomial evaluated at a x by default The library syntax is GEN pollegendre_eval long n GEN a NULL To obtain the n th Legendre polynomial in variable v use GEN pollegendre long n long v 3 7 26 polrecip pol Reciprocal polynomial of pol i e the coefficients are in reverse order pol must be a polynomial The library syntax is GEN polrecip GEN pol 3 7 27 polresultant z y v flag 0 Resultant of the two polynomials x and y with exact entries with respect to the main variables of x and y if v is omitted with respect to the variable v otherwise The algorithm assumes the base ring is a domain If you also need the u and v such that x x u y x v Res x y use the polresultantext function If flag 0 default uses the the algorithm best suited to the inputs either the subresultant algorithm Lazard Ducos variant generic case a modular algorithm inputs in Q X or Sylvester s matrix inexact inpu
417. on as one of the dimensions gets large m or n is larger than 20 say it will often be much faster to use mathnf B 1 or mathnf B 4 directly The library syntax is GEN detint GEN B 235 3 8 17 matdiagonal x x being a vector creates the diagonal matrix whose diagonal entries are those of x matdiagonal 1 2 3 41 1 0 0 0 2 0 0 0 3 Block diagonal matrices are easily created using matconcat 7 U 1 2 3 4 V 11 2 3 4 5 637 8 91 matconcat matdiagonal U V 1 12000 3 400 0 o 0 1 2 3 0 0 4 5 6 0 07 8 9 The library syntax is GEN diagonal GEN x 3 8 18 mateigen z flag 0 Returns the complex eigenvectors of x as columns of a matrix If flag 1 return L H where L contains the eigenvalues and H the corresponding eigenvectors multiple eigenvalues are repeated according to the eigenspace dimension which may be less than the eigenvalue multiplicity in the characteristic polynomial This function first computes the characteristic polynomial of x and approximates its complex roots A then tries to compute the eigenspaces as kernels of the x A This algorithm is ill conditioned and is likely to miss kernel vectors if some roots of the characteristic polynomial are close in particular if it has multiple roots A 13 2 10 14 mateigen A 1 1 2 2 5 Et 1 L H mateigen A 1 7 L 3 9 18 7 H 4 1 2 2 5 1 1 For symmetric ma
418. on evaluation in this context hence to get for example the standard exponential of a square matrix you would need to implement a different function 10 1 5 3 Transcendental functions They usually operate on any complex number power series and some also on p adics The list is ever expanding and of course contains all the elementary functions exp log trigonometric func tions plus many others modular functions Bessel functions polylogarithms Recall that by extension PARI usually allows a transcendental function to operate componentwise on vectors or matrices 1 5 4 Arithmetic functions Apart from a few like the factorial function or the Fibonacci numbers these are functions which explicitly use the prime factor decomposition of integers The standard functions are included A number of factoring methods are used by a rather sophisticated factoring engine to name a few Shanks s SQUFOF Pollard s rho Lenstra s ECM the MPQS quadratic sieve These routines output strong pseudoprimes which may be certified by the APRCL test There is also a large package to work with algebraic number fields All the usual operations on elements ideals prime ideals etc are available More sophisticated functions are also implemented like solving Thue equations finding integral bases and discriminants of number fields computing class groups and fundamental units computing in relative number field extensions Galois and class field
419. on lines e g in between braces or after a line terminating backslash Everything that applies to prompt applies to prompt_cont as well The defaut value is 3 14 31 psfile Name of the default file where gp is to dump its PostScript drawings these are appended so that no previous data are lost Environment and time expansion are performed The default value is pari ps 3 14 32 readline Switches readline line editing facilities on and off This may be useful if you are running gp in a Sun cmdtool which interacts badly with readline Of course until readline is switched on again advanced editing features like automatic completion and editing history are not available The default value is 1 310 3 14 33 realprecision The number of significant digits used to convert exact inputs given to transcendental functions see Section 3 3 or to create absolute floating point constants input as 1 0 or Pi for instance Unless you tamper with the format default this is also the number of significant digits used to print a t_REAL number format will override this latter behaviour and allow you to have a large internal precision while outputting few digits for instance Note that PARI s internal precision works on a word basis by increments of 32 or 64 bits hence may be a little larger than the number of decimal digits you expected For instance to get 2 decimal digits you need one word of precision which on a 64 bit machine actua
420. on name The equation must be integral at p but need not be minimal at p of course a minimal model will be more efficient E ellinit 0 11 y 2 x 3 0 x 1 defined over Q ellap E 7 7 necessary here 42 4 E F_7 7 1 4 12 ellcard E 7 3 12 OK E ellinit 0 1 11 defined over F_11 ellap E no need to repeat 11 14 0 ellap E 11 M1 but it also works 15 0 ellgroup E 13 ouch inconsistent input kk at top level ellap E 13 4K ellap inconsistent moduli in Rg_to_Fp 11 13 Fq ffgen ffinit 11 3 a defines F_q F_ 1173 7 E ellinit ati a Fq y 2 x 3 a 1 x a defined over F_q ellap E 48 3 134 Algorithms used If E F has CM by a principal imaginary quadratic order we use a fast explicit formula involving essentially Kronecker symbols and Cornacchia s algorithm in O log q Otherwise we use Shanks Mestre s baby step giant step method which runs in time q p using O q 4 storage hence becomes unreasonable when q has about 30 digits If the seadata package is installed the SEA algorithm becomes available heuristically in log q and primes of the order of 200 digits become feasible In very small characteristic 2 3 5 7 or 13 we use Harley s algorithm The library syntax is GEN ellap GEN E GEN p NULL 3 5 8 ellbil E 21 22 If z1 and 22 are points on the elliptic curve E assumed to
421. on of current seq and immediately exits from the n innermost enclosing loops within the current function call or the top level loop the integer n must be positive If n is greater than the number of enclosing loops all enclosing loops are exited 3 11 2 breakpoint Interrupt the program and enter the breakloop The program continues when the breakloop is exited N x my z x 2 1 breakpoint gcd N z 2 1 z 221 3 at top level 221 3 xk Sees in function f my z x 2 1 breakpoint gcd N z eK Break loop type lt Return gt to continue break to go back to GP break gt z 10 break gt 42 13 278 3 11 3 dbg down n 1 In the break loop go down n frames This allows to cancel a previous call to dbg_up 3 11 4 dbg_err In the break loop return the error data of the current error if any See iferr for details about error data Compare iferr 1 Mod 2 12019 6 1 E Vec E 1 e_INV Fp_inv Mod 119 12019 1 Mod 2 12019 6 1 kk at top level 1 Mod 2 12019 6 4K kk _ _ impossible inverse in Fp_inv Mod 119 12019 Break loop type break to go back to GP prompt break gt Vec dbg_err e_INV Fp_inv Mod 119 12019 3 11 5 dbg_up n 1 In the break loop go up n frames This allows to inspect data of the parent function To cancel a dbg_up call use dbg_down 3 11 6 dbg_x A n Print t
422. onal coeffs P 2 time 1 392 ms c content P c72 P c 72 MAN same computation in integers time 1 116 ms And much more efficient but harder to setup to use homomorphic imaging schemes and modular computations As the simple example below indicates if you only need modular information it is very worthwhile to work with t_INTMODs directly rather than deal with t_FRACs all the way through p nextprime 10 7 sum i 1 1075 1 i p 18 time 13 288 ms 1 2759492 sum i 1 1075 Mod 1 i p time 60 ms 2 Mod 2759492 10000019 2 3 5 Finite field elements t_FFELT Let T F X be a monic irreducible polynomial defining your finite field over F for instance obtained using ffinit Then the ffgen function creates a generator of the finite field as an F algebra namely the class of X in F X T from which you can build all other elements For instance to create the field F s we write T ffinit 2 8 y ffgen T y y70 the unit element in the field 43 1 y78 74 y6 yS5t y4 y3 yti The second optional parameter to ffgen is only used to display the result it is customary to use the name of the variable we assign the generator to If g is a t_FFELT the following member functions are defined g pol the polynomial with reduced integer coefficients expressing g in term of the field generator g p the characteristic of the finite field g f the dimension of
423. onductor given as a 3 component vector h cyc gen as usual for a group and subgroup is a matrix in HNF defining the subgroup of the ray class group on the given generators gen The library syntax is GEN rnfconductor GEN bnf GEN pol 3 6 129 rnfdedekind nf pol pr flag 0 Given a number field K coded by nf and a monic polynomial P Zk X irreducible over K and thus defining a relative extension L of K applies Dedekind s criterion to the order Zx X P at the prime ideal pr It is possible to set pr to a vector of prime ideals test maximality at all primes in the vector or to omit altogether in which case maximality at all primes is tested in this situation flag is automatically set to 1 The default historic behavior flag is 0 or omitted and pr is a single prime ideal is not so useful since rnfpseudobasis gives more information and is generally not that much slower It returns a 3 component vector maz basis v e basis is a pseudo basis of an enlarged order O produced by Dedekind s criterion containing the original order Zk X P with index a power of pr Possibly equal to the original order e maz is a flag equal to 1 if the enlarged order O could be proven to be pr maximal and to 0 otherwise it may still be maximal in the latter case if pr is ramified in L e vis the valuation at pr of the order discriminant If flag is non zero on the other hand we just return 1 if the order Zk X P is pr maximal res
424. onductor of the curve e v gives the coordinate change for E over Q to the minimal integral model see ellmini malmodel e cis the product of the local Tamagawa numbers c a quantity which enters in the Birch and Swinnerton Dyer conjecture e F is the factorization of N over Z e Lisa vector whose i th entry contains the local data at the i th prime divisor of N i e L i elllocalred E F i 1 where the local coordinate change has been deleted and replaced by a 0 The library syntax is GEN ellglobalred GEN E 3 5 20 ellgroup E p flag Let E be an ell structure as output by ellinit defined over Q or a finite field F The argument p is best left omitted if the curve is defined over a finite field and must be a prime number otherwise This function computes the structure of the group E F Z d Z x Z d2Z with dy d If the curve is defined over Q p must be explicitly given and the function computes the structure of the reduction over F the equation need not be minimal at p but a minimal model will be more efficient The reduction is allowed to be singular and we return the structure of the cyclic group of non singular points in this case If the flag is 0 default return d or d1 d2 if da gt 1 If the flag is 1 return a triple h cyc gen where h is the curve cardinality cyc gives the group structure as a product of cyclic groups as per flag 0 More precisely if da gt 1 the output is d
425. one wants more information one could do instead nf nfinit x 2 1 L ideallist nf 100 0 1 L 251 vector 1 i 1 i clgp 43 20 20 16 4 41 20 2011 1 1 mod 74 25 18 0 1 O 1 2 mod 5 5 0 0 5 11 1 3 mod 46 25 7 0 11 1 where we ask for the structures of the Z i I for all three ideals of norm 25 In fact for all moduli with finite part of norm 25 and trivial Archimedean part as the last 3 commands show See ideallistarch to treat general moduli The library syntax is GEN ideallistO GEN nf long bound long flag 3 6 58 ideallistarch nf list arch list is a vector of vectors of bid s as output by ideallist with flag 0 to 3 Return a vector of vectors with the same number of components as the original list The leaves give information about moduli whose finite part is as in original list in the same order and Archimedean part is now arch it was originally trivial The information contained is of the same kind as was present in the input see ideallist in particular the meaning of flag bnf bnfinit x 2 2 bnf sign 2 2 0 two places at infinity L ideallist bnf 100 0 1 L 98 vector 1 i 1 i clgp 4 42 42 36 6 6 42 42 La ideallistarch bnf L 1 1 add them to the modulus 1 Lal98 vector 1 i 1 i clgp 6 168 42 2 2 144 6 6 2 2 168 42 2 2 Of
426. onent when the message x is short and the padding P is known to the attacker We use the same RSA modulus N as in the first example 127 setrand 1 P random N known padding e 3 small public encryption exponent X floor N 0 3 N 1 e epsilon x0 random X unknown short message C 1ift Mod x0 N P e known ciphertext with padding P zncoppersmith P x 73 C N X result in 244ms 3 265174753892462432 11 x0 74 1 We guessed an integer of the order of 1018 almost instantly The library syntax is GEN zncoppersmith GEN P GEN N GEN X GEN B NULL 3 4 87 znlog z g o Discrete logarithm of x in Z NZ in base g The result is when x is not a power of g If present o represents the multiplicative order of g see Section 3 4 2 the preferred format for this parameter is ord factor ord where ord is the order of g This provides a definite speedup when the discrete log problem is simple p nextprime 10 4 g znprimroot p o p 1 factor p 1 for i 1 10 4 znlog i g 0 time 205 ms for i 1 10 4 znlog i g time 244 ms a little slower The result is undefined if g is not invertible mod N or if the supplied order is incorrect This function uses e a combination of generic discrete log algorithms see below e in Z NZ when N is prime a linear sieve index calculus method suitable for N lt 10 say is used for large
427. onents of x When z and y are both given and one of them is a vector matrix type the LCM is again taken recursively on each component but in a different way If y is a vector resp matrix then the result has the same type as y and components equal to 1cm x y i resp 1cm x y i Else if x is a vector matrix the result has the same type as x and an analogous definition Note that for these types 1cm is not commutative Note that 1cm v is quite different from 1 v 1 for Gi 1 v 1 1cm 1 v i Indeed 1cm v is a scalar but 1 may not be if one of the v i is a vector matrix The computa tion uses a divide conquer tree and should be much more efficient especially when using the GMP multiprecision kernel and more subquadratic algorithms become available y vector 10 4 i random lcm v time 323 ms 1 v 1 for i 1 tv 1 1lem 1 v i time 833 ms The library syntax is GEN glcmO GEN x GEN y NULL 3 4 49 logint x b amp z Return the largest integer e so that b lt x where the parameters b gt 1 and x gt 0 are both integers If the parameter z is present set it to b logint 1000 2 11 9 7 279 72 512 logint 1000 2 amp z 13 9 Zz 4 512 The number of digits used to write b in base x is 1 logint x b digits 1000 10 5 2568 logint 1000 10 46 2567 This function may conveniently replace floor log x log b which may n
428. ood relatively to N in particular if the infinite place does not divide the module e g if m is an integer then it is not a subgroup of Z nZ but of its quotient by 1 If fl 0 compute a polynomial in the variable v defining the the subfield of Q fixed by the subgroup H of Z nZ If fl 1 compute only the conductor of the abelian extension as a module If fl 2 output pol N where pol is the polynomial as output when fl 0 and N the conductor as output when fl 1 The following function can be used to compute all subfields of Q C of exact degree d if d is set polsubcyclo n d 1 my bnr L IndexBound IndexBound if d lt 0 n d bnr bnrinit bnfinit y In 1 1 L subgrouplist bnr IndexBound 1 vector L i galoissubcyclo bnr L i Setting L subgrouplist bnr IndexBound would produce subfields of exact conductor noo The library syntax is GEN galoissubcyclo GEN N GEN H NULL long fl long v 1 where v is a variable number 3 6 43 galoissubfields G flags 0 v Outputs all the subfields of the Galois group G as a vector This works by applying galoisfixedfield to all subgroups The meaning of the flag ff is the same as for galoisfixedfield The library syntax is GEN galoissubfields GEN G long flags long v 1 where v is a variable number 172 3 6 44 galoissubgroups G Outputs all the subgroups of the Galois group gal A subgroup is a
429. oorossoss T19 NUMdIV 24064 RPO eR aA e a ee 119 D ET mw HE se RO ee a ee 82 mumerator 33 82 numerical derivation 28 numerical integration 257 mumtoperm 83 NUPOW 6 be ee REESE Ee Ee 123 O Geek eae oa aces ae ems 215 QMEYA be ae ee ee Se ee we ae a 124 OMEQA a6 Aw eo ak a Ae we ae 119 131 ONCUTVE 635 e re Pe RSS OE 141 Operator como 27 OF fare 6 toe E ee Ae ale oe we 70 OE i Acie ee hes GO Go oe ee 77 78 Order e iiss ee a Ba ew A 129 orderell ssas BGS eR ee ee 145 G D L ashen arena ath we a ara a 57 308 P Dy oe eee foe te eg EE SOE ech ee T T31 p adic number 7 8 19 padicappr 2 4 5 26 we dee a 218 219 padicfields sasaaa ee ewan 219 padicfieldsO 219 335 padicprec 2 564655 8 2 Re oa ra 83 padic_lindep 232 parametric Plot o i conne E eua a a 274 panapply oa sree eais paa wi 302 303 Pareval saue do bee doe O a 303 Parlor boicot os on Go s er oe a 303 parforprime gt o lt s 68s ee eee es 303 Paribmacs o rog eT ora Dae eb a 315 PAarISIZO penae Be ds ke me Bee S 308 pari malloc 2 2 see eve as retipa 285 pari_reallocl Ve oe Be ee ww a 285 pari_version 301 parselect po uo 24s he be eee ko 303 PACET amp 2 00 4 eto Gee 303 partitions s lada ee tk ee es 119 120 Parvector sesse he mares 303 Pascal triangle 241 PACH 6 ota Be ace dea Ais Ae a Ge E E Hes 309 Pauli 2 252
430. oots As in polroots 0 E has one component 1 t_STR the function name s e e SQRTN Trying to compute an n th root of x which does not exist in function s As in sqrt Mod 1 3 E has two components 1 t_STR the function name s 2 the argument z 286 3 11 18 next n 1 Interrupts execution of current seq resume the next iteration of the innermost enclosing loop within the current function call or top level loop If n is specified resume at the n th enclosing loop If n is bigger than the number of enclosing loops all enclosing loops are exited 3 11 19 return z 0 Returns from current subroutine with result x If x is omitted return the void value return no result like print 3 11 20 until a seq Evaluates seq until a is not equal to 0 i e until a is true If a is initially not equal to 0 seq is evaluated once more generally the condition on a is tested after execution of the seq not before as in while 3 11 21 while a seq While a is non zero evaluates the expression sequence seg The test is made before evaluating the seq hence in particular if a is initially equal to zero the seg will not be evaluated at all 3 12 Programming in GP other specific functions In addition to the general PARI functions it is necessary to have some functions which will be of use specifically for gp though a few of these can be accessed under library mode Before we start describing these we recall the
431. or with strictly increasing entries with respect to the somewhat arbitrary universal comparison function cmp Standard container types t_VEC t_COL t_LIST and t_VECSMALL are converted to the set with corresponding elements All others are converted to a set with one element Set 1 2 4 2 1 3 1 1 2 3 4 Set x 12 x Set Vecsmall 1 3 2 1 3 73 1 2 3 The library syntax is GEN gtoset GEN x NULL 3 2 11 Str x x Converts its argument list into a single character string type t_STR the empty string if x is omitted To recover an ordinary GEN from a string apply eval to it The arguments of Str are evaluated in string context see Section 2 9 x2 0 i 2 Str x i vi xo eval 7 42 0 This function is mostly useless in library mode Use the pair strtoGEN GENtostr to convert between GEN and char The latter returns a malloced string which should be freed after usage 3 2 12 Strchr x Converts x to a string translating each integer into a character Strchr 97 1 a Vecsmall hello world 72 Vecsmal11 104 101 108 108 111 32 119 111 114 108 100 Strchr 3 hello world The library syntax is GEN Strchr GEN x 75 3 2 13 Strexpand x Converts its argument list into a single character string type t_STR the empty string if x is omitted Then perform environment expansion see Section 2 12 This feature can be used to read environment var
432. or gp old timers Thus to encourage switching to compatible 0 it is not possible to disable the warning compatible 2 use only the old function naming scheme as used up to version 1 39 15 but taking case into account Thus I y 1 is not the same as i user variable unbound by default and you won t get an error message using i as a loop index as used to be the case compatible 3 try to mimic exactly the former behavior This is not always possible when functions have changed in a fundamental way But these differences are usually for the better they were meant to anyway and will probably not be discovered by the casual user One adverse side effect is that any user functions and aliases that have been defined before changing compatible will get erased if this change modifies the function list i e if you move between groups 0 1 and 2 3 variables are unaffected We of course strongly encourage you to try and get used to the setting compatible 0 Note that the default new_galois_format is another compatibility setting which is completely independent of compatible The default value is 0 3 14 5 datadir The name of directory containing the optional data files For now this includes the elldata galdata galpol seadata packages The default value is home kb PARI pari GP share pari the location of installed precom puted data can be specified via Configure datadir 3 14 6 debug Debugging level If it is no
