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1. BRHM Area of cyclic quadrilateral JM Baillard THV Tetrahedron Volume JM Baillard This is the small set of geometry functions in the SandMath just a token to glimpse the subject not a comprehensive implementation The VECTOR ANALYSIS module contains many more as well as a full featured 3D Vector Calculator see overlay below It is a 4k module that can be used independently from the SandMath but sure it is a powerful complement for these specific subjects Distance between two points PP2 The Euclidean distance between two points p and q is the length of the line segment connecting them In the Euclidean plane if p p1 p2 and q q1 q2 then the distance is given by d p q y p 91 P2 q2 expects the coordinates of the two points stored in the stack y1 x1 y2 x2 in T Z Y and X or vice versa The distance will be placed in X upon completion Example Calculate the distance between the points a 3 5 and b 6 2 from the figure below Y Type 5 ENTERAS 3 ENTERAS 2 ENTERS 6 ENTER EF PP2 gt 11 40175425 B 3 9 Note A similar function exists in the 41Z module ZWDIST which basically calculates the same thing albeit done in a complex number context A 6 2 3D Vector Modulus Magnitude VMOD With the 3 coordinates stored in the stack registers Z Y and X VMOD calculates the vector modulus The result is returned in X but the stack is ot
2. eS oe H S HMKEYS assigns to the SHIFT key for convenience HSIN HASIN HCOS HACOS HTAN HATAN 1 761594156 1 001 176744862 1 001000000 1 001000000 1 001000000 7 0 010000000 0 009999958 0 009999667 0 010000000 0 0 0 1 0 01 0 01000016 0 0001 0 000100000 0 000100000 1 000000005 0 000100000 0 000100000 0 000100000 10 11013 23287 10 00000000 11013 23292 10 00000000 0 999999996 10 00271302 By now you ve become an expert in the HYP launcher and for sure appreciate its compactness lots of keystrokes With a couple of exceptions it s a100 accuracy to 10 decimal places and really the only sore point is in the point 0 001 for HACOS But don t worry there s no bugs creating havoc here it s just the nature of the beast bound to occur with the limited precision even using 13 digits in the Coconut CPU No wonder you re going to repeat the same table for the trigonometric functions and see how it stacks up right While you re at it go ahead and calculate the power of two of the square root pressing Ex 9 2 r but don t call HP to report a bug For very small arguments the accuracy of SINH and COSH will also start showing incorrect digits However HTAN and HATAN use an enhanced formula that will hold the accuracy regardless of how small the argument is Note Make sure that revision N or higher of the Library 4 module is installed c Angel M Ma
3. 1o r y A aid TE 10 j 10 j pecese nz ni n as a function of n red dots compared to n Both plots are logarithmic Natural primorial right plot In general for a positive integer n such a primorial n can also be defined namely as the product of those primes lt n a n n J Pi Prin F i i The FOCAL programs below can be used to calculate both flavors of primorials Note the primordial pun intended role of function PRIME which effectively makes this a simple application as opposed to a full fledged program from the scratch Examples Calculate both primorials for the first 20 natural numbers See the solutions on the table next page c ngel M Martin Page 73 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 Table of primorials n nit Pn Prt 0 1 no prime 1 1 2 2 2 2 3 6 3 6 5 30 4 6 7 210 5 30 11 2310 6 30 13 30030 7 210 17 510510 8 210 19 9699690 9 210 23 223092870 10 210 29 6469693230 11 2310 31 200560490130 12 2310 37 7420738134810 13 30030 41 304250263527210 14 30030 43 13082761331670030 15 30030 47 614889782588491410 16 30030 53 32589158477190044730 17 510510 59 1922760350154212639070 18 510510 61 11728838 1359406970983270 19 9699690 67 7858321551080267055879090 20 9699690 71 5579408301266989609674 15390 01 LBL NPRML 01 LBL PPRML 02 ABS 03 INT 04 E 05 X gt Y 06 RTN 07 STO Z PRIME INC