433. or of permutations of L gal gen gal 8 contains the relative orders o1 0y of the generators of S gal orders Let H be as above we have the following properties e if G H A4 then 01 0 ends by 2 2 3 e if G H S4 then 01 0g ends by 2 2 3 2 e for 1 lt i lt g the subgroup of G generated by s1 Sg is normal with the exception of i g 2 in the Aq case and of i g 3 in the Sa case e the relative order o of s is its order in the quotient group G s1 s 1 with the same exceptions e for any x G there exists a unique family e1 ey such that no exceptions for 1 lt i lt g we have 0 lt e lt oi T Gigs Gn If present den must be a suitable value for gal 5 The library syntax is GEN galoisinit GEN pol GEN den NULL 3 6 39 galoisisabelian gal flag 0 gal being as output by galoisinit return 0 if gal is not an abelian group and the HNF matrix of gal over gal gen if fl 0 1 if fl 1 This command also accepts subgroups returned by galoissubgroups The library syntax is GEN galoisisabelian GEN gal long flag 3 6 40 galoisisnormal gal subgrp gal being as output by galoisinit and subgrp a subgroup of gal as output by galoissubgroups return 1 if subgrp is a normal subgroup of gal else return 0 This command also accepts subgroups returned by galoissubgroups The library syntax is long galoisisnormal GEN gal GEN subgrp 3 6 41 galoi
434. or of polynomial with integer coefficients d is a denominator and the integer basis is given by A d rnf 3 rnf disc is a two component row vector 0 L K s where 0 L K is the relative ideal discriminant of L K and s is the discriminant of L K viewed as an element of K K in other words it is the output of rnfdisc rnf 4 rnf index is the ideal index f i e such that d pol Zx Pd L K nf rnf 6 is currently unused 3 5 is currently unused rnf 7 rnf zk is the pseudo basis 4 1 for the maximal order Zz as a Zx module A is the relative integral pseudo basis expressed as polynomials in the variable of pol with polmod coefficients in nf and the second component I is the ideal list of the pseudobasis in HNF rnf 8 is the inverse matrix of the integral basis matrix with coefficients polmods in nf rnf 9 is currently unused rnf 10 rnf nf is nf 209 rnf 11 is the output of rnfequation K pol 1 Namely a vector P a k describing the absolute extension L Q P is an absolute equation more conveniently obtained as rnf polabs a expresses the generator a y mod K pol of the number field K as an element of L i e a polynomial modulo the absolute equation P k is a small integer such that if 6 is an abstract root of pol and a the generator of K given above then P G ka 0 Caveat Be careful if k 0 when dealing simultaneously with absolute and relative quantities since L Q 6 ka K
435. order of nf computes its determinant The library syntax is GEN rnfdet GEN nf GEN M 3 6 131 rnfdisc nf pol Given a number field nf as output by nfinit and a polynomial pol with coefficients in nf defining a relative extension L of nf computes the relative discriminant of L This is a two element row vector D d where D is the relative ideal discriminant and d is the relative discriminant considered as an element of nf nf The main variable of nf must be of lower priority than that of pol see Section 2 5 3 The library syntax is GEN rnfdiscf GEN nf GEN pol 3 6 132 rnfeltabstorel rnf x rnf being a relative number field extension L K as output by rnfinit and x being an element of L expressed as a polynomial modulo the absolute equation rnf pol computes x as an element of the relative extension L K as a polmod with polmod coefficients K nfinit y 2 1 L rnfinit K x 2 y L pol 12 x 4 1 rnfeltabstorel L Mod x L pol 13 Mod x x 2 Mod y y 2 1 rnfeltabstorel L Mod 2 L pol 14 2 rnfeltabstorel L Mod x x 2 y EK at top level rnfeltabstorel L Mod OK rnfeltabstorel inconsistent moduli in rnfeltabstorel x 2 y x 4 1 The library syntax is GEN rnfeltabstorel GEN rnf GEN x 204 3 6 133 rnfeltdown rnf x rnf being a relative number field extension L K as output by rn finit and zx being an element of L expressed as a polynomial or polmod with polm
436. ork Bees Gwe ee aS 44 151 burclassno gt a e Bee a o a aa 163 165 burclassno0 s ae a do ae a 163 bnrclassnolist 163 164 177 bnrconductor 164 bnreonductord s se se eae a aw a 164 bnrconductorofchar 164 DOAS oos soa Bae we eS 164 165 bnrdiscO secos ae ae oo i Ee a 164 bnrdisclist iee 4 285 26 sees 165 177 bnrdisclistO 165 DOPING cose ee ee aoe ea 154 164 165 POTANICO se e eS eae ee Ge ee 166 bnrisconductor 166 bnrisconductorO 166 bnrisprincipal 159 166 DOT 04 04 2 4 bbe Ba ADe wa 162 163 bnrnewprec o oo o 195 bnrrootnumber 166 167 bnrstark 125 167 168 214 Boolean operators 0 70 boundfact ura is 108 brace characters 16 break LOOP s a Kb we ee ee 50 break 24 4 226 2 weed eee hes 50 278 breakloop ss a e ea roro anie 4 52 304 breakpoint so co saa ao oe E 278 Buchall sr i i 6 Gig i aa e 159 326 Buchall_param 159 Buchmann 156 158 181 Buchmann McCurley 124 buchnarrowW p00 6 8 es ee we 162 Buchquad ea tou 4 oe k Shee a E 124 Buchray cop cris ea eed es 166 C GA a ei Oe ek Oy a Ee e A 130 GO e a a ps a pled Gl a e aao oS 130 Cantor Zassenhaus 109 Caract osag ea bss ae ERN ps 229 carad sawp ao a a 229 carberkowitz 229 Carhess a sag dih n 24 84 2 au be e
437. orm equations Na a for x in L andain K If flag 0 do not care whether L K is Galois or not If flag 1 L K is assumed to be Galois unchecked which speeds up rnfisnorm If flag 2 let the routine determine whether L K is Galois The library syntax is GEN rnfisnorminit GEN pol GEN polrel long flag 3 6 154 rnfkummer bnr subgp d 0 bnr being as output by bnrinit finds a relative equation for the class field corresponding to the module in bnr and the given congruence subgroup the full ray class field if subgp is omitted If d is positive outputs the list of all relative equations of degree d contained in the ray class field defined by bnr with the same conductor as bnr subgp Warning This routine only works for subgroups of prime index It uses Kummer theory adjoining necessary roots of unity it needs to compute a tough bnfinit here and finds a generator via Hecke s characterization of ramification in Kummer extensions of prime degree If your extension does not have prime degree for the time being you have to split it by hand as a tower compositum of such extensions The library syntax is GEN rnfkummer GEN bnr GEN subgp NULL long d long prec 3 6 155 rnfillgram nf pol order Given a polynomial pol with coefficients in nf defin ing a relative extension L and a suborder order of L of maximal rank as output by rnfpseudobasis nf pol or similar gives neworder U where neworder is a reduced o
438. orms it into an integer or a rational number by chopping away the O p The library syntax is GEN truncO GEN x GEN e NULL The following functions are also available GEN gtrunc GEN x and GEN gcvtoi GEN x long e 86 3 2 55 valuation z p Computes the highest exponent of p dividing x If p is of type integer x must be an integer an intmod whose modulus is divisible by p a fraction a q adic number with q p or a polynomial or power series in which case the valuation is the minimum of the valuation of the coefficients If p is of type polynomial x must be of type polynomial or rational function and also a power series if x is a monomial Finally the valuation of a vector complex or quadratic number is the minimum of the component valuations If x 0 the result is LONG_MAX 2 1 for 32 bit machines or 2 1 for 64 bit machines if x is an exact object If x is a p adic numbers or power series the result is the exponent of the zero Any other type combinations gives an error The library syntax is long gvaluation GEN x GEN p 3 2 56 variable 1 Gives the main variable of the object x the variable with the highest priority used in x and p if x is a p adic number Return 0 if x has no variable associated to it variable x 2 y Zl x variable 1 0 572 42 5 variable x y z t 13 x variable 1 14 0 The construction if Ivariable x can be used to test whether a variable
439. ot give the correct answer since PARI does not guarantee exact rounding The library syntax is long logintO GEN x GEN b GEN z NULL 3 4 50 moebius xz Moebius function of x x must be of type integer The library syntax is long moebius GEN x 118 3 4 51 nextprime x Finds the smallest pseudoprime see ispseudoprime greater than or equal to x x can be of any real type Note that if x is a pseudoprime this function returns x and not the smallest pseudoprime strictly larger than x To rigorously prove that the result is prime use isprime The library syntax is GEN nextprime GEN x 3 4 52 numbpart n Gives the number of unrestricted partitions of n usually called p n in the literature in other words the number of nonnegative integer solutions to a 2b 3c n n must be of type integer and n lt 101 with trivial values p n 0 for n lt 0 and p 0 1 The algorithm uses the Hardy Ramanujan Rademacher formula To explicitly enumerate them see partitions The library syntax is GEN numbpart GEN n 3 4 53 numdiv z Number of divisors of x x must be of type integer The library syntax is GEN numdiv GEN x 3 4 54 omega x Number of distinct prime divisors of x x must be of type integer factor 392 1 2 3 7 2 7 omega 392 2 2 without multiplicity bigomega 392 3 5 3 2 with multiplicity The library syntax is long omega GEN x 3 4 55 partitions k a k
440. otentially confusing concepts Skip to the last subsection for practical advice if the next discussion does not make sense to you After learning about user functions study the example in Section 2 7 3 then come back 34 Definitions A scope is an enclosing context where names and values are associated A user s function body the body of a loop an individual command line all define scopes the whole program defines the global scope The argument of eval is evaluated in the enclosing scope Variables are bound to values within a given scope This is traditionally implemented in two different ways e lexical or static scoping the binding makes sense within a given block of program text The value is private to the block and may not be accessed from outside Where to find the value is determined at compile time e dynamic scoping introducing a local variable say x pushes a new value on a stack associated to the name x possibly empty at this point which is popped out when the control flow leaves the scope Evaluating x in any context possibly outside of the given block always yields the top value on this dynamic stack GP implements both lexical and dynamic scoping using the keywords my lexical and local dynamic x 0 fQ x gQ my x 1 0 h local x 1 fO The function g returns O since the global x binding is unaffected by the introduction of a private variable of the same name in g On the other hand
441. oup Cl K as well as technical data required to solve the two associated discrete logarithm problems e bnr denotes a ray number field i e a data structure output by bnrinit corresponding to the ray class group structure of the field for some modulus f It contains a bnf the modulus f the ray class group Cl K and data associated to the discrete logarithm problem therein 151 3 6 2 Algebraic numbers and ideals An algebraic number belonging to K Q X T is given as e a t_INT t_FRAC or t_POL implicitly modulo T or e a t_POLMOD modulo T or e a t_COL v of dimension N K Q representing the element in terms of the computed integral basis as sum i 1 N v i nf zk i Note that a t_VEC will not be recognized An ideal is given in any of the following ways e an algebraic number in one of the above forms defining a principal ideal e a prime ideal i e a 5 component vector in the format output by idealprimedec or ideal factor e a t_MAT square and in Hermite Normal Form or at least upper triangular with non negative coefficients whose columns represent a Z basis of the ideal One may use idealhnf to convert any ideal to the last preferred format e an extended ideal is a 2 component vector 7 t where J is an ideal as above and t is an algebraic number representing the ideal t J This is useful whenever idealred is involved implicitly working in the ideal class group while keeping track of prin
442. p maximal at all relevant primes as described above and 0 if not This is much faster than the default since the enlarged order is not computed nf nfinit y 2 3 P x73 2 y pr3 idealprimedec nf 3 1 rnfdedekind nf P pr3 2 1 1 O O O 1 O O O 1 1 1 1 8 rnfdedekind nf P pr3 1 13 1 In this example pr3 is the ramified ideal above 3 and the order generated by the cube roots of y is already pr3 maximal The order discriminant has valuation 8 On the other hand the order is not maximal at the prime above 2 pr2 idealprimedec nf 2 1 203 rnfdedekind nf P pr2 1 45 0 rnfdedekind nf P pr2 6 TO 2 O 05 O 1 O O O 1 LM O O 1 1 0 O 1 1 1 2 0 1 2 2 The enlarged order is not proven to be pr2 maximal yet In fact it is it is in fact the maximal order 7 B rnfpseudobasis nf P 7 1 0 0 O 1 0 0 O 1 1 1 1 1 2 O 1 2 162 0 O 1621 1 idealval nf B 3 pr2 14 2 It is possible to use this routine with non monic P J lt a X Zk X if flag 1 in this case we test maximality of Dedekind s order generated by lana anQ Gn 10 F aca iy a Oe The routine will fail if P is 0 on the projective line over the residue field Zg pr FIXME The library syntax is GEN rnfdedekind GEN nf GEN pol GEN pr NULL long flag 3 6 130 rnfdet nf M Given a pseudo matrix M over the maximal
443. passed as the first argument to most nfxrx functions but it contains some information which may be directly useful Access to this information via member functions is preferred since the specific data organization specified below may change in the future Currently nf is a row vector with 9 components nf 1 contains the polynomial pol nf pol nf 2 contains r1 r2 nf sign nf r1 nf r2 the number of real and complex places of K nf 3 contains the discriminant d K nf disc of K nf 4 contains the index of nf 1 nf index i e Zg Z 0 where 0 is any root of nf 1 nf 5 is a vector containing 7 matrices M G roundG T MD TI MDI useful for certain computations in the number field K e M is the r1 72 xn matrix whose columns represent the numerical values of the conjugates of the elements of the integral basis e G is an n x n matrix such that T2 GG where T2 is the quadratic form Ta 1 Y lo x 1 o running over the embeddings of K into C e roundG is a rescaled copy of G rounded to nearest integers e T is the n x n matrix whose coefficients are Tr w w where the w are the elements of the integral basis Note also that det T is equal to the discriminant of the field K Also when understood as an ideal the matrix T generates the codifferent ideal e The columns of MD nf diff express a Z basis of the different of K on the integral basis e TI is equal to the primitive part of T whic
444. pplying automorphisms yields the same elements as a direct ide alprimedec call The automorphism can also be given as a column vector representing the image of Mod x nf pol as an algebraic number This last representation is more efficient and should be preferred if a given automorphism must be used in many such calls nf nfinit x 3 37 x 2 74 x 37 1 nfgaloisconj nf aut 1 2 automorphisms in basistoalg form 2 31 11 x 2 1109 11 x 925 11 L matalgtobasis nf 1 AUT L 2 same in algtobasis form 13 16 6 5 v 1 2 3 nfgaloisapply nf aut v nfgaloisapply nf AUT v 14 1 same result for i 1 10 5 nfgaloisapply nf aut v time 1 451 ms for i 1 10 5 nfgaloisapply nf AUT v time 1 045 ms but the latter is faster The library syntax is GEN galoisapply GEN nf GEN aut GEN x 3 6 101 nfgaloisconj nf flag 0 d nf being a number field as output by nfinit computes the conjugates of a root r of the non constant polynomial x nf 1 expressed as polynomials in r This also makes sense when the number field is not Galois since some conjugates may lie in the field nf can simply be a polynomial If no flags or flag 0 use a combination of flag 4 and 1 and the result is always complete There is no point whatsoever in using the other flags If flag 1 use nfroots a little slow but guaranteed to work in polynomial time 191 If flag 2
445. pr Product of expression expr initialized at 1 i e to a real number equal to 1 to the current realprecision the formal parameter X ranging over the prime numbers between a and b The library syntax is prodeuler void E GEN eval void GEN GEN a GEN b long prec 3 9 17 prodinf X a ezpr flag 0 infinite product of expression expr the formal parameter X starting at a The evaluation stops when the relative error of the expression minus 1 is less than the default precision In particular non convergent products result in infinite loops The expressions must always evaluate to an element of C If flag 1 do the product of the 1 ezpr instead The library syntax is prodinf void E GEN eval void GEN GEN a long prec flag 0 or prodinf1 with the same arguments flag 1 3 9 18 solve X a b expr Find a real root of expression expr between a and b under the con dition expr X a expr X b lt 0 You will get an error message roots must be bracketed in solve if this does not hold This routine uses Brent s method and can fail miserably if expr is not defined in the whole of a b try solve x 1 2 tan x The library syntax is zbrent void E GEN eval void GEN GEN a GEN b long prec 3 9 19 sum X a b expr x 0 Sum of expression expr initialized at x the formal parameter going from a to b As for prod the initialization parameter x may be given to force the type of the ope
446. prime divisors of the order The generic discrete log algorithms are e Pohlig Hellman algorithm to reduce to groups of prime order q where q p 1 and p is an odd prime divisor of N e Shanks baby step giant step q lt 2 is small e Pollard rho method q gt 237 The latter two algorithms require O q operations in the group on average hence will not be able to treat cases where q gt 10 say In addition Pollard rho is not able to handle the case where there are no solutions it will enter an infinite loop g znprimroot 101 hi Mod 2 101 znlog 5 g 12 24 g 24 13 Mod 5 101 128 7 G znprimroot 2 101710 4 Mod 110462212541120451003 220924425082240902002 znlog 5 G 5 76210072736547066624 Gh 76 1 N 274 3 2 5 3 7 4 11 g Mod 13 N znlog g 110 g 77 110 znlog 6 Mod 2 3 no solution 18 O For convenience g is also allowed to be a p adic number g 3 0 5710 znlog 2 g 1 1015243 7 oh 2 2 0 5710 The library syntax is GEN znlog GEN x GEN g GEN o NULL 3 4 88 znorder z o x must be an integer mod n and the result is the order of x in the multiplicative group Z nZ Returns an error if x is not invertible The parameter o if present represents a non zero multiple of the order of x see Section 3 4 2 the preferred format for this parameter is ord factor ord where ord eulerphi n is the cardinality of
447. proach would use precomputations to ensure a given discriminant is computed only once DISC vector v i abs poldisc v i perm vecsort vector v i i x y gt sign DISC x DISC y vecextract v perm Similar ideas apply whenever we sort according to the values of a function which is expensive to compute The binary digits of flag mean 255 e 1 indirect sorting of the vector zx i e if x is an n component vector returns a permutation of 1 2 n which applied to the components of x sorts x in increasing order For example vecextract x vecsort x 1 is equivalent to vecsort x e 4 use descending instead of ascending order e 8 remove duplicate entries with respect to the sorting function keep the first occurring entry For example vecsort Pi Mod 1 2 z x y gt 0 8 make everything compare equal 71 3 141592653589793238462643383 vecsort 2 31 0 11 0 311 2 8 2 0 1 2 31 The library syntax is GEN vecsortO GEN x GEN cmpf NULL long flag 3 8 76 vecsum v Return the sum of the component of the vector v The library syntax is GEN vecsum GEN v 3 8 77 vector n X expr 0 Creates a row vector type t_VEC with n components whose components are the expression expr evaluated at the integer points between 1 and n If one of the last two arguments is omitted fill the vector with zeroes Avoid modifying X within expr if you do the formal variable stil
448. puted In that case a warning is printed and t is set equal to the empty vector If this bit is set increase the precision and recompute needed quantities until t can be computed Warning setting this may induce very lengthy computations The library syntax is GEN bnfisprincipal0 GEN bnf GEN x long flag Instead of the above hardcoded numerical flags one should rather use an or ed combination of the symbolic flags nf_GEN include generators possibly a place holder if too difficult and nf __FORCE insist on finding the generators 3 6 15 bnfissunit bnf sfu x bnf being output by bnfinit sfu by bnfsunit gives the column vector of exponents of x on the fundamental S units and the roots of unity If x is not a unit outputs an empty vector The library syntax is GEN bnfissunit GEN bnf GEN sfu GEN x 3 6 16 bnfisunit bnf x bnf being the number field data output by bnfinit and x being an algebraic number type integer rational or polmod this outputs the decomposition of x on the fundamental units and the roots of unity if x is a unit the empty vector otherwise More precisely if u1 uy are the fundamental units and is the generator of the group of roots of unity bnf tu the output is a vector z1 r amp r 1 such that x uf u r r 1 The x are integers for i lt r and is an integer modulo the order of fori r 1 Note that bnf need not contain the fundamental unit explicitly setrand 1
449. r Vi style editing M C j will toggle you to Vi mode Of course you can change all these default bindings For that you need to create a file named inputrc in your home directory For instance notice the embedding conditional in case you would want specific bindings for gp if Pari GP set show all if ambiguous C h backward delete char e C h backward kill word C xd dump functions C C v C b can be annoying when copy pasting C v C b endif C x C r will re read this init file incorporating any changes made to it during the current session Note By default and are bound to the function pari matched insert which if electric parentheses are enabled default off will automatically insert the matching closure respectively and This behavior can be toggled on and off by giving the numeric argument 2 to M 2 which is useful if you want e g to copy paste some text into the calculator If you do not want a toggle you can use M 0 M 1 to specifically switch it on or off 61 Note In some versions of readline 2 1 for instance the Alt or Meta key can give funny re sults output 8 bit accented characters for instance Tf you do not want to fall back to the Esc combination put the following two lines in your inputrc set convert meta on set output meta off Command completion and online help Hitting lt TAB gt will complete words for you This mechanism is context depen
450. r giving the structure of the class group as a product of cyclic groups e v gen a vector giving generators of those cyclic groups as binary quadratic forms ev reg the regulator computed to an accuracy which is the maximum of an internal accuracy determined by the program and the current default note that once the regulator is known to a small accuracy it is trivial to compute it to very high accuracy see the tutorial The flag is obsolete and should be left alone In older versions it supposedly computed the narrow class group when D gt 0 but this did not work at all use the general function bnfnarrow Optional parameter tech is a row vector of the form c1 c2 where c lt cz are non negative real numbers which control the execution time and the stack size see 3 6 7 The parameter is used as a threshold to balance the relation finding phase against the final linear algebra Increasing the default c means that relations are easier to find but more relations are needed and the linear algebra will be harder The default value for c is 0 and means that it is taken equal to c2 The parameter c2 is mostly obsolete and should not be changed but we still document it for completeness we compute a tentative class group by generators and relations using a factorbase of prime ideals lt c log D then prove that ideals of norm lt c2 log D do not generate a larger group By default an optimal ca is chosen so that the result is
451. r q The behaviour is then as follows e If the reduction is split H tate 2 is a t_PADIC we have an isomorphism E Q gt Q 4 and the function returns p P Qp 145 e If the reduction is not split E tate 2 is a t_POLMOD we only have an isomorphism E K Kk q2 over the unramified quadratic extension K Q In this case the output P K is a t_POLMOD E ellinit 0 1 1 0 0 O 1175 P 0 0 u2 u q E tate type u split multiplicative reduction 2 t_PADIC ellmul E P 5 P has order 5 13 o z ellpointtoz E 0 0 4 3 1172 2x1173 3 1174 0 1175 275 5 1 0 1175 7 E ellinit ellfromj 1 4 0 2 6 x 1 2 y ellordinate E x 1 z ellpointtoz E x y t_POLMOD of t_POL with t_PADIC coeffs liftint z lift all p adics 18 Mod 8 u 7 u 2 437 The library syntax is GEN zell GEN E GEN P long prec 3 5 39 ellpow z n Deprecated alias for el1mul The library syntax is GEN ellmul GEN E GEN z GEN n 3 5 40 ellrootno E pj E being an ell structure over Q as output by ellinit this function computes the local root number of its L series at the place p at the infinite place if p 0 If p is omitted return the global root number Note that the global root number is the sign of the functional equation and conjecturally is the parity of the rank of the Mordell Weil group The equation for E needs not be minimal at p but if
452. r the sake of efficiency 1 3 3 Finite field elements The exact internal format depends of the finite field size but it includes the field characteristic p an irreducible polynomial T F X defining the finite field F X T and the element expressed as a polynomial in the class of X 1 3 4 Complex numbers and quadratic numbers Quadratic numbers are numbers of the form a bw where w is such that Z w Z 2 and more precisely w Vd 2 when d 0mod 4 and w 1 Vd 2 when d 1 mod 4 where d is the discriminant of a quadratic order Complex numbers correspond to the important special case w 1 Complex numbers are partially recursive the two components a and b can be of type t_INT t_REAL t_INTMOD t_FRAC or t_PADIC and can be mixed subject to the limitations mentioned above For example a bi with a and b p adic is in Q i but this is equal to Q when p 1 mod 4 hence we must exclude these p when one explicitly uses a complex p adic type Quadratic numbers are more restricted their components may be as above except that t_REAL is not allowed 1 3 5 Polynomials power series vectors matrices and lists They are completely recur sive their components can be of any type and types can be mixed however beware when doing operations Note in particular that a polynomial in two variables is simply a polynomial with polynomial coefficients In the present version 2 7 0 of PARI it is not possible to handle conveni
453. ra type which is coded as follows 1 means good reduction type Io 2 3 and 4 mean types II III and IV respectively 4 v with v gt 0 means type I finally the opposite values 1 2 etc refer to the starred types Ip II etc The third component v is itself a vector u r s t giving the coordinate changes done during the local reduction u 1 if and only if the given equation was already minimal at p Finally the last component c is the local Tamagawa number Cp The library syntax is GEN elllocalred GEN E GEN p 3 5 29 elllog E P G o Given two points P and G on the elliptic curve E F returns the discrete logarithm of P in base G i e the smallest non negative integer n such that P njG See znlog for the limitations of the underlying discrete log algorithms If present o represents the order of G see Section 3 4 2 the preferred format for this parameter is N factor N where N is the order of G If no o is given assume that G generates the curve The function also assumes that P is a multiple of G a ffgen ffinit 2 8 a E ellinit a 1 0 0 1 AN over F_ 278 x a 3 y ellordinate E x 1 P x y G ellmul E P 113 ord 242 factor 242 P generates a group of order 242 Initialize ellorder E G ord 14 242 e elllog E P G ord 5 15 ellmul E G e 6 1 The library syntax is GEN elllog GEN E GEN P GEN G GEN o NULL 3 5 30 ellls
454. rations being performed As an extreme example compare sum i 1 1074 1 1 rational number denominator has 4345 digits time 236 ms sum i 1 5000 1 i 0 time 8 ms 12 9 787606036044382264178477904 The library syntax is somme GEN a GEN b char expr GEN x 3 9 20 sumalt X a expr flag 0 Numerical summation of the series expr which should be an alternating series the formal variable X starting at a Use an algorithm of Cohen Villegas and Zagier Experiment Math 9 2000 no 1 3 12 If flag 1 use a variant with slightly different polynomials Sometimes faster The routine is heuristic and a rigorous proof assumes that the values of expr are the moments of a positive measure on 0 1 Divergent alternating series can sometimes be summed by this method as well as series which are not exactly alternating see for example Section 2 7 It should be used to try and guess the value of an infinite sum However see the example at the end of Section 2 7 1 If the series already converges geometrically suminf is often a better choice p28 sumalt i 1 1 i i log 2 time O ms 267 1 2 524354897 E 29 suminf i 1 1 7i i Had to hit C C at top level suminf i 1 1 7i i kK Etoo suminf user interrupt after 10min 20 100 ms p1000 sumalt i 1 1 i i log 2 time 90 ms 12 4 459597722 E 1002 sumalt i 0 1 i i exp 1
455. ray class group gives the conductor of this character as a modulus The library syntax is GEN bnrconductorofchar GEN bnr GEN chi 164 3 6 25 bnrdisc A B C flag 0 A B C defining a class field L over a ground field K of type bnr bnr subgroup bnf modulus or bnf modulus subgroup Section 3 6 5 outputs data N r1 D giving the discriminant and signature of L depending on the binary digits of flag e 1 if this bit is unset output absolute data related to L Q N is the absolute degree L Q r the number of real places of L and D the discriminant of L Q Otherwise output relative data for L K N is the relative degree L K r is the number of real places of K unramified in L so that the number of real places of L is equal to r times N and D is the relative discriminant ideal of L K e 2 if this bit is set and if the modulus is not the conductor of L only return 0 The library syntax is GEN bnrdiscO GEN A GEN B NULL GEN C NULL long flag 3 6 26 bnrdisclist bnf bound arch bnf being as output by bnfinit with units computes a list of discriminants of Abelian extensions of the number field by increasing modulus norm up to bound bound The ramified Archimedean places are given by arch all possible values are taken if arch is omitted The alternative syntax bnrdisclist bnf list is supported where list is as output by ideal list or ideallistarch with units in which case arch
456. rder and U is the unimodular transformation matrix The library syntax is GEN rnflllgram GEN nf GEN pol GEN order long prec 3 6 156 rnfnormgroup bnr pol bnr being a big ray class field as output by bnrinit and pol a relative polynomial defining an Abelian extension computes the norm group alias Artin or Takagi group corresponding to the Abelian extension of bnf bnr bnf defined by pol where the module corresponding to bnr is assumed to be a multiple of the conductor i e pol defines a subextension of bnr The result is the HNF defining the norm group on the given generators of bnr gen Note that neither the fact that pol defines an Abelian extension nor the fact that the module is a multiple of the conductor is checked The result is undefined if the assumption is not correct The library syntax is GEN rnfnormgroup GEN bnr GEN pol 211 3 6 157 rnfpolred nf pol THIS FUNCTION IS OBSOLETE use rnfpolredbest instead Rel ative version of polred Given a monic polynomial pol with coefficients in nf finds a list of relative polynomials defining some subfields hopefully simpler and containing the original field In the present version 2 7 0 this is slower and less efficient than rnfpolredbest Remark this function is based on an incomplete reduction theory of lattices over number fields implemented by rnflllgram which deserves to be improved The library syntax is GEN rnfpolred GEN nf GEN pol long prec 3 6 158 rnfpolreda
457. re GEN redimag GEN x for definite x and for indefinite forms GEN redreal GEN x GEN rhoreal GEN x qfbred x 1 GEN redrealnod GEN x GEN isd qfbred x 2 isd GEN rhorealnod GEN x GEN isd qfbred x 3 isd 123 3 4 69 qfbsolve Q p Solve the equation Q x y p over the integers where Q is a binary quadratic form and p a prime number Return z y as a two components vector or zero if there is no solution Note that this function returns only one solution and not all the solutions Let D discQ The algorithm used runs in probabilistic polynomial time in p through the computation of a square root of D modulo p it is polynomial time in D if Q is imaginary but exponential time if Q is real through the computation of a full cycle of reduced forms In the latter case note that bnfisprincipal provides a solution in heuristic subexponential time in D assuming the GRH The library syntax is GEN qfbsolve GEN Q GEN p 3 4 70 quadclassunit D flag 0 tech Buchmann McCurley s sub exponential algo rithm for computing the class group of a quadratic order of discriminant D This function should be used instead of qfbclassno or quadregula when D lt 10 D gt 10 or when the structure is wanted It is a special case of bnfinit which is slower but more robust The result is a vector v whose components should be accessed using member functions e v no the class number e v cyc a vecto
458. reates the canonical quadratic polynomial in the variable v corresponding to the discriminant D i e the minimal polynomial of quadgen D D must be an integer congruent to 0 or 1 modulo 4 which is not a square The library syntax is GEN quadpoly0 GEN D long v 1 where v is a variable number 3 4 75 quadray D f Relative equation for the ray class field of conductor f for the quadratic field of discriminant D using analytic methods A bnf for x D is also accepted in place of D For D lt 0 uses the o function and Schertz s method For D gt 0 uses Stark s conjecture and a vector of relative equations may be returned See bnrstark for more details The library syntax is GEN quadray GEN D GEN f long prec 3 4 76 quadregulator x Regulator of the quadratic field of positive discriminant x Returns an error if x is not a discriminant fundamental or not or if x is a square See also quadclassunit if x is large The library syntax is GEN quadregulator GEN x long prec 3 4 77 quadunit D Fundamental unit of the real quadratic field Q V D where D is the positive discriminant of the field If D is not a fundamental discriminant this probably gives the funda mental unit of the corresponding order D must be an integer congruent to 0 or 1 modulo 4 which is not a square the result is a quadratic number see Section 3 4 72 The library syntax is GEN quadunit GEN D 3 4 78 randomprime N 2 Returns a s
459. recursively independently of the surrounding context For instance a b c gt a b c left associative 27 a b c gt a b cC XX right associative Assuming that 0p op2 0p3 are binary operators with increasing priorities think of 7 T 0P1 Y 0P2 Z 0P2 Ops Y is equivalent to T op y Opa 2 opa x ops y GP contains many different operators either unary having only one argument or binary plus a few special selection operators Unary operators are defined as either prefix or postfix meaning that they respectively precede op x and follow x op their single argument Some symbols are syntactically correct in both positions like but then represent different operators the symbol represents the negation and factorial operators when in prefix and postfix position respectively Binary operators all use the infix syntax x op y Most operators are standard some are borrowed from the C language lt lt and a few are specific to GP Beware that some GP operators differ slightly from their C counterparts For instance GP s postfix returns the new value like the prefix of C and the binary shifts lt lt gt gt have a priority which is different from higher than that of their C counterparts When in doubt just surround everything by parentheses besides your code will be more legible Here is the list of available operators ordered by decreasing priority binary and left
460. ressed The various print commands see Section 3 12 are unaffected so you can always type print or Na to view the full result If the actual screen width cannot be determined a line is assumed to be 80 characters long The default value is 0 3 14 19 linewrap If set to a positive value gp wraps every single line after printing that many characters The default value is O unset 307 3 14 20 log This can be either 0 off or 1 2 3 on see below for the various modes When logging mode is turned on gp opens a log file whose exact name is determined by the logfile default Subsequently all the commands and results will be written to that file see 1 In case a file with this precise name already existed it will not be erased your data will be appended at the end The specific positive values of log have the following meaning 1 plain logfile 2 emit color codes to the logfile if colors is set 3 write LaTeX output to the logfile can be further customized using TeXstyle The default value is 0 3 14 21 logfile Name of the log file to be used when the log toggle is on Environment and time expansion are performed The default value is pari log 3 14 22 nbthreads Number of threads to use for parallel computing The exact meaning an default depend on the mt engine used e single not used always one thread e pthread number of threads unlimited default number of core e mpi number of MPI proce
461. rge ellap E time 4 440 ms 42 1376268269510579884904540406082 ellcard E now instantaneous time O ms ellgenerators E time 5 965 ms ellgenerators E second time instantaneous time O ms See the description of member functions related to elliptic curves at the beginning of this section The library syntax is GEN ellinit GEN x GEN D NULL long prec 3 5 26 ellisoncurve E z Gives 1 i e true if the point z is on the elliptic curve E 0 otherwise If E or z have imprecise coefficients an attempt is made to take this into account i e an imprecise equality is checked not a precise one It is allowed for z to be a vector of points in which case a vector of the same type is returned The library syntax is GEN ellisoncurve GEN E GEN z Also available is int oncurve GEN E GEN z which does not accept vectors of points 141 3 5 27 ellj x Elliptic j invariant must be a complex number with positive imaginary part or convertible into a power series or a p adic number with positive valuation The library syntax is GEN jell GEN x long prec 3 5 28 elllocalred E p Calculates the Kodaira type of the local fiber of the elliptic curve E at the prime p E must be an ell structure as output by ellinit and is assumed to have all its coefficients a in Z The result is a 4 component vector f kod v c Here f is the exponent of p in the arithmetic conductor of E and kod is the Kodai
462. riants ga or 93 such that y 42 goa gs is a Weierstrass equation for E The library syntax is GEN elleisnum GEN w long k long flag long prec 3 5 16 elleta w Returns the quasi periods 71 172 associated to the lattice basis w w1 w2 Alternatively w can be an elliptic curve E as output by ellinit in which case the quasi periods associated to the period lattice basis E omega namely E eta are returned elleta 1 IJ 1 3 141592653589793238462643383 9 424777960769379715387930149 I The library syntax is GEN elleta GEN w long prec 3 5 17 ellfromj j Returns the coefficients a az az as ag of a fixed elliptic curve with j invariant 7 The library syntax is GEN ellfromj GEN j 3 5 18 ellgenerators E If E is an elliptic curve over the rationals return a Z basis of the free part of the Mordell Weil group associated to E This relies on the elldata database being installed and referencing the curve and so is only available for curves over Z of small conductors If E is an elliptic curve over a finite field F as output by ellinit return a minimal set of generators for the group E F The library syntax is GEN ellgenerators GEN E 137 3 5 19 ellglobalred E Calculates the arithmetic conductor the global minimal model of E and the global Tamagawa number c E must be an ell structure as output by ellinit defined over Q The result is a vector N v c F L where e N is the arithmetic c
463. riggering an error message not yet available for this architecture e If when running gp dyn you get a message of the form ld so warning libpari so xrx has older revision than expected xxx possibly followed by more errors you already have a dynamic PARI library installed and a broken local configuration Either remove the old library or unset the LD_LIBRARY_PATH environment variable Try to disable this variable in any case if anything very wrong occurs with the gp dyn binary like an Illegal Instruction on startup It does not affect gp sta e Some implementations of the diff utility on HPUX for instance output No differences encountered or some similar message instead of the expected empty input thus producing a spurious BUG message 3 6 2 Some more testing Optional You can test gp in compatibility mode with make test compat If you want to test the graphic routines use make test ploth You will have to click on the mouse button after seeing each image There will be eight of them probably shown twice try to resize at least one of them as a further test The make bench and make test compat runs produce a Postscript file pari ps in Orxx which you can send to a Postscript printer The output should bear some similarity to the screen images 3 6 3 Heavy duty testing Optional There are a few extra tests which should be useful only for developers make test kernel checks whether the low level kernel seems to wor
464. riginal and the rounded value the fractional part If the exponent of x is too large compared to its precision i e e gt 0 the result is undefined and an error occurs if e was not given Important remark Contrary to the other truncation functions this function operates on every coefficient at every level of a PARI object For example 2 4 X 1 7 t t runcate Xx 24ex whereas d 24x X 1 7 2x X 2 roun ou 7 ae An important use of round is to get exact results after an approximate computation when theory tells you that the coefficients must be integers The library syntax is GEN roundO GEN x GEN e NULL Also available are GEN grnd toi GEN x long e and GEN ground GEN x 85 3 2 51 simplify x This function simplifies x as much as it can Specifically a complex or quadratic number whose imaginary part is the integer 0 i e not Mod 0 2 or 0 E 28 is converted to its real part and a polynomial of degree 0 is converted to its constant term Simplifications occur recursively This function is especially useful before using arithmetic functions which expect integer argu ments x 2 y y Wi 2 isprime x eK at top level isprime x WRK Fae xxx isprime not an integer argument in an arithmetic function type x 2 t_POL type simplify x 3 t_INT Note that GP results are simplified as above before they are stored in the history Unless you disable automatic
465. rimedec nf p j m can be decoded using bnfdecodemodule Note that to compute such data for a single field either bnrclassno or bnrdisc is more efficient The library syntax is GEN bnrdisclistO GEN bnf GEN bound GEN arch NULL 165 3 6 27 bnrinit bnf f flag 0 bnf is as output by bnfinit f is a modulus initializes data linked to the ray class group structure corresponding to this module a so called bnr structure One can input the associated bid with generators for f instead of the module itself saving some time As in idealstar the finite part of the conductor may be given by a factorization into prime ideals as produced by idealfactor The following member functions are available on the result bnf is the underlying bnf mod the modulus bid the bid structure associated to the modulus finally clgp no cyc gen refer to the ray class group as a finite abelian group its cardinality its elementary divisors its generators only computed if flag 1 The last group of functions are different from the members of the underlying bnf which refer to the class group use bnr bnf xxx to access these e g bnr bnf cyc to get the cyclic decomposition of the class group They are also different from the members of the underlying bid which refer to Zx f use bnr bid xxx to access these e g bnr bid no to get o f If flag 0 default the generators of the ray class group are not computed which saves ti