4. If we re to believe that behind a great man there is often an even greater woman then the greatest idea behind all these functions is the implementation of the Generalized Hyper geometric function A general purpose definition requires the use of data registers for the parameters al am and b1 bn and expects the argument x in the X register with the number of parameters m and n stored in Z and Y for the generic expression mip a1 42 54m 3 by by b X Yie0 12 ar K a2 ke Am k b1 k b2 g bp k x k Ifm p 0 HGF returns exp x The program doesn t check if the series are convergent or not Even when they are convergent execution time may be prohibitive press any key to stop Stack register T is saved and x is saved in the L register ROO is unused The alpha register is cleared The original HGF was written by Jean Marc Baillard Only small changes have been made to the version in the SandMath optimizing the code and checking for ALPHA DATA in all registers used as well as for the argument x Function Description Author Al WLO Lambert s W main branch Angel Martin WL1 Lambert s W secondary branch Angel Martin Al ZETA Riemann s Zeta direct method Angel Martin ZETAX Riemann s Zeta Borwein algorithm JM Baillard c Angel M Martin Page 100 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 3 4 1 Exponential Integral and associat
5. which results in DATA ERROR with the standard function but here returns 1 This has practical applications in FOCAL programs where the all zero case is just to be ignored and not the cause for an error is an extended range factorial capable of displaying results over the standard numeric range of th calculator Like Y X above it returns the mantissa to X and the exponent to the Y register This function resides in the secondary FAT and therefore needs to be called using any of the launchers The implementation is just a particular case of the super factorial with the repeat factor p 1 This will be described in the corresponding section later on Example to calculate 70 and 120 just type using FIX 6 for display formatting 70 EF XFCT gt 1 197857 E100 120 AAi gt 6 689503 E198 The full value of the mantissa is left in the X register c ngel M Martin Page 23 of 167 December 2014 SandMath_44 Manual Revision 3x3 2 1 2 Number Displaying Bases and Coordinate Conversions A basic set of base conversions and diverse number displaying functions round up the elementary set _ Function Description Author XDGT Sum of Mantissa digits Angel Martin AINT A fixture appends integer part of X to ALPHA Frits Ferwerda DSP Shows current decimal digits setting Angel Martin HMS HMS Division by scalar Tom Bruns HMS HMS Multiplication by scalar
6. 1 gt RUNNING followed by 0 567143290 Example 3 Calculate both branches of W for x 1 2e 1 EX CHS ST X 1 X gt Wp 1 2e 0 231960953 LASTX EF WL1 gt W 1 1 2e 2 678346990 And here s a 3D representation of the i fe complex Lambert to end this section with a e graphical splash Enough to make you want so tf to start using your 41Z Module isn t it PL TES fae c ngel M Martin Page 99 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 3 4 Remaining Special Functions in the Main FAT The third and last chapter of the Special functions in the main FAT comprises other Hyper geometric derived functions plus one notable exception not easy to associate LINX Function Description Author Al CI Cosine Integral JM Baillard Al EI Exponential Integral JM Baillard F LI Logarithmic Integral ngel Martin RF ELIPF Eliptic Integral ngel Martin Al ERF Error Function JM Baillard H HCI Hyperbolic Cosine Integral JM Baillard HGF Generalized Hyper geometric Function JM Baillard H HSI Hyperbolic Sine Integral JM Baillard LINX Polylogarithm function ngel Martin Al SI Sine Integral JM Baillard Notable examples of multi purposed function are also the Carlson Integrals themselves a generator for several other functions like the Elliptic Integrals More about these later on in the corresponding sections of the manual The unsung hero HGF