466. rization Note that including a C such that C does not divide D is useless If neither happen then the computed basis need not generate the maximal order Here is an example B 1075 P factor poldisc T B 1 primes lt B dividing D cofactor basis nfbasis T listP disc nfdisc T listP NN NN We obtain the maximal order and its discriminant if the field discriminant factors completely over the primes less than B together with the primes contained in the addprimes table This can be tested as follows check factor disc B lastp check 1 1 1 if lastp gt B amp amp setsearch addprimes lastp warning nf may be incorrect This is a sufficient but not a necessary condition hence the warning instead of an error N B lastp is the last entry in the first column of the check matrix i e the largest prime dividing nf disc if lt B or if it belongs to the prime table The function nfcertify speeds up and automates the above process B 1075 nf nfinit T B nfcertify nf 13 nf is unconditionally correct basis nf zk disc nf disc The library syntax is nfbasis GEN T GEN d GEN listP NULL which returns the order basis and where d receives the order discriminant 186 3 6 78 nfbasistoalg nf x Given an algebraic number x in the number field nf transforms it into t_POLMOD form nf nfinit y 2 4 nf zk 12 1 1 2x y nfbas
467. rmed using binary left shift powering techniques In particular in this case x cannot be a vector or matrix unless it is a square matrix invertible if the exponent is negative If x is a p adic number its precision will increase if v n gt 0 Powering a binary quadratic form types t_QFI and t_QFR returns a reduced representative of the class provided the input is reduced In particular x71 is identical to x PARI is able to rewrite the multiplication x x of two identical objects as x or sqr x Here identical means the operands are two different labels referencing the same chunk of memory no equality test is performed This is no longer true when more than two arguments are involved If the exponent is not of type integer this is treated as a transcendental function see Sec tion 3 3 and in particular has the effect of componentwise powering on vector or matrices As an exception if the exponent is a rational number p q and x an integer modulo a prime or a p adic number return a solution y of y x if it exists Currently q must not have large prime factors Beware that Mod 7 19 1 2 66 1 Mod 11 19 is any square root sqrt Mod 7 19 12 Mod 8 19 is the smallest square root Mod 7 19 7 3 5 13 Mod 1 19 1 37 5 3 14 Mod 1 19 Mod 7 19 is just another cubic root If the exponent is a negative integer an inverse must be computed For non invertible t_INTMOD
468. rns a preimage z of y by x if one exists i e such that xz y an empty vector or matrix otherwise The complete inverse image is z Kerz where a basis of the kernel of x may be obtained by matker M 1 2 2 4 matinverseimage M 1 2 12 1 0 matinverseimage M 3 4 13 O XV no solution matinverseimage M 1 3 6 2 6 12 Y4 1 3 6 o 0 0 matinverseimage M 1 2 3 4 45 3 no solution K matker M 16 2 1 The library syntax is GEN inverseimage GEN x GEN y 240 3 8 32 matisdiagonal x Returns true 1 if x is a diagonal matrix false 0 if not The library syntax is GEN isdiagonal GEN x 3 8 33 matker z flag 0 Gives a basis for the kernel of the matrix x as columns of a matrix The matrix can have entries of any type provided they are compatible with the generic arithmetic operations x and If x is known to have integral entries set flag 1 The library syntax is GEN matker0 GEN x long flag Also available are GEN ker GEN x flag 0 GEN keri GEN x flag 1 3 8 34 matkerint z flag 0 Gives an LLL reduced Z basis for the lattice equal to the kernel of the matrix x as columns of the matrix x with integer entries rational entries are not permitted If flag 0 uses an integer LLL algorithm If flag 1 uses matrixqz x 2 Many orders of magnitude slower than the default never use this The library syntax is GEN matkerintO G
469. ror type is given by errname E and other data can be accessed using the component function The code seq2 should check whether the error is the one expected In the negative the error can be rethrown using error E and possibly caught by an higher iferr instance The following uses iferr to implement Lenstra s ECM factoring method ecm N B 1000 nb 100 for a 1 nb iferr ellmul ellinit a 1 Mod 1 N 0 1 Mod 1 N B E return gcd lift component E 2 N errname E e_INV amp amp type component E 2 t_INTMOD ecm 27101 1 2 7432339208719 The return value of iferr itself is the value of seq2 if an error occurs and the value of seq otherwise We now describe the list of valid error types and the associated error data E in each case we list in order the components of E accessed via component E 1 component E 2 etc Internal errors system errors e e_ARCH A requested feature s is not available on this architecture or operating system E has one component t_STR the missing feature name s e e BUG A bug in the PARI library in function s E has one component t_STR the function name s e e FILE Error while trying to open a file E has two components 1 t_STR the file type input output etc 2 t_STR the file name e e_IMPL A requested feature s is not implemented E has one component 1 t_STR the feature name s e e PACKAGE Missing optional packa
470. rrors to forget for instance that a real number is given as r 2 where r is a rational approximation e a binary exponent and e is a nondescript real number less than 1 in absolute value Hence any number less than 2 may be treated as an exact zero 0 E 28 1 E 100 1 0 E 28 0 E100 1 92 0 E100 As an exercise if a 27 100 why doa O anda 1 differ The second principle is that PARI operations are in general quite permissive For instance taking the exponential of a vector should not make sense However it frequently happens that one wants to apply a given function to all elements in a vector This is easily done using a loop or using the apply built in function but in fact PARI assumes that this is exactly what you want to do when you apply a scalar function to a vector Taking the exponential of a vector will do just that so no work is necessary Most transcendental functions work in the same way In the same spirit when objects of different types are combined they are first automatically mapped to a suitable ring where the computation becomes meaningful 1 3 Mod 1 5 1 Mod 3 5 I 0 579 12 2 5 2 572 573 3x5 4 4x5 5 2x5 6 3x5 7 0 579 Mod 1 15 Mod 1 10 13 Mod 2 5 The first example is straightforward since 3 is invertible mod 5 1 3 is easily mapped to Z 5Z In the second example I stands for the customary square root of 1 we obtain a 5 adic number 5 adi
471. s express generators of H on the given generators g Note that the determinant of that matrix is equal to the index G H 152 3 6 4 Relative extensions We now have a look at data structures associated to relative extensions of number fields L K and to projective Zx modules When defining a relative extension L K the nf associated to the base field K must be defined by a variable having a lower priority see Section 2 5 3 than the variable defining the extension For example you may use the variable name y to define the base field K and x to define the relative extension L K 3 6 4 1 Basic definitions e rnf denotes a relative number field i e a data structure output by rnfinit associated to the extension L K The nf associated to be base field K is rnf nf e A relative matrix is an m x n matrix whose entries are elements of K in any form Its m columns A represent elements in K e An ideal list is a row vector of fractional ideals of the number field nf e A pseudo matriz is a 2 component row vector A J where A is a relative m x n matrix and I an ideal list of length n If J a a and the columns of A are 4y An this data defines the torsion free projective Zx module a A O an An e An integral pseudo matrix is a 3 component row vector w A I J where A a j is an mx n relative matrix and I b1 bm J a1 an are ideal lists such that a j bia for all i j This data defines
472. s matrix and character string n is meaningless and must be omitted or an error is raised Otherwise if n is given 0 entries are appended at the end of the vector if n gt 0 and prepended at the beginning if n lt 0 The dimension of the resulting vector is n Note that the function Colrev does not exist use Vecrev The library syntax is GEN gtocol0 GEN x long n GEN gtocol GEN x is also available 3 2 2 Colrev z n As Col x n then reverse the result In particular The library syntax is GEN gtocolrevO GEN x long n GEN gtocolrev GEN x is also avail able 3 2 3 List x Transforms a row or column vector x into a list whose components are the entries of x Similarly for a list but rather useless in this case For other types creates a list with the single element x Note that except when x is omitted this function creates a small memory leak so either initialize all lists to the empty list or use them sparingly The library syntax is GEN gtolist GEN x NULL The variant GEN listcreate void cre ates an empty list 71 3 2 4 Mat x Transforms the object x into a matrix If x is already a matrix a copy of x is created If x is a row resp column vector this creates a l row resp 1 column matrix unless all elements are column resp row vectors of the same length in which case the vectors are concatenated sideways and the associated big matrix is returned If x is a binary quadratic form
473. s a not necessarily square matrix of maximal rank with integer entries and d is a multiple of the non zero determinant of the lattice spanned by the columns of x finds the upper triangular Hermite normal form of z If the rank of x is equal to its number of rows the result is a square matrix In general the columns of the result form a basis of the lattice spanned by the columns of x Even when d is known this is in general slower than mathnf but uses much less memory The library syntax is GEN hnfmod GEN x GEN d 3 8 24 mathnfmodid x d Outputs the upper triangular Hermite normal form of x concate nated with the diagonal matrix with diagonal d Assumes that x has integer entries Variant if d is an integer instead of a vector concatenate d times the identity matrix m 0 7 1 0 1 1 41 o 7 1 0 1 1 mathnfmodid m 6 2 2 12 2 1 1 o 1 0 0 O 1 mathnfmodid m 10 13 10 7 3 010 00 1 The library syntax is GEN hnfmodid GEN x GEN d 3 8 25 mathouseholder Q v applies a sequence Q of Householder transforms as returned by matqr M 1 to the vector or matrix v The library syntax is GEN mathouseholder GEN Q GEN v 3 8 26 matid n Creates the n x n identity matrix The library syntax is GEN matid long n 239 3 8 27 matimage z flag 0 Gives a basis for the image of the matrix x as columns of a matrix A priori the matrix can have entries of any type If f
474. s elements are given as elements in Q X T nfbasis x 2 1 1 1 x This function uses a modified version of the round 4 algorithm due to David Ford Sebastian Pauli and Xavier Roblot Local basis orders maximal at certain primes Obtaining the maximal order is hard it requires factoring the discriminant D of T Obtaining an order which is maximal at a finite explicit set of primes is easy but if may then be a strict suborder of the maximal order To specify that we are interested in a given set of places only we can replace the argument T by an argument T listP where listP encodes the primes we are interested in it must be a factorization matrix a vector of integers or a single integer e Vector we assume that it contains distinct prime numbers e Matrix we assume that it is a two column matrix of a partial factorization of D namely the first column contains primes and the second one the valuation of D at each of these primes e Integer B this is replaced by the vector of primes up to B Note that the function will use at least O B time a small value about 10 should be enough for most applications Values larger than 2 are not supported In all these cases the primes may or may not divide the discriminant D of T The function then returns a Z basis of an order whose index is not divisible by any of these prime numbers The result is actually a global integral basis if all prime divisors of the field discrim
475. s for numerical summation using sumnum with sgn 1 or sumnumalt with sgn 1 sig is the abscissa of integration coded as in sumnum and m is as in intnuminit The library syntax is GEN sumnuminit GEN sig long m long sgn long prec 3 9 27 sumpos X a expr flag 0 Numerical summation of the series expr which must be a series of terms having the same sign the formal variable X starting at a The algorithm used is Van Wijngaarden s trick for converting such a series into an alternating one then we use sumalt For regular functions the function sumnum is in general much faster once the initializations have been made using sumnuminit The routine is heuristic and assumes that expr is more or less a decreasing function of X In particular the result will be completely wrong if expr is 0 too often We do not check either that all terms have the same sign As sumalt this function should be used to try and guess the value of an infinite sum If flag 1 use slightly different polynomials Sometimes faster The library syntax is sumpos void E GEN eval void GEN GEN a long prec Also available is sumpos2 with the same arguments flag 1 271 3 10 Plotting functions Although plotting is not even a side purpose of PARI a number of plotting functions are provided Moreover a lot of people suggested ideas or submitted patches for this section of the code There are three types of graphic functions 3 10 1 Hig
476. s generators on bnf tufu tpuexpo bnf my S d K S bnfsignunit bnf d matsize S S matrix d 1 d 2 i j if S i j lt 0 1 0 S concat vectorv d 1 i 1 S add sign 1 K lift matker S Mod 1 2 if K mathnfmodid K 2 2 matid d 1 totally positive units tpu bnf my vu bnf tufu ex tpuexpo bnf vector ex 1 i factorback vu ex i 1 AN exI 1 is 1 The library syntax is GEN signunits GEN bnf 3 6 19 bnfsunit bnf S Computes the fundamental S units of the number field bnf output by bnfinit where S is a list of prime ideals output by idealprimedec The output is a vector v with 6 components v 1 gives a minimal system of integral generators of the S unit group modulo the unit group v 2 contains technical data needed by bnfissunit v 3 is an empty vector used to give the logarithmic embeddings of the generators in v 1 in version 2 0 16 v 4 is the S regulator this is the product of the regulator the determinant of v 2 and the natural logarithms of the norms of the ideals in 5 v 5 gives the S class group structure in the usual format a row vector whose three components give in order the S class number the cyclic components and the generators v 6 is a copy of S The library syntax is GEN bnfsunit GEN bnf GEN S long prec 162 3 6 20 bnrL1 bnr H flag 0 Let bnr be the number field data output by bnrinit 1 and H
477. s main variable v t_FFELT are not lifted nor are List elements you may convert the latter to vectors first or use apply 1ift L More generally components for which such lifts are meaningless e g character strings are copied verbatim lift Mod 5 3 1 2 1ift 3 0 379 12 3 lift Mod x x 2 1 13 x lift Mod x x 2 1 94 x Lifts are performed recursively on an object components but only by one level once a t_POLMOD is lifted the components of the result are not lifted further lift x Mod 1 3 Mod 2 3 Y4 x 2 lift x Mod y y 2 1 Mod 2 3 75 y x Mod 2 3 do you understand this one 1ift x Mod y y 2 1 Mod 2 3 x 16 Mod y y72 1 x Mod Mod 2 3 y 2 1 litt y 7 y x Mod 2 3 To recursively lift all components not only by one level but as long as possible use 1iftall To lift only t_INTMODs and t_PADICs components use liftint To lift only t_POLMODs components use liftpol Finally centerlift allows to lift t_INTMODs and t_PADICs using centered residues lift of smallest absolute value The library syntax is GEN 1iftO GEN x long v 1 where v is a variable number Also available is GEN 1ift GEN x corresponding to 1ift0 x 1 81 3 2 39 liftall x Recursively lift all components of x from Z nZ to Z from Q to Q as truncate and polmods to polynomials t_FFELT are not lifted nor are List elements you may convert the latter to vec
478. s possible at least a gt 1 265 Tf flag 3 the function is allowed to be undefined but continuous at a or b for example the function sin x x at z 0 The user should not require too much accuracy 18 or 28 decimal digits is OK but not much more In addition analytical cleanup of the integral must have been done there must be no singularities in the interval or at the boundaries In practice this can be accomplished with a simple change of variable Furthermore for improper integrals where one or both of the limits of integration are plus or minus infinity the function must decrease sufficiently rapidly at infinity This can often be accomplished through integration by parts Finally the function to be integrated should not be very small compared to the current precision on the entire interval This can of course be accomplished by just multiplying by an appropriate constant Note that infinity can be represented with essentially no loss of accuracy by an appropriate huge number However beware of real underflow when dealing with rapidly decreasing functions For example in order to compute the tee e7 dx to 28 decimal digits then one can set infinity equal to 10 for example and certainly not to 1e1000 The library syntax is intnumromb void E GEN eval void GEN GEN a GEN b long flag long prec where eval x E returns the value of the function at x You may store any additional information required by eval in E or
479. s we must use a default value When no input arguments are left the defaults are used instead to fill in remaining formal parameters A final example f x y 2 z 3 print x y z defines a function which prints its arguments at most three of them separated by colons 6 7 Ai eo 26 5 523 0 20 oN ON DN If strictargs is set recommended x is now a mandatory argument and the above becomes default strictargs 1 6 7 27 3 f 5 kk at top level f 5 RR Te NON y Hook in function f x y 2 z 3 KK sae eee ere re missing mandatory argument x in user function 40 Example We conclude with an amusing example intended to illustrate both user defined func tions and the power of the sumalt function Although the Riemann zeta function is included as zeta among the standard functions let us assume that we want to check other implementations Since we are highly interested in the critical strip we use the classical formula 217 1 s Y 1 J n7 Rs gt 0 n gt 1 The implementation is obvious ZETA s sumalt n 1 1 n n s 27 1 8 1 Note that n is automatically lexically scoped to the sumalt loop so that it is unnecessary to add a my n declaration to the function body Surprisingly this gives very good accuracy in a larger region than expected check z gt ZETA z zeta z check 2 1 1 000000000000000000000000000 check 200
480. s y Compare 7x 4 5 x 1 11 5 x Mod 4 5 x 1 92 Mod 0 5 The library syntax is GEN gmodulo GEN a GEN b 3 2 6 Pol t v x Transforms the object t into a polynomial with main variable v If t is a scalar this gives a constant polynomial If t is a power series with non negative valuation or a rational function the effect is similar to truncate i e we chop off the O X or compute the Euclidean quotient of the numerator by the denominator then change the main variable of the result to v The main use of this function is when t is a vector it creates the polynomial whose coefficients are given by t with t 1 being the leading coefficient which can be zero It is much faster to evaluate Pol on a vector of coefficients in this way than the corresponding formal expression an X ap which is evaluated naively exactly as written linear versus quadratic time in n Polrev can be used if one wants x 1 to be the constant coefficient Pol 1 2 3 fi x72 Qex 3 Polrev 1 2 3 12 3 x 2 2 x 1 The reciprocal function of Pol resp Polrev is Vec resp Vecrev Vec Po1 1 2 31 1 1 2 3 Vecrev Polrev 1 2 3 12 1 2 3 73 Warning This is not a substitution function It will not transform an object containing variables of higher priority than v Pol x y y at top level Pol x y y CK Pol variable must have higher priority in gtopoly
481. safe to use and will not be mistaken for I y 1 analogously o is not synonymous to O In GP you can use up to 16383 variable names up to 65535 on 64 bit machines If you ever need thousands of variables and this becomes a serious limitation you should probably be using vectors instead e g instead of variables X1 X2 X3 you might equally well store their values in X 1 X 2 X 3 2 5 2 Variables and polynomials What happens when you use a valid variable name t say for the first time before assigning a value into it This registers a new free variable with the interpreter which will be written as t and evaluates to a monomial of degree 1 in the said variable t It is important to understand that PARI GP is not a symbolic manipulation package even free variables already have default values there is no such thing as an unbound variable in GP You have access to this default value using the quote operator t always evaluates to the above monomial of degree 1 independently of assignments made since then e g t 1 2t72 1 1 t72 1 tS 2 t 2 4 1 More generally any expression has a value and is replaced by its value as soon as it is read it never stays in an abstract form 31 12 5 hi 13 t72 1 eval 41 14 5 In the above t is initially a free variable later bound to 2 We see that assigning a value to a variable does not affect previous expressions involving it to take into acco
482. sequence rec is executed if the error occurs and the evaluation of rec becomes the result of the command If e is omitted all exceptions are trapped See Section 2 10 2 for an introduction to error recovery under gp trap division by 0 inv x trap e_INV INFINITY 1 x inv 2 wi 1 2 inv 0 72 INFINITY Note that seq is effectively evaluated up to the point that produced the error and the recovery sequence is evaluated starting from that same context it does not undo whatever happened in the other branch restore the evaluation context x 1 trap recover x try x 0 1 x 141 0 Note The interface is currently not adequate for trapping individual exceptions In the current version 2 7 0 the following keywords are recognized but the name list will be expanded and changed in the future all library mode errors can be trapped it s a matter of defining the keywords to gp e_ALARM alarm time out e_ARCH not available on this architecture or operating system e_STACK the PARI stack overflows e_INV impossible inverse e_IMPL not yet implemented e OVERFLOW all forms of arithmetic overflow including length or exponent overflow when a larger value is supplied than the implementation can handle e_SYNTAX syntax error e_MISC miscellaneous error e_TYPE wrong type e_USER user error from the error function The library syntax is GEN trapO const char e NULL GEN rec NULL
483. set 84 using setrand from a given seed and given sequence of randoms the exact same values will be generated The same seed is used at each startup reseed the generator yourself if this is a problem Note that internal functions also call the random number generator adding such a function call in the middle of your code will change the numbers produced Technical note Up to version 2 4 included the internal generator produced pseudo random numbers by means of linear congruences which were not well distributed in arithmetic pro gressions We now use Brent s XORGEN algorithm based on Feedback Shift Registers see http wwwmaths anu edu au brent random html The generator has period 24096 1 passes the Crush battery of statistical tests of L Ecuyer and Simard but is not suitable for cryptographic purposes one can reconstruct the state vector from a small sample of consecutive values thus predicting the entire sequence The library syntax is GEN genrand GEN N NULL Also available GEN ellrandom GEN E and GEN ffrandom GEN a 3 2 49 real x Real part of x In the case where x is a quadratic number this is the coefficient of 1 in the canonical integral basis 1 w The library syntax is GEN greal GEN x 3 2 50 round z amp e If x is in R rounds x to the nearest integer rounding to 00 in case of ties then and sets e to the number of error bits that is the binary exponent of the difference between the o
484. set it to NULL 3 9 14 intnumstep Give the value of m used in all the intnum and sumnum programs hence such that the integration step is equal to 1 2 The library syntax is long intnumstep long prec 3 9 15 prod X a b erpr x 1 Product of expression expr initialized at x the formal parameter X going from a to b As for sum the main purpose of the initialization parameter x is to force the type of the operations being performed For example if it is set equal to the integer 1 operations will start being done exactly If it is set equal to the real 1 they will be done using real numbers having the default precision If it is set equal to the power series 1 O X for a certain k they will be done using power series of precision at most k These are the three most common initializations As an extreme example compare prod i 1 100 1 X7i MAN this has degree 5050 time 128 ms 2 prod ind 100 1 X i 1 0 X 101 time 8 ms A2 1 X X72 X75 X 7 X712 X 15 X 22 X 26 X 35 X740 X 51 X 57 X 70 X 77 X792 X7100 O X7101 Of course in this specific case it is faster to use eta which is computed using Euler s formula prod i 1 1000 1 X i 1 O X 1001 time 589 ms ps1000 seriesprecision 1000 significant terms eta X time 8ms 4 0 X71001 The library syntax is produit GEN a GEN b char expr GEN x 266 3 9 16 prodeuler X a b ex
485. sfy eh gt l e R 0 2 gt 1 5 Here s how to solve the Thue equation 21 5y 4 tnf thueinit x 13 5 thue tnf 4 41 1 1 In this case one checks that bnfinit x713 5 no is 1 Hence the only solution is x y 1 1 and the result is unconditional On the other hand P x73 2xx 2 3 x 17 tnf thueinit P thue tnf 15 2 1 111 a priori conditional on the GRH K bnfinit P K no 13 3 K reg 14 2 8682185139262873674706034475498755834 This time the result is conditional All results computed using this particular tnf are likewise conditional except for a right hand side of 1 The above result is in fact correct so we did not just disprove the GRH tnf thueinit x 3 2 x 2 3 x 17 1 unconditional thue tnf 15 4 1 1 Note that reducible or non monic polynomials are allowed tnf thueinit 2 x 1 5 4 x 3 2xx 2 3x x 17 1 thue tnf 128 2 1 0 1 0 Reducible polynomials are in fact much easier to handle The library syntax is GEN thue GEN tnf GEN a GEN sol NULL 227 3 7 47 thueinit P flag 0 Initializes the tnf corresponding to P a univariate polynomial with integer coefficients The result is meant to be used in conjunction with thue to solve Thue equations P X Y Y48P a where a is an integer If flag is non zero certify results unconditionally Otherwise assume GRH this being much faster of
486. sic internal type t_COMPLEX exists However the components i e the real and imaginary part of such a complex number can be of any type The only sensible ones are integers we are then in Z i rational numbers Q i real numbers R C or even elements of Z nZ in Z nZ t t2 1 or p adic numbers when p 3mod4 Q i This feature must not be used too rashly in library mode for example you are in principle allowed to create objects which are complex numbers of complex numbers This is not possible under gp But do not expect PARI to make sensible use of such objects you will mainly get nonsense On the other hand it is allowed to have components of different but compatible types which can be freely mixed in basic ring operations or X For example taking again complex numbers the real part could be an integer and the imaginary part a rational number On the other hand if the real part is a real number the imaginary part cannot be an integer modulo n Let us now describe the types As explained above they are built recursively from basic types which are as follows We use the letter T to designate any type the symbolic names t_xxx correspond to the internal representations of the types type t_INT Z Integers with arbitrary precision type t_REAL R Real numbers with arbitrary precision type t_INTMOD Z nZ Intmods integers modulo n type t_FRAC Q Rational numbers in irreducible form type t_FFELT F
487. simplification with y that is In particular type 1 YA t INT The library syntax is GEN simplify GEN x 3 2 52 sizebyte x Outputs the total number of bytes occupied by the tree representing the PARI object x The library syntax is long gsizebyte GEN x Also available is long gsizeword GEN x returning a number of words 3 2 53 sizedigit x Outputs a quick bound for the number of decimal digits of the components of x off by at most 1 If you want the exact value you can use Str x which is slower The library syntax is long sizedigit GEN x 3 2 54 truncate z amp e Truncates x and sets e to the number of error bits When z is in R this means that the part after the decimal point is chopped away e is the binary exponent of the difference between the original and the truncated value the fractional part Tf the exponent of x is too large compared to its precision i e e gt 0 the result is undefined and an error occurs if e was not given The function applies componentwise on vector matrices e is then the maximal number of error bits If x is a rational function the result is the integer part Euclidean quotient of numerator by denominator and e is not set Note a very special use of truncate when applied to a power series it transforms it into a polynomial or a rational function with denominator a power of X by chopping away the O X Similarly when applied to a p adic number it transf
488. sion If a gp binary file see Section 3 12 44 is read using this command it is silently loaded without cluttering the history Assuming gp figures how to decompress files on your machine this command accepts com pressed files in compressed Z or gzipped gz or z format They will be uncompressed on the fly as gp reads them without changing the files themselves 2 13 19 s Prints the state of the PARI stack and heap This is used primarily as a debugging device for PARI 2 13 20 t Prints the internal longword format of all the PARI types The detailed bit or byte format of the initial codeword s is explained in Chapter 4 but its knowledge is not necessary for a gp user 2 13 21 u Prints the definitions of all user defined functions 2 13 22 um Prints the definitions of all user defined member functions 2 13 23 v Prints the version number and implementation architecture 680x0 Sparc Alpha other of the gp executable you are using 2 13 24 Ww n filename Writes the object number n n into the named file in raw format If the number n is omitted writes the latest computed object If filename is omitted appends to logfile the GP function write is a trifle more powerful as you can have arbitrary file names 2 13 25 x n Prints the complete tree with addresses and contents in hexadecimal of the internal representation of the object number n n If the number n is omitted uses the latest compu
489. spect to the simple variable t x can be of any reasonable type for example a rational function Contrary to Ser which takes the valuation into account this function adds O t to all components of x taylor x 1 y y 5 1 y74 y 3 y 2 y 1 x x 0 y75 Ser x 1l y y 5 at top level Ser x 1l y y 5 do Ser main variable must have higher priority in gtoser The library syntax is GEN tayl GEN x long t long precdl where t is a variable number 226 3 7 46 thue tnf a sol Returns all solutions of the equation P x y a in integers x and y where tnf was created with thueinit P If present sol must contain the solutions of Norm a modulo units of positive norm in the number field defined by P as computed by bnfisintnorm If there are infinitely many solutions an error will be issued It is allowed to input directly the polynomial P instead of a tnf in which case the function first performs thueinit P 0 This is very wasteful if more than one value of a is required If tnf was computed without assuming GRH flag 1 in thueinit then the result is uncondi tional Otherwise it depends in principle of the truth of the GRH but may still be unconditionally correct in some favourable cases The result is conditional on the GRH if a 4 1 and P has a single irreducible rational factor whose associated tentative class number h and regulator R as computed assuming the GRH sati
490. spermtopol gal perm gal being a Galois group as output by galoisinit and perm a element of gal group return the polynomial defining the Galois automorphism as output by nfgaloisconj associated with the permutation perm of the roots gal roots perm can also be a vector or matrix in this case galoispermtopol is applied to all components recursively Note that G galoisinit pol galoispermtopol G G 6 is equivalent to nfgaloisconj pol if degree of pol is greater or equal to 2 The library syntax is GEN galoispermtopol GEN gal GEN perm 171 3 6 42 galoissubcyclo N H fl 0 v Computes the subextension of Q C fixed by the subgroup H C Z nZ By the Kronecker Weber theorem all abelian number fields can be generated in this way uniquely if n is taken to be minimal The pair n H is deduced from the parameters N H as follows e N an integer then n N H is a generator i e an integer or an integer modulo n or a vector of generators e N the output of znstar n H as in the first case above or a matrix taken to be a HNF left divisor of the SNF for Z nZ of type N cyc giving the generators of H in terms of N gen e N the output of bnrinit bnfinit y m 1 where m is a module H as in the first case or a matrix taken to be a HNF left divisor of the SNF for the ray class group modulo m of type N cyc giving the generators of H in terms of N gen In this last case beware that H is underst
491. ss of functions Except when stated otherwise it is understood that a function or operation which should make natural sense is legal In this chapter we will describe the functions according to a rough classification The general entry looks something like foo x flag 0 short description The library syntax is GEN foo GEN x long fl 0 This means that the GP function foo has one mandatory argument x and an optional one flag whose default value is 0 The should not be typed it is just a convenient notation we will use throughout to denote optional arguments That is you can type foo x 2 or foo x which is then understood to mean foo x 0 As well a comma or closing parenthesis where an optional argument should have been signals to GP it should use the default Thus the syntax foo x is also accepted as a synonym for our last expression When a function has more than one optional argument the argument list is filled with user supplied values in order When none are left the defaults are used instead Thus assuming that foo s prototype had been foo x 1 y 2 z 3 typing in foo 6 4 would give you foo 6 4 3 In the rare case when you want to set some far away argument and leave the defaults in between as they stand you can use the empty arg trick alluded to above foo 6 1 would yield foo 6 2 1 By the way foo by itself yields foo 1 2 3 as was to be expected In this rather special case of
492. ss to use limited to the number allocated by mpirun default use all allocated process 3 14 23 new_galois_format This toggle is either 1 on or 0 off If on the polgalois command will use a different more consistent naming scheme for Galois groups This default is provided to ensure that scripts can control this behavior and do not break unexpectedly The defaut value is 0 This value will change to 1 set in the next major version 3 14 24 output There are three possible values 0 raw 1 prettymatriz or 3 external prettyprint This means that independently of the default format for reals which we explained above you can print results in three ways e raw format i e a format which is equivalent to what you input including explicit multiplica tion signs and everything typed on a line instead of two dimensional boxes This can have several advantages for instance it allows you to pick the result with a mouse or an editor and to paste it somewhere else e prettymatrix format this is identical to raw format except that matrices are printed as boxes instead of horizontally This is prettier but takes more space and cannot be used for input Column vectors are still printed horizontally e external prettyprint pipes all gp output in TeX format to an external prettyprinter according to the value of prettyprinter The default script tex2mail converts its input to readable two dimensional text Independently
493. ssor e three versions for the Sparc architecture version 7 version 8 with SuperSparc processors and version 8 with MicroSparc I or II processors UltraSparcs use the MicroSparc II version e the DEC Alpha 64 bit processor e the Intel Itanium 64 bit processor e the PowerPC equipping old macintoshs G3 G4 etc e the HPPA processors both 32 and 64 bit A third version uses the GNU MP library to implement most of its multiprecision kernel It improves significantly on the native one for large operands say 100 decimal digits of accuracy or more You should enable it if GMP is present on your system Parts of the first version are still in use within the GMP kernel but are scheduled to disappear A historical version of the PARI GP kernel written in 1985 was specific to 680x0 based computers and was entirely written in MC68020 assembly language It ran on SUN 3 xx Sony News NeXT cubes and on 680x0 based Macs It is no longer part of the PARI distribution to run PARI with a 68k assembler micro kernel use the GMP kernel 1 3 The PARI types The GP language is not typed in the traditional sense in particular variables have no type In library mode the type of all PARI objects is GEN a generic type On the other hand it is dynamically typed each object has a specific internal type depending on the mathematical object 1t represents The crucial word is recursiveness most of the PARI types are recursive For example the ba
494. st be in Q X If they are not the output is the number 0 If they are the output is a vector of polynomials each polynomial a representing an embedding of K into L i e being such that y xo a If y is a number field nf a much faster algorithm is used factoring x over y using nffactor Before version 2 0 14 this wasn t guaranteed to return all the embeddings hence was triggered by a special flag This is no more the case The library syntax is GEN nfisincl GEN x GEN y 3 6 108 nfisisom x y As nfisincl but tests for isomorphism If either x or y is a number field a much faster algorithm will be used The library syntax is GEN nfisisom GEN x GEN y 3 6 109 nfkermodpr nf x pr Kernel of the matrix a in Zg pr where pr is in modpr format see nfmodprinit The library syntax is GEN nfkermodpr GEN nf GEN x GEN pr This function is normally useless in library mode Project your inputs to the residue field using nfM_to_FqM then work there 3 6 110 nfmodprinit nf pr Transforms the prime ideal pr into modpr format necessary for all operations modulo pr in the number field nf The library syntax is GEN nfmodprinit GEN nf GEN pr 3 6 111 nfnewprec nf Transforms the number field nf into the corresponding data using current usually larger precision This function works as expected if nf is in fact a bnf update bnf to current precision but may be quite slow many generators of principal ideals have to be computed
495. sted in the degrees of the simple factors the rnfequation instruction can be replaced by a trivial poldegree P poldegree L i If flag 1 outputs a vector of 4 component vectors R a b k where R ranges through the list of all possible compositums as above and a resp b expresses the root of P resp Q as an element of Q X R Finally k is a small integer such that b ka X modulo R A compositum is often defined by a complicated polynomial which it is advisable to reduce before further work Here is an example involving the field Q 5 51 L polcompositum x 5 5 polcyclo 5 1 list of R a b k 197 R a L 1 pick the single factor extract R a ignore b k R defines the compositum 3 x720 5 x719 15x xx 18 35 x 17 70 x716 141 x 15 260 x 14 355 x713 95 x712 1460 x711 3279 x 10 3660 x79 2005 x78 705 x77 9210 x76 13506 x75 7145 x74 2740 x73 1040 x72 320 x 256 a 5 5 a fifth root of 5 74 0 T X polredbest R 1 T simpler defining polynomial for Q x R 6 x720 25 x710 5 X root of R in Qly T y 7 Mod 1 11 x 15 1 11 x 14 1 22 x 10 47 22xx 5 29 11 x 4 7 22 x 20 25 x710 5 a subst a pol x X Ala in the new coordinates 18 Mod 1 11 x 14 29 11 x 4 x 20 25 x 10 5 a5 5 49 0 The library syntax is GEN polcompositum0 GEN P GEN Q long flag Also available are GEN
496. subst or by computing the roots of the polynomial given by algdep use polroots Internally lindep 1 2 2 flag is used A non zero value of flag may improve on the default behavior if the input number is known to a huge accuracy and you suspect the last bits are incorrect this truncates the number throwing away the least significant bits but default values are usually sufficient p200 algdep 2 1 6 37 1 5 30 wrong in 0 8s algdep 2 1 6 37 1 5 30 100 wrong in 0 4s algdep 2 1 6 37 1 5 30 170 right in 0 8s algdep 2 1 6 37 1 5 30 200 wrong in 1 0s p250 algdep 2 1 6 37 1 5 30 right in 1 0s algdep 2 1 6 37 1 5 30 200 right in 1 0s p500 algdep 2 1 6 37 1 5 30 right in 2 9s p1000 algdep 2 1 6 37 1 5 30 right in 10 6s The changes in defaultprecision only affect the quality of the initial approximation to 21 6431 5 algdep itself uses exact operations the size of its operands depend on the accuracy of the input of course more accurate input means slower operations 228 Proceeding by increments of 5 digits of accuracy algdep with default flag produces its first correct result at 205 digits and from then on a steady stream of correct results The above example is the test case studied in a 2000 paper by Borwein and Lisonek Appli cations of integer relation algorithms Discrete Math 217 p 65 82 T
497. such that x 0 is not a pole of gamma compute the Taylor expansion PARI only knows about regular power series and can t include logarithmic terms Ingamma 1 x 0 x72 hi 0 57721566490153286060651209008240243104 x 0 x72 Ingamma x 0 x72 kk at top level Ingamma x 0 x 2 kkk Ingamma domain error in lngamma valuation 0 Ingamma 1 x 0 x72 Ingamma Warning normalizing a series with O leading term kk at top level Ingamma 1 x 0 x 2 OK kk Ingamma domain error in intformal residue series pole 0 The library syntax is GEN glngamma GEN x long prec 3 3 42 log x Principal branch of the natural logarithm of x C i e such that Im log 7 7 The branch cut lies along the negative real axis continuous with quadrant 2 i e such that limp_ 9 log a bi log a for a R The result is complex with imaginary part equal to mw if Rand z lt 0 In general the algorithm uses the formula T m log 2 2agm 1 4 s A log x if s 12 is large enough The result is exact to B bits provided s gt 28 2 At low accuracies the series expansion near 1 is used p adic arguments are also accepted for x with the convention that log p 0 Hence in particular exp log 1 x is not in general equal to 1 but to a p 1 th root of unity or 1 if p 2 times a power of p The library syntax is GEN glog GEN x long prec For
498. sym Defines the symbol newsym as an alias for the the symbol sym alias det matdet det 1 2 3 4 1 2 You are not restricted to ordinary functions as in the above example to alias from to member functions prefix them with _ to alias operators use their internal name obtained by writing _ in lieu of the operators argument for instance _ and _ are the internal names of the factorial and the logical negation respectively alias mod _ mod alias add _ _ alias _ sin sin mod Mod x x 4 1 12 x 4 1 288 add 4 6 13 10 Pi sin 94 0 E 37 Alias expansion is performed directly by the internal GP compiler Note that since alias is performed at compilation time it does not require any run time processing however it only affects GP code compiled after the alias command is evaluated A slower but more flexible alternative is to use variables Compare with fun sin g a b intnum t a b fun t g 0 Pi 3 2 0000000000000000000000000000000000000 fun cos g 0 Pi 5 1 8830410776607851098 E 39 alias fun sin g a b intnum t a b fun t g 0 Pi 2 2 0000000000000000000000000000000000000 alias fun cos Oops Does not affect previous definition 7 g 0 Pi 3 2 0000000000000000000000000000000000000 g a b intnum t a b fun t Redefine taking new alias into account 7 g 0 Pi 5 1 8830410776607851