7. Output Examples 0 4 ENTER 1 3 ENTER 0 7 EF ALF gt P1 3 0 4 0 7 0 274932821 0 6 ENTER 1 7 ENTER 4 8 lt XEFL gt P1 7 0 6 4 8 10 67810281 Alternatively M81 H SHIFT L using the main launcher instead c Angel M Martin Page 143 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Whittaker Function WHIM In mathematics a Whittaker function is a special solution of Whittaker s equation a modified form of the confluent hypergeometric equation introduced by Whittaker 1904 to make the formulas involving the solutions more symmetric Whittaker s equation is 2 f A _ 1 2 m I 1 4 HY w 0 dz ma mi z It has a regular singular point at 0 and an irregular singular point at co Two solutions are given by the Whittaker functions Mk u z WK u z defined in terms of Kummer s confluent hypergeometric functions M and U by 1 Mun 2 exp z 2 z 2M x K 5 1 2u 1 Wu 2 exp z 2 2 7 2U x K 1 2p The graphics below show both functions for the particular case k 2 and m 0 5 DATA REGISTERS ROO thru R02 Flags none Output 2 SQRT 3 SQRT PI F WHIM gt W sqrt 2 sqrt 3 z 5 612426206 Example c ngel M Martin Page 144 of 167 December 2014 SandMath 44 Manual Revision_3x3 Toronto Function TNMR In mathematics the Toronto function T m n r is a modification of the co
8. Several refinements make the algorithm more effective For instance instead of using uniformly spaced samples which can induce a kind of resonance producing misleading results when the integrand is periodic FINTG uses samples that are spaced nonuniformly Another refinement is that FNTG uses extended precision 13 significant digits to accumulate the internal sums This allows thousands of samples to be accurately accumulated if necessary A calculator using numerical integration can almost never calculate an integral precisely However there is a convenient way for you to specify how much error is tolerable You can set the display format according to how many figures are accurate in the integrand f x A setting of FIX 2 tells the calculator that decimal digits beyond the second one can t matter so the calculator need not waste time estimating the integral with unwarranted preciSion Refer to the heading Accuracy of FINTG Instructions In calculating integrals FINTG repeatedly executes a program that you write for evaluating f x You must also provide FINTG with two limits for x providing an interval of integration e FINTG requires 32 unused program registers If enough spare program registers are nor available FINTG will not run and the error NO ROOM results Execute PACK in Program mode to see how many program registers are available e Before running FINTG you must have a progra m stored in program memory or a plug in mo
9. X the root this is what is shown in the display If the function that you are analyzing equals zero at more than one value of x FROOT stops when it finds anyone of these values To find additional values key in different initial estimates and execute FROOT again When no root is found It is possible that an equation has no real roots In this case the calculator displays NO instead of a numeric result This would happen for example if you tried to solve the equation Ixl 1 which has no solution since the absolute value funct ion is never negative There are three general types of errors that stop FROOT from running e If repeated iterat ions seeking a root produce a constant nonzero va lue for the specified function the calculator displays NO e If numerous samples indicate that the magnitude of the function appears to have a nonzero minimum value in the area being searched the calculator displays NO e If an improper argument is used in a mathematical operation as part of your program the calculator displays DATA ERROR Programming Information You can incorporate FROOT as part of a larger program you create Be sure that your program provides initial estimates in Ihe X and Y regislers just before it executes Remember also that FROOT will look in the Alpha register for the name of the program that calculates your function If the execution of FROOT in your program produces a root then your p
10. n 5 2 x T va 1 4 x 72 lL 2 2 1 n 1 oF n n 2 z 7 1 r 1 0 1 0 I 2 0 5 0p 05 LO I 0 05 OD Qa TD X X The first few Chebyshev polynomials of the first kindin o The first few Chebyshev polynomials of the second kind the domain 1 lt x lt 1 The flat To T T2 T3 74 and Ts in the domain 1 lt x lt 1 The flat Up Ur Up Us Ua and Us Although not visible in the image U 1 n 1 and Un 1 n 1 1 The functions are CHB for Tn x and CHB2 for Un x both are located in the secondary FAT and thus require XF to execute them Note that the iterative method is used slower but more accurate for small values of the argument and that it returns both P n x in X and P n 1 x in Y Examples Calculate T7 0 314 and U7 0 314 7 ENTER 0 314 F CHB gt 0 786900700 in X and 0 338782777 in Y 7 ENTER 0 314 XFS CHB2 gt 0 582815681 in X and 0 649952293 in Y c Angel M Martin Page 113 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Taylor Series and Polynomials TAYLOR In mathematics a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function s derivatives at a single point The concept of a Taylor series was formally introduced by the English mathematician Brook Taylor in 1715 If the Taylor series is centered at z
11. 0 7 2 0 564297007 sn 0 7 3 0 759113421 cn 0 7 1 0 796705460 cn 0 7 2 0 825571855 cn 0 7 3 0 650958382 dn 0 7 1 0 796705460 dn 0 7 2 0 602609138 dn 0 7 3 1 651895746 Jacobian Theta Functions THETA There are several closely related functions called Jacobi theta functions and many different and incompatible systems of notation for them One Jacobi theta function named after Carl Gustav Jacob Jacobi is a function defined for two complex variables z and T where z can be any complex number and T is confined to the upper half plane which means it has positive imaginary part It is given by the formula Plz T gt exp min r 27inz 13 gt e cos 2rrnz moo n The SandMath uses the following definitions as per JM Baillard with q ePi K K 0 lt q lt 1 Thetal x q 2 q Xiseo 1 q t sin 2k 1 x Theta2 x q 2 q f Xiseo qh cos 2k 1 x Theta3 x3q 1 2 Xs 1 d cos 2kx Theta4 x q 1 2 Yiee 1 q cos 2kx Use the program THETA to calculate any of these using the function index in Z and the two arguments in Y and X The result is returned in X Example Compute Thetal x q Theta2 x q Theta3 x q Theta4 x g for x 2 q 0 3 1 ENTER 0 3 ENTER 2 F THETA gt 1 382545289 2 ENTER 0 3 ENTER 2 EF LastF gt 0 488962527 3 ENTER 0 3 ENTER 2 F LastF gt 0 605489938 4 ENTER 0 3
12. 15 Example 4 Calculate the unknown characteristics of the triangle above right side Solution This is a SSA problem so the input order is S1 S2 A1 Once you have input this problem the calculator displays the warning ANGL SIDE SIDE indicating that this combination of inputs can result in more than one solution The calculator solves for the case where A2 is acute The keystrokes to input the triangle are 435 7 A 452 9 B 67 a After the calculator displays ANGL SIDE SIDE you can solve for the unknowns 73 11 A2 39 89 A3 303 58 S3 63 280 40 AREA Moving between SARR and TRIA Pressing the d key from the TRIA menu will execute the SARR function and pressing d from SARR will execute TRIA Data are transferred between the two functions as follows When going from TRIA to SARR S1 becomes RUN and S2 becomes RISE The slope and angle are calculated accordingly When going from SARR to TRIA RUN becomes SI RISE becomes S2 and A3 is set to 90 If you transfer between SARR and TRIA using the XEQ SOLVER process all the data are cleared c ngel M Martin Page 43 of 167 December 2014 SandMath_44 Manual Revision 3x3 3 CIRC THE CIRCLE SOLVER The CIRC solver allows you to calculate properties of a circle and a sector of that Circle from a known radius and central angle Like the SARR and TRIA cases the CIRC solver is menu driven The following diagram shows you
13. 6 CLX 51 8 RDN 53 9 CF 00 54 X lt gt Z 10 INT 55 MSRt O 11 ABS 56 FS C 00 was it prime 12 PRIME 57 GTO 01 yes wrap up 13 SF 00 58 X lt gt Y no swap things 14 59 GTO 05 and do next PF 16 GTO 01 61 SF 04 17 FS C 00 if prime we re done 62 SF 10 18 GTO 01 63 PRMF 19 STO 01 save for grouping 64 20 ST L 65 E 21 LASTX reduced number 66 FC 10 23 E 68 STO 00 re build the number 24 STO 00 reset counter 69 MSUA 25 RDN 70 CLA 27 RCL 01 previous prime factor 72 28 X lt gt Y 73 29 PRIME 74 30 SF 00 75 31 XHY different PF 76 32 3TO 02 YES 77 33 ISG 00 increase counter 78 34 NOP SAME pf 79 35 FS C 00 was it prime 80 36 GTO 03 Skip if Prime 81 37 ST L 82 38 LASTX 83 ST 00 39 GTO 00 84 FC 10 40 85 J 41 86 FC 10 42 87 GTO 06 43 88 RCL 00 44 f 89 PROMPT 45 STO 01 90 END c ngel M Martin Page 61 of 167 December 2014 SandMath_44 Manual Revision 3x3 Curve Fitting at its best i r Perhaps few other subjects have been so thoroughly covered and repeatedly implemented on programmable calculators as Curve Fitting Certainly the 41 is no exception to this see the excellent examples from the Advantage Module and the PPC ROM or the standard setting macro program from W Kolb on the same subject Revision 3x3 of the SandMath includes the excellent implementation of the Curve Fitting functions from the AECROM enhanced to use 13 digit math routines Both the program in FOCAL