499. syntax is GEN idealfactorback GEN nf GEN f GEN e NULL long flag 3 6 53 idealfrobenius nf gal pr Let K be the number field defined by nf and assume K Q be a Galois extension with Galois group given gal galoisinit nf and that pr is the prime ideal in prid format and that is unramified This function returns a permutation of gal group which defines the automorphism o a i e the Frobenius element associated to If p is the unique prime number in then o a x modP for all x Zg nf nfinit polcyclo 31 gal galoisinit nf pr idealprimedec nf 101 1 g idealfrobenius nf gal pr galoispermtopol gal g 15 x78 NNN NY 175 This is correct since 101 8 mod 31 The library syntax is GEN idealfrobenius GEN nf GEN gal GEN pr 3 6 54 idealhnf nf u v Gives the Hermite normal form of the ideal uZx Z kx where u and v are elements of the number field K defined by nf nf nfinit y 3 2 idealhnf nf 2 y 1 2 1 0 0 o 1 0 0 0 1 idealhnf nf y 2 0 0 1 3 3 1 3 0 0 o 1 6 0 0 0 1 6 If b is omitted returns the HNF of the ideal defined by u u may be an algebraic number defining a principal ideal a maximal ideal as given by idealprimedec or idealfactor or a matrix whose columns give generators for the ideal This last format is a little complicated but useful to reduce general modules to the canonical form once in a while e if s
500. t A set of default colors for dark reverse video or PC console and light backgrounds respectively is activated when colors is set to darkbg resp lightbg or any proper prefix d is recognized as an abbreviation for darkbg A bold variant of darkbg called boldfg is provided if you find the former too pale In the present version this default is incompatible with PariEmacs Changing it will just fail silently the alternative would be to display escape sequences as is since Emacs will refuse to interpret them You must customize color highlighting from the PariEmacs side see its documentation The default value is no colors 304 3 14 4 compatible The GP function names and syntax have changed tremendously between versions 1 xx and 2 00 To help you cope with this we provide some kind of backward compatibility depending on the value of this default compatible 0 no backward compatibility In this mode a very handy function to be described in Section 3 12 41 is whatnow which tells you what has become of your favourite functions which gp suddenly can t seem to remember compatible 1 warn when using obsolete functions but otherwise accept them The output uses the new conventions though and there may be subtle incompatibilities between the behavior of former and current functions even when they share the same name the current function is used in such cases of course We thought of this one as a transitory help f
501. t and bnr is as output by bnrinit gives only the subgroups whose modulus is the conductor Otherwise the modulus is not taken into account If bound is present and is a positive integer restrict the output to subgroups of index less than bound If bound is a vector containing a single positive integer B then only subgroups of index exactly equal to B are computed For instance 213 subgrouplist 6 2 1 6 0 O 2 2 0 0 2 6 3 0O 1 2 1 O 1 3 07 0 2 1 0 0 2 6 0 0 1 2 O O 1 3 O O 1 1 0 O 1 subgrouplist 6 2 3 index less than 3 2 2 1 O 1 1 0 0 2 2 0 O 1 3 O O 1 1 O O 1 subgrouplist 6 2 3 Ml index 3 3 3 0 0 11 bnr bnrinit bnfinit x 120 1 1 L subgrouplist bnr 8 In the last example L corresponds to the 24 subfields of Q C120 of degree 8 and conductor 12000 by setting flag we see there are a total of 43 subgroups of degree 8 vector L i galoissubcyclo bnr L i will produce their equations For a general base field you would have to rely on bnrstark or rnfkummer The library syntax is GEN subgrouplistO GEN bnr GEN bound NULL long flag 3 6 163 zetak nfz x flag 0 znf being a number field initialized by zetakinit not by nfinit computes the value of the Dedekind zeta function of the number field at the complex number x If flag 1 computes Dedekind A function instead
502. t A B my Ker matkerint concat A B mathnf A Ker 1 A The library syntax is GEN idealintersect GEN nf GEN A GEN B 3 6 56 idealinv nf x Inverse of the ideal x in the number field nf given in HNF If x is an extended ideal its principal part is suitably updated i e inverting J t yields 17 1 1 The library syntax is GEN idealinv GEN nf GEN x 3 6 57 ideallist nf bound flag 4 Computes the list of all ideals of norm less or equal to bound in the number field nf The result is a row vector with exactly bound components Each component is itself a row vector containing the information about ideals of a given norm in no specific order depending on the value of flag The possible values of flag are 0 give the bid associated to the ideals without generators 1 as 0 but include the generators in the bid 2 in this case nf must be a bnf with units Each component is of the form bid U where bid is as case 0 and U is a vector of discrete logarithms of the units More precisely it gives the ideallogs with respect to bid of bnf tufu This structure is technical and only meant to be used in conjunction with bnrclassnolist or bnrdisclist 3 as 2 but include the generators in the bid 4 give only the HNF of the ideal nf nfinit x 2 1 L ideallist nf 100 177 L 1 3 1 0 0 1 A single ideal of norm 1 L 65 74 4 There are 4 ideals of norm 4 in Zli If
503. t twice will not only send you back to the gp prompt but to your shell prompt Since C d at the gp prompt exits the gp session If the break loop was started by a user interrupt Control C and not by an error inputting an empty line i e hitting the lt Return gt key at the break gt prompt resumes the temporarily interrupted computation A single empty line has no effect in case of a fatal error to avoid getting get out of the loop prematurely thereby losing valuable debugging data Any of next break return or C d will abort the computation and send you back to the gp prompt as above Break loops are useful as a debugging tool You may inspect the values of gp variables to understand why an error occurred or change gp s state in the middle of a computation increase debugging level start storing results in a log file set variables to different values hit C c type in your modifications then let the computation go on as explained above A break loop looks like this v 0 1 v k at top level v 0 1 v 4K x _ _ impossible inverse in gdiv 0 kk Break loop type break to go back to the GP prompt break gt So the standard error message is printed first The break gt at the bottom is a prompt and hitting v then lt Return gt we see break gt v 0 explaining the problem We could have typed any gp command not only the name of a variable of course Lexically scoped variables are accessible to the evalua
504. t your inputs to the residue field using nf_to_Fq then work there 188 3 6 90 nfeltnorm nf x Returns the absolute norm of x The library syntax is GEN nfnorm GEN nf GEN x 3 6 91 nfeltpow nf x k Given an element x in nf and a positive or negative integer k computes x in the number field nf The library syntax is GEN nfpow GEN nf GEN x GEN k GEN nfinv GEN nf GEN x cor respond to k 1 and GEN nfsqr GEN nf GEN x tok 2 3 6 92 nfeltpowmodpr nf x k pr Given an element x in nf an integer k and a prime ideal pr in modpr format see nfmodprinit computes z modulo the prime ideal pr The library syntax is GEN nfpowmodpr GEN nf GEN x GEN k GEN pr This function is normally useless in library mode Project your inputs to the residue field using nf_to_Fq then work there 3 6 93 nfeltreduce nf a id Given an ideal id in Hermite normal form and an element a of the number field nf finds an element r in nf such that a r belongs to the ideal and r is small The library syntax is GEN nfreduce GEN nf GEN a GEN id 3 6 94 nfeltreducemodpr nf x pr Given an element x of the number field nf and a prime ideal pr in modpr format compute a canonical representative for the class of x modulo pr The library syntax is GEN nfreducemodpr GEN nf GEN x GEN pr This function is nor mally useless in library mode Project your inputs to the residue field using nf_to_Fq then work there 3 6 95 nfelttrace nf
505. ted object in gp As for s this is used primarily as a debugging device for PARI and the format should be self explanatory The underlying GP function dbg_x is more versatile since it can be applied to other objects than history entries 2 13 26 Xy nj Switches simplify on 1 or off 0 If n is explicitly given set simplify to n 2 13 27 Switches the timer on or off 2 13 28 Prints the time taken by the latest computation Useful when you forgot to turn on the timer 58 2 14 The preferences file This file called gpre in the sequel is used to modify or extend gp default behavior in all gp sessions e g customize default values or load common user functions and aliases gp opens the gprc file and processes the commands in there before doing anything else e g creating the PARI stack If the file does not exist or cannot be read gp will proceed to the initialization phase at once eventually emitting a prompt If any explicit command line switches are given they override the values read from the preferences file 2 14 1 Syntax The syntax in the gprc file and valid in this file only is simple minded but should be sufficient for most purposes The file is read line by line as usual white space is ignored unless surrounded by quotes and the standard multiline constructions using braces or are available multiline comments between are also recognized 2 14 1 1 Preprocessor Two types of lines are first de
506. ted using repeated independent calls to bernfrac which is faster than the standard recursion in exact arithmetic It is only kept for backward compatibility it is not faster than individual calls to bernfrac its output uses a lot of memory space and coping with the index shift is awkward The library syntax is GEN bernvec long x 3 3 19 besselh1 nu x H Bessel function of index nu and argument z The library syntax is GEN hbessel1 GEN nu GEN x long prec 3 3 20 besselh2 nu x H Bessel function of index nu and argument z The library syntax is GEN hbessel2 GEN nu GEN x long prec 3 3 21 besseli nu x Bessel function of index nu and argument x If x converts to a power series the initial factor 1 2 TD v 1 is omitted since it cannot be represented in PARI when v is not integral The library syntax is GEN ibessel GEN nu GEN x long prec 91 3 3 22 besselj nu x J Bessel function of index nu and argument x If x converts to a power series the initial factor 1 2 TD v 1 is omitted since it cannot be represented in PARI when v is not integral The library syntax is GEN jbessel GEN nu GEN x long prec 3 3 23 besseljh n x J Bessel function of half integral index More precisely besseljh n x computes J 1 2 1 where n must be of type integer and x is any element of C In the present version 2 7 0 this function is not very accurate when x is small The library syntax is GEN jbesselh GEN n G
507. ten significant digits is always printed in style e e In style g non zero real numbers are printed in f format except when their decimal exponent is lt 4 in which case they are printed in e format Real zeroes of arbitrary exponent are printed in e format The precision n is the number of significant digits printed for real numbers except if n lt 0 where all the significant digits will be printed initial default 28 or 38 for 64 bit machines For more powerful formatting possibilities see printf and Strprintf The default value is g 28 and g 38 on 32 bit and 64 bit machines respectively 306 3 14 13 graphcolormap A vector of colors to be used by hi res graphing routines Its length is arbitrary but it must contain at least 3 entries the first 3 colors are used for background frame ticks and axes respectively All colors in the colormap may be freely used in plotcolor calls A color is either given as in the default by character strings or by an RGB code For valid character strings see the standard rgb txt file in X11 distributions where we restrict to lowercase letters and remove all whitespace from color names An RGB code is a vector with 3 integer entries between 0 and 255 For instance 250 235 215 and antiquewhite represent the same color RGB codes are cryptic but often easier to generate The default value is white black blue violetred red green grey sainsboro 3 14 14 grap
508. the Gram Schmidt coefficients for the final basis satisfy 1 lt 0 51 and the Lov sz s constant is 0 99 If flag 0 default assume that x has either exact integral or rational or real floating point entries The matrix is rescaled converted to integers and the behavior is then as in flag 1 If flag 1 assume that x is integral Computations involving Gram Schmidt vectors are approximate with precision varying as needed Lehmer s trick as generalized by Schnorr Adapted from Nguyen and Stehl s algorithm and Stehl s code fp111 1 3 If flag 2 x should be an integer matrix whose columns are linearly independent Returns a partially reduced basis for x using an unpublished algorithm by Peter Montgomery a basis is said to be partially reduced if v v gt v for any two distinct basis vectors v vj This is faster than flag 1 esp when one row is huge compared to the other rows knapsack style and should quickly produce relatively short vectors The resulting basis is not LLL reduced in general If LLL reduction is eventually desired avoid this partial reduction applying LLL to the partially reduced matrix is significantly slower than starting from a knapsack type lattice If flag 4 as flag 1 returning a vector K T of matrices the columns of K represent a basis of the integer kernel of x not LLL reduced in general and T is the transformation matrix such that x T is an LLL reduced Z basis
509. the group The library syntax is GEN znorder GEN x GEN o NULL Also available is GEN order GEN x 3 4 89 znprimroot n Returns a primitive root generator of Z nZ whenever this latter group is cyclic n 4 or n 2p or n p where p is an odd prime and k gt 0 If the group is not cyclic the result is undefined If n is a prime power then the smallest positive primitive root is returned This may not be true for n 2p p odd Note that this function requires factoring p 1 for p as above in order to determine the exact order of elements in Z nZ this is likely to be costly if p is large The library syntax is GEN znprimroot GEN n 3 4 90 znstar n Gives the structure of the multiplicative group Z nZ as a 3 component row vector v where v 1 p n is the order of that group v 2 is a k component row vector d of integers d i such that d i gt 1 and dli d i 1 for i gt 2 and Z nZ T Z d i Z and v 3 is a k component row vector giving generators of the image of the cyclic groups Z dli Z G znstar 40 1 16 4 2 2 Mod 17 40 Mod 21 40 Mod 11 40 G no eulerphi 40 2 16 G cyc cycle structure 3 4 2 2 G gen generators for the cyclic components 4 Mod 17 40 Mod 21 40 Mod 11 40 apply znorder G gen 129 45 4 2 2 According to the above definitions znstar 0 is 2 2 1 corresponding to Z The library syntax
510. the various libraries Type make all in the examples directory to see how they perform on the extgcd c program which is commented in the manual This should produce a statically linked binary extgcd sta standalone a dynamically linked binary extgcd dyn loads libpari at runtime and a shared library libextgcd which can be used from gp to install your new extgcd command The standalone binary should be bulletproof but the other two may fail for various reasons If when running extgcd dyn you get a message of the form DLL not found then stick to statically linked binaries or look at your system documentation to see how to indicate at linking time where the required DLLs may be found E g on Windows you will need to move libpari d11 somewhere in your PATH 322 5 3 GP scripts Several complete sample GP programs are also given in the examples directory for example Shanks s SQUFOF factoring method the Pollard rho factoring method the Lucas Lehmer primality test for Mersenne numbers and a simple general class group and fundamental unit algorithm See the file examples EXPLAIN for some explanations 5 4 The PARI Community PARI s home page at the address http pari math u bordeaux fr maintains an archive of mailing lists dedicated to PARI documentation including Frequently Asked Questions a download area and our Bug Tracking System BTS Bug reports should be submitted online to the BTS which may be accessed from the
511. then converted to a string as if you had typed it directly For instance a 1 1 b is equivalent to a2b three strings get created the middle one being the expansion of 1 1 and these are then concatenated according to the rule described above Another tricky point here assume you did not assign a value to aaa in a GP expression before Then typing aaa by itself in a string context will actually produce the correct output i e the string whose content is aaa but in a fortuitous way This aaa gets expanded to the monomial of degree one in the variable aaa which is of course printed as aaa and thus will expand to the three letters you were expecting Warning Expression involving strings are not handled in a special way even in string context the largest possible expression is evaluated hence print a 1 is incorrect since a is not an object whose first component can be extracted On the other hand print a 1 is correct two distinct argument each converted to a string and so is print a 1 since a 1 is not a valid expression only a gets expanded then 1 and the result is concatenated as explained above 2 9 2 Keywords Since there are cases where expansion is not desirable we now distinguish between Keywords and Strings String is what has been described so far Keywords are special relatives of Strings which are automatically assumed to be quoted whether you actually type in the quotes or not Thus expansion
512. tor during the break loop 50 for v 2 2 print 1 v 1 2 1 at top level for v 2 2 print 1 v kK tasa xxx _ _ impossible inverse in gdiv 0 KK Break loop type break to go back to the GP prompt break gt v 0 Even though loop indices are automatically lexically scoped and no longer exist when the break loop is run enough debugging information is retained in the bytecode to reconstruct the evaluation context Of course when the error occurs in a nested chain of user function calls lexically scoped variables are available only in the corresponding frame f x 1 x g x for i 1 10 f xti for j 5 5 g j kk at top level for j 5 5 g j kkk L Ta KK in function g for i 1 10 f x i CK ETENN xxx in function f 1 x CK x _ _ impossible inverse in gdiv 0 Break loop type break to go back to GP prompt break gt i j x the x in f s body i j O break gt dbg_up go up one frame at top level for j 5 5 g j xk masias KK in function g for i 1 10 f x i xk ERES break gt i j x the x in g s body i in the for loop 5 j 5 The following GP commands are available during a break loop to help debugging dbg_up n go up n frames as seen above dbg_down n go down n frames cancelling previous dbg_up s dbg_x t examine t as x but more flexible dbg_err returns the current error context t_ERROR The error components often provide
513. torel rnf x Let rnf be a relative number field extension L K as output by rnfinit and x be an ideal of the absolute extension L Q given by a Z basis of elements of L Returns the relative pseudo matrix in HNF giving the ideal x considered as an ideal of the relative extension L K i e as a Zg module The reason why the input does not use the customary HNF in terms of a fixed Z basis for Zz is precisely that no such basis has been explicitly specified On the other hand if you already computed an absolute nf structure Labs associated to L and m is in HNF defining an absolute ideal with respect to the Z basis Labs zk then Labs zk mis a suitable Z basis for the ideal and rnfidealabstorel rnf Labs zk m converts m to a relative ideal K nfinit y 2 1 L rnfinit K x 2 y Labs nfinit L pol m idealhnf Labs 17 x73 2 B rnfidealabstorel L Labs zk m 3 1 8 0 1 17 4 O 1 111 pseudo basis for m as Z_K module A rnfidealreltoabs L B 4 17 x72 4 x 8 x73 8xx 2 Z basis for m in Q x L pol mathnf matalgtobasis Labs A 75 17 8 4 2 0100 0010 000 1 m 76 1 The library syntax is GEN rnfidealabstorel GEN rnf GEN x 207 3 6 141 rnfidealdown rnf x Let rnf be a relative number field extension L K as output by rnfinit and zx an ideal of L given either in relative form or by a Z basis of elements of L see Section 3 6 140 This function r
514. tors first or use apply liftall L More generally components for which such lifts are meaningless e g character strings are copied verbatim liftall x 1 0 3 Mod 2 3 41 x 2 liftall x Mod y y 2 1 Mod 2 3 Mod z z72 12 y x 2 z The library syntax is GEN liftall GEN x 3 2 40 liftint x Recursively lift all components of x from Z nZ to Z and from Qp to Q as truncate t_FFELT are not lifted nor are List elements you may convert the latter to vectors first or use apply liftint L More generally components for which such lifts are meaningless e g character strings are copied verbatim liftint x 1 0 3 Mod 2 3 41 x 2 liftint x Mod y y 2 1 Mod 2 3 Mod z z72 12 Mod y y 2 1 x Mod Mod 2 z z72 y 2 1 The library syntax is GEN liftint GEN x 3 2 41 liftpol x Recursively lift all components of x which are polmods to polynomials t_FFELT are not lifted nor are List elements you may convert the latter to vectors first or use ap ply liftpol L More generally components for which such lifts are meaningless e g character strings are copied verbatim liftpol x 1 0 3 Mod 2 3 1 1 0 3 x Mod 2 3 liftpol x Mod y y72 1 Mod 2 3 Mod z z72 12 y x Mod 2 3 z The library syntax is GEN liftpol GEN x 3 2 42 norm x Algebraic norm of zx i e the product of x with its conjugate no square roots are taken or conjugates for po
515. trices use qf jacobi instead for Hermitian matrices compute A real x B imag x y matconcat A B B A and apply gfjacobi to y 236 The library syntax is GEN mateigen GEN x long flag long prec Also available is GEN eigen GEN x long prec flag 0 3 8 19 matfrobenius M flag v x Returns the Frobenius form of the square matrix M If flag 1 returns only the elementary divisors as a vector of polynomials in the variable v If flag 2 returns a two components vector F B where F is the Frobenius form and B is the basis change so that M B7 FB The library syntax is GEN matfrobenius GEN M long flag long v 1 where v isa variable number 3 8 20 mathess x Returns a matrix similar to the square matrix x which is in upper Hessenberg form zero entries below the first subdiagonal The library syntax is GEN hess GEN x 3 8 21 mathilbert n x being a long creates the Hilbert matrixof order x i e the matrix whose coefficient i j is 1 j 1 The library syntax is GEN mathilbert long n 3 8 22 mathnf M flag 0 Let R be a Euclidean ring equal to Z or to K X for some field K If M is a not necessarily square matrix with entries in R this routine finds the upper triangular Hermite normal form of M If the rank of M is equal to its number of rows this is a square matrix In general the columns of the result form a basis of the R module spanned by the columns of M Th