14. Ln T z Ln X z i i 1 2 n Notice also that the same error will occur when trying to calculate Logcamma when Gamma is negative which occurs between even negative numbers and their immediately lower inferior one see the plot in page 27 Example 1000 XEQ LNGM yields In 1000 5 905 220423 therefore 1000 4 02387 10 See the following link for a detailed description of another implementation using Lanczos for both cases to calculate Gamma and LogGamma on the 41 by steven Thomas Smith An excellent implementation of Gamma and related functions for the 41 is available on the following link written by Jean Marc Baillard very complete and detailed http www hpmuseum org sottware 41 4lgamdgm htm c Angel M Martin Page 82 of 167 December 2014 SandMath_ 44 Manual Revision 44_3x3 3 3 5 Digamma and Polygamma functions P In mathematics the digamma function is defined as the logarithmic derivative of the gamma function V x E log T x aa It is the first of the polygamma functions Its relationship to the harmonic 5 numbers is shown in that for natural numbers Pin H Y where Hn is the n th harmonic number and y is the Euler Mascheroni constant As can be seen in the figure above plotting the digamma function it s an interesting behavior showing the same poles and other singularities to worry about It should be possible to find an approximation valid for all th
15. c ngel M Martin Page 66 of 167 December 2014 SandMath_44 Manual Revision 3x3 More Triangles and Tetrahedrons r r A short reminder section to reflect the popularity of these topics so common in the early days of programmable calculators See JM Baillard s pages on the subjects posted at http hp41programs yolasite com polygon php and http hp41programs yolasite com heron ph calculates the area of a triangle knowing its three sides using Heron s formula Just enter the sides values in the stack and execute the function located in the auxiliary FAT The result is stored in X with the original side saved in LastX The rest of the stack is unchanged Let the triangle ABC with 3 known sides a b c and s at b c 2 the semi perimeter Heron s formula is Area s s a s b s c 1 2 Examples a 2 b 3 c 4 Type 2 ENTER 3 ENTER 4 F HERON gt Area 2 904737510 is related to it but the calculation for the area of the cyclic quadrilateral using Brhamagupta s formula Just enter the four values in the stack and execute the function in the secondary FAT The result is stored in X with the original side saved in LastX The rest of the stack is unchanged Let a b c and d be its sides lengths and the semi perimeter s a b c d 2 The area A of the cyclic quadrilaterais A s a s b S c s d 1 2 Example a 4 b 5 c 6 d 7 Type 4 ENTER 5 ENTER 6 ENTER 7 xF BRHM
16. gt Area 28 98275349 calculates the volume of a tetrahedron using Francesca s formula with edges values stored in registers RO1 to RO6 provided that the edges a b c intersect at the same vertex and the edges d e f are respectively opposite to the edgesa b c thus aandd respectively b and e c and f must be non coplanar Here too you can use IN or INPUT to conveniently store those values in the registers see DHST description section for details Examplel a 3 b 5 c 7 d 6 e 8 f 4 Store these 6 numbers into RO1 thru R06 then XF THV gt V 8 426149773 The exact value is sqrt 71 all digits correct Example a 120 b 160 c 153 d 25 e 39 f 56 Store these 6 numbers into RO1 thru R06 T hen F THV gt V 8 063 999998 the exact result is 8 064 The second tetrahedron is a heronian tetrahedron the edges lengths the faces areas amp the volume are all integers So not even the 13 digit math routines return exact results in difficult cases like example2 c ngel M Martin Page 67 of 167 December 2014 SandMath_44 Manual Revision 3x3 de FRETORIALS Quick recap a summary table of the different factorial functions available in the SandMath Function Description Author 2 APNB Apery Numbers JM Baillard BN2 Bernouilli Numbers Angel Martin MFCT Multifactorial JM Baillard LOGMF Logarithm Multifactorial Angel Martin SFCT Superfac
17. 1 000049974E 09 0 000000000E 00 1 000000472E 09 1 000000133E 09 1 000000088E 09 December 2014 SandMath 44 Manual Revision 44_3x3 3 4 3 How many logarithms did you say LINX LINX calculates the polylogarithm function also known as Jonquiere s function a special function defined by the infinite sum or power series oO k A Lisl z i Z Ta g z a a5 Js k i t Only for special values of the order s does the polylogarithm reduce to an elementary function such as the logarithm function The above definition is valid for all complex orders s and for all complex arguments z with z lt 1 it can be extended to z 1 by the process of analytic continuation For particular cases the polylogarithm may be expressed in terms of other functions see below Particular values for the polylogarithm may thus also be found as particular values of these other functions For integer values of the polylogarithm order the following explicit expressions are known Li z 20 z z Li_ z Oz yy _ ziti z 1 2 i a z 1 oe 2 o n Li_4 04 atl 1024 2 l Lid 1 2 The SandMath implementation is an MCODE function that uses direct series summation adding terms until their contribution to the sum is negligible Convergence is very slow especially for small arguments Its usage expects n to be in register Y and x in register X The result is saved in X and X is mo
18. 42 ELP 43 HERON 44 JEF 45 LE 46 L I2 47 LEIB 48 PP2 49 SAE 50 THETA 51 THV 52 VMOD 53 VXA 54 V A Toronto function Weber and Anger functions Whittaker M function Lambert W1 Output Complex to ALPHA Section Header Aux for JEF Area of cyclic quadrilateral Bhramagupta Clausen Function Carlson Integral 1st Kind Carlson Integral 2nd Kind Carlson Integral 3rd Kind Fresnel Integrals C x amp S x Elliptic Integral Perimeter of Ellipse Area of Triangle Heron formula Jacobian Elliptic functions Legendre Elliptic Integral 1st Kind Legendre Elliptic Integral 2nd Kind Legendre Elliptic Integral 3rd Kind Point to Point Distance Surface Area of an Ellipsoid Theta Functions 1 2 3 4 Tetrahedron Volume Vector Module Vector Cross Product Vector Dot Product JM Baillard JM Baillard ngel Martin ngel Martin ngel Martin n a JM Baillard JM Baillard JM Baillard JM Baillard JM Baillard JM Baillard JM Baillard ngel Martin ngel Martin JM Baillard JM Baillard JM Baillard JM Baillard JM Baillard ngel Martin JM Baillard JM Baillard JM Baillard ngel Martin ngel Martin ngel Martin The following section groups the factorial functions circling back from the special functions into the number theory field a timid foray to say the most index Name 55 FACTORIAL 56 AGM 57 APNB 58 BN2 59 CPF 60 DSP 61 ERFN 62 FFCT 63 ICPF 64 LOGH
19. 6 SQRT 13 14 15 16 17 1 X Pl X lt gt Y 19 20 21 22 END And here are some results compared to the values obtained using ELIPF As you can expect the execution is substantially faster using the AGM approach x ELIPK x ELIPF 1 2 x 1 612441348 1 659623599 1 713889448 1 777519373 1 854074677 1 949567749 2 075363134 2 257205326 2 578092113 1 612441348 1 659623598 1 713889447 1 777519371 1 854074677 1 949567749 2 075363135 2 257205326 2 578092113 6 02546E 10 5 83468E 10 1 12516E 09 0 0 4 81843E 10 0 0 Let s now continue with the not so simple functions still remaining where some of them will not surprisingly be based on the Hyper geometric functions again c Angel M Martin Page 150 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Debye Function The family of Debye functions is defined by fe gh Dalz dt O g ef The functions are named in honor of Peter Debye who came across this function with n 3 in 1912 when he analytically computed the heat capacity of what is now called the Debye model The formula used for n positive integers and X gt 0 is db x n Eo e xk x ke n k ix Pa 20 40 Despite being a FOCAL program DBY pretty much behaves like an MCODE function no data registers are used only the stack and ALPHA and the original argument is p