516. trictly less than N K Q generators are given u is the Zx module they generate e if N or more are given it is assumed that they form a Z basis of the ideal in particular that the matrix has maximal rank N This acts as mathnf since the Zx module structure is taken for granted hence not taken into account in this case idealhnf nf idealprimedec nf 2 1 4 2 0 0 o 1 0 0 0 1 idealhnf nf 1 2 2 3 3 4 5 1 0 0 o 1 0 0 0 1 Finally when K is quadratic with discriminant Dg we allow u Qfb a b c provided b 4ac Dx As usual this represents the ideal aZ 1 2 b VDk Z K nfinit x 2 60 K disc 1 60 idealhnf K qfbprimeform 60 2 4 2 2 1 o 1 176 idealhnf K Qfb 1 2 3 KK at top level idealhnf K Qfb 1 2 3 kK taaan ee ee eee idealhnf Qfb 1 2 3 has discriminant 60 in idealhnf The library syntax is GEN idealhnfO GEN nf GEN u GEN v NULL Also available is GEN idealhnf GEN nf GEN a 3 6 55 idealintersect nf A B Intersection of the two ideals A and B in the number field nf The result is given in HNF nf nfinit x 2 1 idealintersect nf 2 x 1 12 2 0 0 2 This function does not apply to general Z modules e g orders since its arguments are replaced by the ideals they generate The following script intersects Z modules A and B given by matrices of compatible dimensions with integer coefficients ZM_intersec
517. trolling the absolute error This is faster than repeatedly calling eint1 i x but less precise The library syntax is GEN veceint1 GEN x GEN n NULL long prec Also available is GEN eint1 GEN x long prec 92 3 3 31 erfc x Complementary error function analytic continuation of 2 y7 e dt incgam 1 2 2 T where the latter expression extends the function definition from real x to all complex x 0 The library syntax is GEN gerfc GEN x long prec 3 3 32 eta z flag 0 Variants of Dedekind s 7 function If flag 0 return 1 q where q depends on x in the following way e q e if x is a complex number which must then have positive imaginary part notice that the factor q 2 is missing e q x if x is a t_PADIC or can be converted to a power series which must then have positive valuation If flag is non zero x is converted to a complex number and we return the true 7 function q 4 TA q where q e277 The library syntax is GEN eta0 GEN z long flag long prec Also available is GEN trueeta GEN x long prec flag 1 3 3 33 exp 1 Exponential of x p adic arguments with positive valuation are accepted The library syntax is GEN gexp GEN x long prec For a t_PADIC x the function GEN Qp_exp GEN x is also available 3 3 34 expm1 z Return exp x 1 computed in a way that is also accurate when the real part of x is near 0 Only accept real or complex arg
518. trong pseudo prime see ispseudoprime in 2 N 1 At_VEC N a b is also allowed with a lt bin which case a pseudo prime a lt p lt bis returned if no prime exists in the interval the function will run into an infinite loop If the upper bound is less than 264 the pseudo prime returned is a proven prime The library syntax is GEN randomprime GEN N NULL 125 3 4 79 removeprimes x Removes the primes listed in x from the prime number table In particular removeprimes addprimes empties the extra prime table x can also be a single integer List the current extra primes if x is omitted The library syntax is GEN removeprimes GEN x NULL 3 4 80 sigma z k 1 Sum of the kt powers of the positive divisors of x 2 and k must be of type integer The library syntax is GEN sumdivk GEN x long k Also available is GEN sumdiv GEN n for k 1 3 4 81 sqrtint z Returns the integer square root of x i e the largest integer y such that y lt z where x a non negative integer N 120938191237 sqrtint N 1 347761 sqrt N 2 347761 68741970412747602130964414095216 The library syntax is GEN sqrtint GEN x 3 4 82 sqrtnint z n Returns the integer n th root of x i e the largest integer y such that y lt x where x is a non negative integer N 120938191237 sqrtnint N 5 1 164 N7 1 5 2 164 63140849829660842958614676939677391 The special case n 2 is sqrtint
519. try is described by a function f i 7 Mat 1 ALE 1 matrix 2 2 i j 2 i j 12 3 4 5 6 Let the variable M contain a matrix and let i j k l denote four integers e M i j refers to its i j th entry you can assign any result to M i j e M i refers to its i th row you can assign a t_VEC of the right dimension to M i e ML j refers to its j th column you can assign a t_COL of the right dimension to M j But M i is meaningless and triggers an error The range 2 j and caret c notations are available as for vectors you can not assign to any of these eM i j k 1 7 lt j k lt l returns the submatrix built from the rows to j and columns k tol of M assign to Mli j j 1 e MLi j returns the submatrix built from the rows to j of M e M i j returns the submatrix built from the columns i to j of M eM i j k i lt j returns the submatrix built from the rows 7 to j and column k removed e M k returns the submatrix with row k removed e M 7k returns the submatrix with column k removed Finally e M i j k returns the t_COL built from the k th column entries 7 to 7 e M i k returns the t_COL built from the k th column entry i removed e M x i j returns the t_VEC built from the k th row entries i to j e M x i returns the t_VEC built from the k th row entry i removed M 1 2 3 4 5 6 7 8 9 M 1 2 2 3 f 2 3 5 6 M 1 2
520. ts If flag 1 uses the determinant of Sylvester s matrix instead this should always be slower than the default The library syntax is GEN polresultantO GEN x GEN y long v 1 long flag where v is a variable number 222 3 7 28 polresultantext A B v Finds polynomials U and V such that AxU BxV R where R is the resultant of U and V with respect to the main variables of A and B if v is omitted and with respect to v otherwise Returns the row vector U V R The algorithm used subresultant assumes that the base ring is a domain A x y B xty 72 U V R polresultantext A B 12 y x 2 y72 y 2 y 4 A U B V 13 y 4 U V R polresultantext A B y 14 2 x72 y x x72 x 4 A U Bx V 45 x 4 The library syntax is GEN polresultantextO GEN A GEN B long v 1 where visa variable number Also available is GEN polresultantext GEN x GEN y 3 7 29 polroots x Complex roots of the polynomial pol given as a column vector where each root is repeated according to its multiplicity The precision is given as for transcendental functions in GP it is kept in the variable realprecision and is transparent to the user but it must be explicitly given as a second argument in library mode The algorithm used is a modification of A Sch nhage s root finding algorithm due to and originally implemented by X Gourdon Barring bugs it is guaranteed to converge and to give the roots to t
521. ts are optional The files argument is a list of files written in the GP scripting language which will be loaded on startup There can be any number of arguments of the form D key val setting some internal parameters of gp or defaults each sets the default key to the value val See Section 2 12 below for a list and explanation of all defaults These defaults can be changed by adding parameters to the input line as above or interactively during a gp session or in a preferences file also known as gprc If a preferences file to be discussed in Section 2 14 is found gp then reads it and executes the commands it contains This provides an easy way to customize gp The files argument is processed right after the gprc A copyright banner then appears which includes the version number and a lot of useful tech nical information After the copyright the computer writes the top level help information some initial defaults and then waits after printing its prompt which is by default Whether ex tended on line help and line editing are available or not is indicated in this gp banner between the version number and the copyright message Consider investigating the matter with the person who installed gp if they are not Do this as well if there is no mention of the GMP kernel 13 2 1 2 Getting help To get help type a and hit return A menu appears describing the main categories of available functions and how to get more detailed
522. tten tend to be shorter and clearer than C programs and are certainly easier to write maintain or debug You don t need to deal with memory management garbage collection pointers declarations and so on Because of their intrinsic simplicity they are more robust as well They are unfortunately somewhat slower Thus their use will remain complementary it is suggested that you test and debug your algorithms using scripts before actually coding them in C if speed is paramount The GP2C compiler often eases this part The install command see Section 3 12 21 efficiently imports foreign functions for use under gp which can of course be written using other libraries than PARI Thus you may code only critical parts of your program in C and still maintain most of the program as a GP script We are aware of three PARI related Free Software packages to embed PARI in other languages We neither endorse nor support any of them but you may want to give them a try if you are familiar with the languages they are based on The first is William Stein s Python based SAGE system The second is the Math Pari Perl module see any CPAN mirror written by Ilya Zakharevich Finally Michael Stoll has integrated PARI into CLISP which is a Common Lisp implementation by Bruno Haible Marcus Daniels and others this interface has been updated for pari 2 by Sam Steingold These provide interfaces to gp functions for use in python perl or Lisp programs respect
523. typically involve a lot of trailing zeroes somewhat technical Note As we saw in Section 2 9 the second argument to default is subject to string context expansion which means you can use run time values In other words something like a 3 default logfile file a log logs the output in file3 1log Some special defaults corresponding to file names and prompts expand further the resulting value at the time they are set Two kinds of expansions may be performed e time expansion the string is sent through the library function strftime This means that char combinations have a special meaning usually related to the time and date For instance 4H hour 24 hour clock and M minute 00 59 on a Unix system you can try man str time at your shell prompt to get a complete list This is applied to prompt psfile and logfile For instance default prompt H M will prepend the time of day in the form hh mm to gp s usual prompt e environment expansion When the string contains a sequence of the form SOMEVAR e g HOME the environment is searched and if SOMEVAR is defined the sequence is replaced by the corresponding value Also the symbol has the same meaning as in many shells by itself stands for your home directory and user is expanded to user s home directory This is applied to all file names Available defaults are described in the reference guide Section 3 14 2 13 Simple metacommands S
524. u can add more signs for extended functionality keyword yields the function description as it stands in this manual usually in Chapter 2 or 3 If you re not satisfied with the default chapter chosen you can impose a given chapter by ending the keyword with followed by the chapter number e g Hello 2 will look in Chapter 2 for section heading Hello which doesn t exist by the way All operators e g amp amp etc are accepted by this extended help as well as a few other keywords describing key gp concepts e g readline the line editor integer nf number field as used in most algebraic number theory computations e11 elliptic curves etc In case of conflicts between function and default names e g log simplify the function has higher priority To get the default help use default log default simplify 7 pattern produces a list of sections in Chapter 3 of the manual related to your query As before if pattern ends by followed by a chapter number that chapter is searched instead you also have the option to append a simple without a chapter number to browse through the whole manual If your query contains dangerous characters e g or blanks it is advisable to enclose it within double quotes as for GP strings e g elliptic curve Note that extended help is much more powerful than the short help since it knows about operators as well you can type or amp amp wh
525. ubsequent computations are done using this new polynomial In particular the first component of the result is the modified polynomial If flag 3 apply polredbest as in case 2 but outputs nf Mod a P where nf is as before and Mod a P Mod x pol gives the change of variables This is implicit when pol is not monic first a linear change of variables is performed to get a monic polynomial then polredbest The library syntax is GEN nfinitO GEN pol long flag long prec Also available are GEN nfinit GEN x long prec flag 0 GEN nfinitred GEN x long prec flag 2 GEN nfinitred2 GEN x long prec flag 3 Instead of the above hardcoded numerical flags in nfinito one should rather use 194 GEN nfinitall GEN x long flag long prec where flag is an or ed combination of e nf_RED find a simpler defining polynomial e nf_ORIG if nf_RED set also return the change of variable e nf_ROUND2 Deprecated Slow down the routine by using an obsolete normalization algorithm do not use this one e nf_PARTIALFACT Deprecated Lazy factorization of the polynomial discriminant Result is conditional unless nfcertify can certify it 3 6 106 nfisideal nf x Returns 1 if x is an ideal in the number field nf 0 otherwise The library syntax is long isideal GEN nf GEN x 3 6 107 nfisincl x y Tests whether the number field K defined by the polynomial x is conjugate to a subfield of the field L defined by y where x and y mu
526. uch that p D gt a1 8 the second the corresponding elements a as in Proposition 8 3 1 in GTM 138 and the third the output of isprime p 1 115 The algorithm fails if one of the pseudo prime factors is not prime which is exceedingly unlikely and well worth a bug report Note that if you monitor isprime at a high enough debug level you may see warnings about untested integers being declared primes This is normal we ask for partial factorisations sufficient to prove primality if the unfactored part is not too large and factor warns us that the cofactor hasn t been tested It may or may not be tested later and may or may not be prime This does not affect the validity of the whole isprime procedure If flag 2 use APRCL The library syntax is GEN gisprime GEN x long flag 3 4 42 isprimepower z amp n If z p is a prime power p prime k gt 0 return k else return 0 If a second argument amp n is given and x is indeed the k th power of a prime p sets n to p The library syntax is long isprimepower GEN x GEN n NULL 3 4 43 ispseudoprime z flag True 1 if x is a strong pseudo prime see below false 0 otherwise If this function returns false x is not prime if on the other hand it returns true it is only highly likely that x is a prime number Use isprime which is of course much slower to prove that x is indeed prime The function accepts vector matrices arguments and is then applied component
527. ult binary use make dobench which starts the bench immediately If a BUG message shows up something went wrong The testing utility directs you to files containing the differences between the test output and the expected results Have a look and decide for yourself if something is amiss If it looks like a bug in the Pari system we would appreciate a report see the last section 317 2 5 Cross compiling When cross compiling you can set the environment variable RUNTEST to a program that is able to run the target binaries e g an emulator It will be used for both the Configure tests and make bench 3 Troubleshooting and fine tuning In case the default Configure run fails miserably try Configure a interactive mode and answer all the questions there are about 30 of them and default answers are provided If you accept all default answers Configure will fail just the same so be wary In any case we would appreciate a bug report see the last section 3 1 Installation directories The precise default destinations are as follows the gp binary the scripts gphelp and tex2mail go to prefix bin The pari library goes to prefix lib and include files to prefix include pari Other system dependent data go to prefix lib pari Architecture independent files go to various subdirectories of share_prefix which defaults to prefix share and can be specified via the share prefix argument Man pages go into share_prefix ma
528. umber carrying in its internal representation its own length or precision with the following mild restrictions given for 32 bit machines the restrictions for 64 bit machines being so weak as to be considered nonexistent integers must be in absolute value less than 2536870815 i e roughly 161614219 decimal digits The precision of real numbers is also at most 161614219 significant decimal digits and the binary exponent must be in absolute value less than 2 resp 261 on 32 bit resp 64 bit machines Integers and real numbers are non recursive types 1 3 2 Intmods rational numbers p adic numbers polmods and rational functions These are recursive but in a restricted way For intmods or polmods there are two components the modulus which must be of type integer resp polynomial and the representative number resp polynomial For rational numbers or rational functions there are also only two components the numerator and the denominator which must both be of type integer resp polynomial Finally p adic numbers have three components the prime p the modulus p and an ap proximation to the p adic number Here Z is considered as the projective limit lim Z p Z via its finite quotients and Qp as its field of fractions Like real numbers the codewords contain an exponent giving the p adic valuation of the number and also the information on the precision of the number which is redundant with p but is included fo
529. umber is not supported to factor in Q use t_PADIC coefficients not t_INTMOD modulo p T x72 1 factor T over Q factor T Mod 1 3 over F_3 factor T ffgen ffinit 3 2 t 0 over F_ 3 2 factor T Mod Mod 1 3 t72 t 2 over F_ 3 2 again factor T 1 0 376 over Q_3 precision 6 factor T 1 over R current precision factor T 1 0 I over C factor T Mod 1 y 3 2 over Q 2 1 3 In most cases it is clearer and simpler to call an explicit variant than to rely on the generic factor function and the above detection mechanism factormod T 3 over F_3 factorff T 3 t 2 t 2 over F_1372 factorpadic T 3 6 over Q_3 precision 6 nffactor y 3 2 T over Q 27 1 3 polroots T over C Note that factorization of polynomials is done up to multiplication by a constant In particular the factors of rational polynomials will have integer coefficients and the content of a polynomial or rational function is discarded and not included in the factorization If needed you can always ask for the content explicitly factor t 2 5 2 t 1 1 2xt 1 1 t 2 1 content t 2 5 2 t 1 42 1 2 The irreducible factors are sorted by increasing degree See also nffactor The library syntax is GEN gp_factorO GEN x GEN lim NULL This function should only be used by the gp interface Use directly GEN factor GEN x or GEN boundfact GEN
530. uments A naive direct computation would suffer from catastrophic cancellation PARI s direct computation of exp x alleviates this well known problem at the expense of computing exp to a higher accuracy when x is small Using expm1 is recommanded instead default realprecision 10000 x 1e 100 a expmi x time 4 ms b exp x 1 time 28 ms default realprecision 10040 x 1e 100 c expmi x reference point abs a c c relative error in expm1 x 47 0 E 10017 abs b c c relative error in exp x 1 48 1 7907031188259675794 E 9919 As the example above shows when x is near 0 expm1 is both faster and more accurate than exp x 1 The library syntax is GEN gexpm1 GEN x long prec 93 3 3 35 gamma s For s a complex number evaluates Euler s gamma function T s f tt exp t dt Error if s is a non positive integer where I has a pole For s a t_PADIC evaluates the Morita gamma function at s that is the unique continuous p adic function on the p adic integers extending T k S lt p J Where the prime means that p does not divide 7 gamma 1 4 0 5710 1 1 4 5 34574 5 6 5 7 4x5 9 0 5710 algdep 4 12 x74 4 x72 5 The library syntax is GEN ggamma GEN s long prec For a t_PADIC zv the function GEN Qp_gamma GEN x is also available 3 3 36 gammah x Gamma function evaluated at the argument x 1 2 The library syntax is GEN ggammah GEN
531. unction qfbprimeform which directly creates a prime form of given discriminant 2 3 14 Row and column vectors t_VEC and t_COL To enter a row vector type the com ponents separated by commas and enclosed between brackets and e g 1 2 3 To enter a column vector type the vector horizontally and add a tilde to transpose yields the empty row vector The function Vec can be used to transform any object into a vector see Chap ter 3 The construction i j where i lt j are two integers returns the vector fi i 1 j 1 j 1 2 3 Ji 1 2 3 2 122 8 12 2 1 0 1 2 3 Let the variable v contain a row or column vector e vim refers to its m th entry you can assign any value to vim i e write something like v m expr e v i jl where i lt j returns the vector slice containing elements v i u j you can not assign a result to v i j e v i returns the vector whose i th entry has been removed you can not assign a result to yiil In the last two constructions v i j and v i i and j are allowed to be negative integers in which case we start counting from the end of the vector e g 1 is the index of the last element v 1 2 3 4 v 2 4 42 2 3 4 v 73 43 1 2 4 v 1 1a 23 w 23 11 4 2 3 4 23 Remark vector is the standard constructor for row vectors whose i th entry is given by a simpl