20. 9 u tan u du which using the series expansion for x tan x expressed using the Bernoulli numbers B2n can be written by a sum of the integration of the terms a much easier approach to say the least We ll use the ZETA function to calculate B2n for n gt 3 thus we have all tools required for the task Graphically we see a nice slanted shape compared to the trigonometric functions also notice that they are periodic functions with period x Some special values include Cl2 n 4 G Catalan s constant 0 915 965 594 Graph of the Clausen function Cl 6 The Lobachevsky function A or JI is essentially the same function with a change of variable A 0 log 2sin t dt Cly 20 2 although the name Lobachevsky function is not quite historically accurate as Lobachevsky s formulas for hyperbolic volume used the slightly different function A log sec t dt A 7 2 8 log 2 0 We have also L y 1 2 Im Li2 exp 2i u where Li2 dilogarithm function Using the same method explained above the expression to program becomes A H 2 u In 2p 2 Ek 2 1 1 2k 1 Bax 2k 2 p c Angel M Martin Page 108 of 167 December 2014 SandMath_44 Manual Revision _3x3 LOBACH has an all MCODE implementation also motivated by the need to locate the code in a secondary bank where FOCAL is not supported This provides a fast execution ev
21. Appendix 12 His master s voice or text The following excerpts are taken from the Advantage Manual pages 61 66 Just replaced SOLVE with FROOT and INTEG with FINTG and we re all set Besides the Advantage Pac manual and the HP 15C Advanced Functions Handbook as obvious first references most recommended reading is the description of IG and SV in the PPC_ROM users manual with a thorough description of the methodology and plenty of examples to try your hand and test the functions Finding the roots of an equation f x 0 The FROOT program finds the roots of an equation of the form ffx f x 0 where x represents a real root Note that any equation with one variable can be expressed in this form J ROOT For example f x a is equivalent to f x a 0 and f x g x is equivalent to f x g x 0 Method FROOT normally uses the secant method to iteratively find and test x values as potential roots It takes the program several seconds to several minutes to do this and produce a result iy diet If c isn t a root but f c is closer to zero r than f b then b is relabeled as a c is relabeled as b and the prediction process Jf is repealed Provided the graph of x is smooth and provided the initial values of Pee a and b are close to a simple root the ff secant method rapidly converges to a s root YB Pah ae ae a If the calculated secant is nearly horizontal th
22. IN 2 STO 01 13 RAD 3 X lt gt Y 14 RCL 00 4 TO 00 15 5 EIN 16 X lt gt y 6 0 17 SIN 7 Pl 18 RCL 01 8 FINTG 19 i 9 PI 20 10 21 COs 11 RTN 22 END Which won t compete for the speed award compared with the SandMath JBS function but besides illustrating the example note that it returns more accurate results for large orders and arguments as discussed in the Bessel functions section Example J 50 50 0 121409022 Correct to the 9 decimal place as can be seen using WolframAlpha s result 0 1214090218976150638201083836782773998739591421282135 Note that the iterative JNX1 can also be used for this calculation yielding the exact same result in a comparable execution time 50 ENTER ZFL JNX1 gt 0 121409022 c Angel M Martin Page 159 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Programming Highlights MCODE Cathedrals Often when visiting a landmark or a commemorative building we feel the imposing presence of something that s bigger than what any possible description could convey and we proceed tip toeing speaking in whispers not to disturb the spirit of its creators this is exactly how I felt about the addition of the SOLVE and INTEG cathedral of MCODE to the SandMath Leaving their mathematical prowess and attributes aside as tremendous as they are the housekeeping chores and implementation on the 41 platform are nothing short of spectacular The origin