532. unity congruent to x p modulo p The library syntax is GEN teich GEN x 3 3 53 theta q z Jacobi sine theta function 01 2 0 2 Y ge sin 2n 1 2 n gt 0 The library syntax is GEN theta GEN q GEN z long prec 3 3 54 thetanullk q k k th derivative at z 0 of theta q z The library syntax is GEN thetanullk GEN q long k long prec GEN vecthetanullk GEN q long k long prec returns the vector of all 2 9 0 for all odd i 1 3 2k 1 GEN vecthetanullk_tau GEN tau long k long prec returns vec thetanullk_tau at q exp 2 1tau 3 3 55 weber z flag 0 One of Weber s three f functions If flag 0 returns f a exp im 24 n x 1 2 n x such that j f 16 f where j is the elliptic j invariant see the function e11j If flag 1 returns f z 2 2 n x such that j f7 16 f7 Finally if flag 2 returns falz V2n 2x n x such that j f2 16 f3 Note the identities f8 f f and f fifo V2 The library syntax is GEN weber0 GEN x long flag long prec Also available are GEN weberf GEN x long prec GEN weberf1 GEN x long prec and GEN weberf2 GEN x long prec 3 3 56 zeta s For s a complex number Riemann s zeta function s gt n computed using the Euler Maclaurin summation formula except when s is of type integer in which case it is computed using Bernoulli numbers for s lt 0 or s gt 0 and even and using modular for
533. unt the new variable s value one must force a new evaluation using the function eval see Section 3 7 5 It is preferable to leave alone your polynomial variables never assigning values to them and to use subst and its more powerful variants rather than eval You will avoid the following kind of problems 7p t 2 1 subst p t 2 hi 5 t 2 subst p t 3 t is no longer free it evaluates to 2 kk at top level subst p t 3 4K ALE variable name expected subst p t 3 OK 43 10 A statement like x x in effect restores x as a free variable 2 5 3 Variable priorities multivariate objects A multivariate polynomial in PARI is just a polynomial in one variable whose coefficients are themselves polynomials arbitrary but for the fact that they do not involve the main variable PARI currently has no sparse representation for polynomials listing only non zero monomials All computations are then done formally on the coefficients as if the polynomial was univariate This is not symmetrical So if I enter x y in a clean session what happens This is understood as y 0 y 2 Zly x but how do we know that x is more important than y Why not y xx y which is the same mathematical entity after all The answer is that variables are ordered implicitly by the interpreter when a new identifier e g x or y as above is input the corresponding variable is registered as having
534. uracy if you are looking for true rational functions presumably approximated to sufficient accuracy you should first try that option Otherwise B must be a non negative real impose O lt degree b lt B e If x is a t_RFRAC or t_SER this function uses continued fractions bestapprPade 1 x711 1 x 0 x711 11 1 x 1 bestapprPade 1 1 x 0 x710 x73 2 x73 1 1 2 1 x 1 2 e If x is a t_POLMOD modulo N or a t_SER of precision N t this function performs rational modular reconstruction modulo N The routine then returns the unique rational function a b in coprime polynomials with degree b lt B which is congruent to x modulo N Omitting B amounts to choosing it of the order of N 2 If rational reconstruction is not possible no suitable a b exists returns bestapprPade Mod 1 x x72 x 3 x 4 x74 2 1 x 1 Mod 1 x 4 2 A2 Mod x 3 x 2 x 3 x 4 2 bestapprPade Mod 1 x x7 2 x 3 x 5 x79 12 7 bestapprPade Mod 1 x x7 2 x 3 x 5 x710 13 2xx74 x73 x 1 x75 x73 x72 1 The function applies recursively to components of complex objects polynomials vectors If rational reconstruction fails for even a single entry return The library syntax is GEN bestapprPade GEN x long B 3 4 6 bezout z y Deprecated alias for gedext The library syntax is GEN gcdextO GEN x GEN y 3 4 7 bigomega x Number of prime divisors of the integer x
535. ut rather evaluates sequences of expressions Of course it is by no means a true LISP and has been strongly influenced by C and Perl since then 2 2 1 Introduction User interaction with a gp session proceeds as follows First one types a sequence of characters at the gp prompt see Section 2 15 for a description of the line editor When you hit the lt Return gt key gp gets your input evaluates it then prints the result and assigns it to an history array More precisely the input is case sensitive and outside of character strings blanks are com pletely ignored Inputs are either metacommands or sequences of expressions Metacommands are shortcuts designed to alter gp s internal state such as the working precision or general verbosity level we shall describe them in Section 2 13 and ignore them for the time being The evaluation of a sequence of instructions proceeds in two phases your input is first digested byte compiled to a bytecode suitable for fast evaluation in particular loop bodies are compiled only once but a priori evaluated many times then the bytecode is evaluated An expression is formed by combining constants variables operator symbols functions and control statements It is evaluated using the conventions about operator priorities and left to right associativity An expression always has a value which can be any PARI object fi 1 11 2 XX an ordinary integer TX 12 x a polynomial of degree 1
536. w call your function as in 1 2 supplying values for the formal variables The number of parameters actually supplied may be less than the number of formal variables in the function definition An uninitialized formal variable is given an implicit default value of the integer 0 i e after the definition f x y you may call 1 2 supplying values for the two formal parameters or for example f 2 equivalent to 2 0 39 f f 0 0 f 3 f 0 3 Empty argument trick This implicit default value of 0 is actually deprecated and setting default strictargs 1 allows to disable it see Section 3 14 39 The recommended practice is to explicitly set a default value in the function definition you can append ezpr to a formal parameter to give that variable a default value The expression gets evaluated the moment the function is called and may involve the preceding function parameters a default value for x may involve x for j lt i For instance after f x 1 y 2 z y 1 typing in 3 4 would give you 3 4 5 In the rare case when you want to set some far away argument and leave the defaults in between as they stand use the empty argument trick 6 1 would yield 6 2 1 Of course f by itself yields 1 2 3 as was to be expected In short the argument list is filled with user supplied values in order A comma or closing parenthesis where a value should have been signal
537. wise If flag 0 checks whether x is a Baillie Pomerance Selfridge Wagstaff pseudo prime strong Rabin Miller pseudo prime for base 2 followed by strong Lucas test for the sequence P 1 P smallest positive integer such that P 4 is not a square mod zx There are no known composite numbers passing this test although it is expected that infinitely many such numbers exist In particular all composites lt 264 are correctly detected checked using http www cecm sfu ca Pseudoprimes index 2 to 64 html If flag gt O checks whether x is a strong Miller Rabin pseudo prime for flag randomly chosen bases with end matching to catch square roots of 1 The library syntax is GEN gispseudoprime GEN x long flag 3 4 44 issquare z amp n True 1 if x is a square false 0 if not What being a square means depends on the type of x all t_COMPLEX are squares as well as all non negative t_REAL for exact types such as t_INT t_FRAC and t_INTMOD squares are numbers of the form s with s in Z Q and Z NZ respectively issquare 3 as an integer hi 0 issquare 3 as a real number 42 1 issquare Mod 7 8 in Z 8Z 43 0 issquare 5 0 1374 Ml in Q_13 14 0 If n is given a square root of x is put into n issquare 4 amp n 1 1 n 116 12 2 For polynomials either we detect that the characteristic is 2 and check directly odd and even power monomials or we assume that
538. with compressed files the online help system can handle it 4 1 Static binaries and libraries By default if a dynamic library libpari so can be built the gp binary we install is gp dyn pointing to libpari so On the other hand we can build a gp binary into which the libpari is statically linked the library code is copied into the binary that binary is not independent of the machine it was compiled on and may still refer to other dynamic libraries than libpari You may want to compile your own programs in the same way using the static libpari a instead of libpari so By default this static library libpari a is not created If you want it as well use the target make install lib sta You can install a statically linked gp with the target make install bin sta As a rule programs linked statically with libpari a may be slightly faster about 5 gain possibly up to 20 when using pthreads but use more disk space and take more time to compile They are also harder to upgrade you will have to recompile them all instead of just installing the new dynamic library On the other hand there is no risk of breaking them by installing a new pari library 4 2 Extra packages The following optional packages endow PARI with some extra capabilities e elldata This package contains the elliptic curves in John Cremona s database It is needed by the functions ellidentify ellsearch forell and can be used by ellinit to initialize a curve given by
539. with respect to the variable v Beware that divrem x y 2 is in general not the same as x y no GP operator corresponds to it divrem 1 2 3 2 41 1 2 2 1 2 3 12 2 divrem Mod 2 9 3 2 KK at top level divrem Mod 2 9 3 2 RRR Fa kkk forbidden division t_INTMOD t_INT Mod 2 9 6 3 Mod 2 3 The library syntax is GEN divrem GEN x GEN y long v 1 where v is a variable num ber Also available is GEN gdiventres GEN x GEN y when v is not needed 68 3 1 12 lex x y Gives the result of a lexicographic comparison between x and y as 1 0 or 1 This is to be interpreted in quite a wide sense It is admissible to compare objects of different types scalars vectors matrices provided the scalars can be compared as well as vectors matrices of different lengths The comparison is recursive In case all components are equal up to the smallest length of the operands the more complex is considered to be larger More precisely the longest is the largest when lengths are equal we have matrix gt vector gt scalar For example lex 1 3 1 2 5 1 1 lex 1 3 1 3 1 42 1 lex 11 1 13 1 lex 1 1 74 0 The library syntax is GEN lexcmp GEN x GEN y 3 1 13 max x y Creates the maximum of x and y when they can be compared The library syntax is GEN gmax GEN x GEN y 3 1 14 min x y Creates the minimum of x and y when they can be compare
540. x p a Factors the polynomial x in the field F defined by the irreducible polynomial a over F The coefficients of x must be operation compatible with Z pZ The result is a two column matrix the first column contains the irreducible factors of x and the second their exponents If all the coefficients of x are in F a much faster algorithm is applied using the computation of isomorphisms between finite fields Either a or p can omitted in which case both are ignored if x has t_FFELT coefficients the function then becomes identical to factor factortf x 2 1 5 y72 3 MAN over F_5ly1 y72 3 F_25 1 Mod Mod 1 5 Mod 1 5 ry 2 Mod 3 5 x Mod Mod 2 5 Mod 1 5 y72 Mod 3 5 1 Mod Mod 1 5 Mod 1 5 y72 Mod 3 5 x Mod Mod 3 5 Mod 1 5 y72 Mod 3 5 1 t ffgen y 2 Mod 3 5 t A a generator for F_25 as a t_FFELT factorff x 2 1 not enough information to determine the base field kk at top level factorff x 2 1 109 RRR Teso x factorff incorrect type in factorff factorff x 2 t70 make sure a coeff is a t_FFELT 3 x 2 1 x 3 1 factorff x 2 t 1 11 x 2xt 1 1 x 3x t 4 1 Notice that the second syntax is easier to use and much more readable The library syntax is GEN factorff GEN x GEN p NULL GEN a NULL 3 4 24 factorial x Factorial of x The expression x gives a result which is an integer while factoria
541. y flag 4 will be fastest When M has maximal rank then H mathnfmod M matdetint M 237 will be even faster You can then recover U as M7 H M matrix 3 4 i j random 5 5 1 02 3 0 5 3 5 5 43 5 4 H U mathnf M 1 U 3 1 0 1 0 05 32 03 11 10 00 H hd 19 9 7 09 1 00 1 M U 6 0 19 9 7 o 0 9 1 lo 00 1 For convenience M is allowed to be a t_VEC which is then automatically converted to a t_MAT as per the Mat function For instance to solve the generalized extended gcd problem one may use v 116085838 181081878 314252913 10346840 H U mathnf v 1 U 2 103 603 15 88 146 13 1208 352 58 220 678 167 362 144 381 101 vxU 13 O O O 1 This also allows to input a matrix as a t_VEC of t_COLs of the same length which Mat would concatenate to the t_MAT having those columns v 1 0 4 3 3 4 0 4 5 mathnf v 41 47 32 12 o 1 0 238 Oo o 4 The library syntax is GEN mathnfO GEN M long flag Also available are GEN hnf GEN M flag 0 and GEN hnfall GEN M flag 1 To reduce huge relation matrices sparse with small entries say dimension 400 or more you can use the pair hnfspec hnfadd Since this is quite technical and the calling interface may change they are not documented yet Look at the code in basemath hnf_snf c 3 8 23 mathnfmod z d If x i
542. y increasing size To avoid storing all divisors possibly using a lot of memory the following much slower routine loops over the divisors using essentially constant space FORDIV N 1 my P E P factor N E P 2 P P 1 forvec v vector E i 0 E i X factorback P v ANE nee for i 1 10 5 FORDIV i time 3 445 ms for i 1 10 5 fordiv i d time 490 ms 3 11 10 forell E a b seq Evaluates seq where the formal variable E name M G ranges through all elliptic curves of conductors from a to b In this notation name is the curve name in Cremona s elliptic curve database M is the minimal model G is a Z basis of the free part of the Mordell Weil group E Q forell E 1 500 my name M G E A if G gt 1 print name 389al 433a1 446d1 The elldata database must be installed and contain data for the specified conductors The library syntax is forell void data long cal1 void GEN long a long b 3 11 11 forpart X k seq a k n k Evaluate seg over the partitions X x1 1 of the integer k i e increasing sequences 11 lt 2 lt Lp of sum z1 8 n k By convention 0 admits only the empty partition and negative numbers have no partitions A partition is given by a t_VECSMALL where parts are sorted in nondecreasing order forpart X 3 print X Vecsmall 3 Vecsmal1 1 2 Vecsmal1 1 1 1 Optional parameters n a
543. y single variable objects It is awkward to handle many variables at once since PARI s implementation cannot handle multivariate error terms like O x y It can handle the polyno mial O y x x which is a very different thing see below The basic assumption in our model is that if variable x has higher priority than y then y does not depend on x setting y to a function of x after some computations with bivariate power series does not make sense a priori This is because implicit constants in expressions like O x depend on y whereas in O y they can not depend on x For instance O x y 1 0 x 0 y x 2 OCy x Here is a more involved example A 1 x72 1 0 x B 1 x 1 0 x73 subst z A z B 42 x 3 x 2 x 1 1 OCx 7 Be A 13 x 3 x7 2 x 1 0 1 zak A L4 z x 2 z 0 x The discrepancy between 2 and 3 is surprising Why does 2 contain a spurious constant term which cannot be deduced from the input Well we ignored the rule that forbids to substitute an expression involving high priority variables to a low priority variable The result 4 is correct according to our rules since the implicit constant in O x may depend on z It is obviously wrong if z is allowed to have negative valuation in x Of course the correct error term should be O x2 but this is not possible in PARI 2 6 Variables and Scope This section is rather technical and strives to explain p
544. y syntax is GEN ellchangecurve GEN E GEN v 3 5 11 ellchangepoint x v Changes the coordinates of the point or vector of points x using the vector v u r s tJ i e if x and y are the new coordinates then x u z r y uy su x t see also ellchangecurve EO ellinit 1 1 PO 0 1 v 1 2 3 4 E ellchangecurve E0 v P ellchangepoint P0 v 43 2 3 ellisoncurve E P 14 1 ellchangepointinv P v 5 0 1 The library syntax is GEN ellchangepoint GEN x GEN v The reciprocal function GEN ellchangepointinv GEN x GEN ch inverts the coordinate change 135 3 5 12 ellchangepointinv x v Changes the coordinates of the point or vector of points x using the inverse of the isomorphism associated to v u r s t i e if x and y are the old coordinates then x u22 r y uy su z t inverse of ellchangepoint EO ellinit 1 1 1 PO 0 1 v 1 2 3 4 E ellchangecurve E0 v P ellchangepoint P0 v 13 2 3 ellisoncurve E P 44 1 ellchangepointinv P v 5 0 11 we get back PO The library syntax is GEN ellchangepointinv GEN x GEN v 3 5 13 ellconvertname name Converts an elliptic curve name as found in the elldata database from a string to a triplet conductor isogeny class index It will also convert a triplet back to a curve name Examples ellconvertname 123b1 1 123 1 1 ellconvertname 2 123b1 Th
545. y72 Mod 3 4 y Mod 2 4 This is not guaranteed to work either since it must invert pivots See Section 3 8 The library syntax is GEN gpow GEN x GEN n long prec for xn 67 3 1 10 cmp x y Gives the result of a comparison between arbitrary objects x and y as 1 0 or 1 The underlying order relation is transitive the function returns 0 if and only if y and its restriction to integers coincides with the customary one Besides that it has no useful mathematical meaning In case all components are equal up to the smallest length of the operands the more complex is considered to be larger More precisely the longest is the largest when lengths are equal we have matrix gt vector gt scalar For example cmp 1 2 1 1 cmp 2 1 12 1 cmp 1 1 0 note that 1 1 0 but 1 1 0 is false 43 1 cmp x Pi 14 1 This function is mostly useful to handle sorted lists or vectors of arbitrary objets For instance if v is a vector the construction vecsort v cmp is equivalent to Set v The library syntax is GEN cmp_universal GEN x GEN y 3 1 11 divrem z y v Creates a column vector with two components the first being the Euclidean quotient x y the second the Euclidean remainder x a y y of the division of x by y This avoids the need to do two divisions if one needs both the quotient and the remainder If v is present and x y are multivariate polynomials divide
546. yping a whose content is merely the one letter string a You can concatenate two strings using the concat function If either argument is a string the other is automatically converted to a string if necessary it will be evaluated first concat ex 1 1 1 ex2 a 2 b ex concat b a 42 ex2 concat a b 13 2ex Some functions expect strings for some of their arguments print would be an obvious example Str is a less obvious but useful one see the end of this section for a complete list While typing in such an argument you will be said to be in string context The rest of this section is devoted to special syntactical tricks which can be used with such arguments and only here you will get an error message if you try these outside of string context e Writing two strings alongside one another will just concatenate them producing a longer string Thus it is equivalent to type in a b or a b A little tricky point in the first expression the first whitespace is enclosed between quotes and so is part of a string while the second before the b is completely optional and gp actually suppresses it as it would with any number of whitespace characters at this point i e outside of any string e If you insert any expression when a string is expected it gets expanded it is evaluated as a standard GP expression and the final result as would have been printed if you had typed it by itself is
547. ysterious because gp cannot guess what you were trying to do and the error may occur once gp has been sidetracked The first error is straightforward factor has one mandatory argument which is missing The other two are simple typos involving an ill formed addition 1 missing its second operand The error messages differ because the parsing context is slightly different in the first case we reach the end of input end while still expecting a token and in the second one we received an unexpected token the comma Here is a more complicated one factor x kk syntax error unexpected end expecting gt or or factor x OK 3a The error is a missing parenthesis but from gp s point of view you might as well have intended to give further arguments to factor this is possible and useful see the description of the function In fact gp expected either a closing parenthesis or a second argument separated from the first by a comma And this is essentially what the error message says we reached the end of the input end while expecting a ora Actually a third possibility is mentioned in the error message gt which could never be valid in the above context but a subexpression like x gt sin x defining an inline closure would be valid and the parser is not clever enough to rule that out so we get the same message as in x eK syntax error unexpected end expecting gt or or
548. z is a rational number or function If x is an integer or a polynomial it is treated as a rational number or function respectively and the result is equal to 1 For polynomials you probably want to use denominator content x instead As for modular objects t_INTMOD and t_PADIC have denominator 1 and the denominator of a t_POLMOD is the denominator of its minimal degree polynomial representative If x is a recursive structure for instance a vector or matrix the lcm of the denominators of its components a common denominator is computed This also applies for t_COMPLEXs and t_QUADs 79 Warning Multivariate objects are created according to variable priorities with possibly surprising side effects x y is a polynomial but y x is a rational function See Section 2 5 3 The library syntax is GEN denom GEN x 3 2 32 digits x b 10 Outputs the vector of the digits of x in base b where x and b are integers The library syntax is GEN digits GEN x GEN b NULL 3 2 33 floor x Floor of x When z is in R the result is the largest integer smaller than or equal to x Applied to a rational function floor x returns the Euclidean quotient of the numerator by the denominator The library syntax is GEN gfloor GEN x 3 2 34 frac x Fractional part of x Identical to x floor x If x is real the result is in 0 1 The library syntax is GEN gfrac GEN x 3 2 35 hammingweight x If x is a t_INT return the binary H

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