23. coefficients indexed by n is a rational function of n The series if convergent defines a generalized hypergeometric function which may then be defined over a wider domain of the argument by analytic continuation We ve already described the pivotal role of this function in the multiple ways it s used to calculate many of the special functions but so far haven t used it by itself Let s complete the description with a few examples all taken from JM Baillard web pages as it s become customary already The first remark is about the parameter entry Being a generalized function it takes a variable number of arguments which are to be stored in the corresponding data registers starting with RO1 The total number of arguments is specified by the function s indexes m and p as they are provided in the function s name mF p Besides those register ROO is reserved for the principal argument x The usage requires m p and x in the Stack in registers Z Y and X respectively Last k val The second remark is the dual character of the implementation it can compute the standard or the regularized function the latter has all the coefficients divided by products of the Gamma function The option is indicated by the sign in the second parameter p in the Y register positive for the standard and negative for the reguralized Examplei Calculate 3F4 1 4 7 2 3 6 5 7 and 3F4 1 4 7 2 3 6 5 7 1 007
24. gt 0 834626842 A piece of trivia the Gauss constant is a transcendental number and appears in the calculation of several integrals such as those below il m 2 l m2 _ sin x dx f cos x dzr G 0 0 5o ax c 0 cosh 72 c Angel M Martin Page 149 of 167 December 2014 SandMath_44 Manual Revision _3x3 Example 3 Complete Elliptic Integral of 1 Kind Using AGM it s a convenient way to calculate the Complete Elliptic Integral of the first kind ELIPK k by means of the following relationship where M x y represents the AGM won 3 f do 2 cos 0 y sin 0 cos 0 y Nere y sin 0 8 where K k is the Complete Elliptic Integral of the first kind K f As usual the conventions used for the input parameters get in the way so paying special attention to this we can re write the expresion using the Incomplete Elliptic Integral instead as follows ELIPF 7 2 a b a b x a b 4 AGM a b o o Woo 1 k sin 0 ELIPF lt 7 2 a b a b 2 z a b 4 AGM a b The idea is to find two values a b derived from the argument x a b a b 2 The easiest approach is to choose a 1 and therefore b 1 sqr x 1 sqr x which is the same as Al t Y Here s the FOCAL program used for the calculation Note the first step needed to get the square root of the argument to harmonize both conventions used 1 LBL ELIPK 2 3 4 5
25. k 6 Continuous Fractions Windschitl Stirling x 1 x Result error Result error_ Result er 120 720 5040 40320 362880 3628800 39916800 1 000000001 1 000000001 2 000000001 1 000000012 1 000000012 2 000000024 24 00000028 120 0000014 1 2E 08 1 2E 08 1 2E 08 1 18333E 08 1 16667E 08 1 16667E 08 1 20833E 08 1 19048E 08 1 19048E 08 3 30688E 09 1 37787E 08 2 25469E 08 4 1336E 08 2 52127E 08 Enhanced 13 digit Implementation Reference x 1 120 720 5040 40320 362880 3628800 39916800 O oOo N WU BRWN EF pary 11 3628800 12 39916800 479 001 600 13 479001600 6 227 020 800 14 6227020800 c Angel M Martin 5040 40320 362880 oo oo oo oO 00000 0 1 000000001 2 6 000000004 362880 3628800 39916800 479001600 6227020800 Page 80 of 167 1 1 362 880 3 628 800 Lanczos k 6 Continuous Fractions Windschitl Stirling Result error error December 2014 SandMath_ 44 Manual Revision 44_3x3 3 3 2 Reciprocal Gamma function 1 GMF The reciprocal Gamma function is the function flz m Zh 4 T z y where z denotes the Gamma function Since the 2 Gamma function is meromorphic and nonzero everywhere in the complex plane its reciprocal is an entire function The reciprocal is sometimes used as a starting point for numerical computation of the Gamma function and a few software libraries provide it separately from the regu
26. key you told the CURVE program to determine which of the sixteen curves fits this data best The calculator displays each curve name as it is fitting the data to that curve When the search is completed the name LIN EXPON_ is shown in the display to indicate that the data fits best to a linear exponential curve The equation for the LIN EXPON curve is y ax b x and the values for a and b are stored in registers 01 and 02 respectively If you press RCL 01 you will see 4 3859 which is the calculated value for a RCL 02 will show you 1 1476 which is b Goodness of Fit The coefficient of determination RR is stored in register 04 As you know this number is a score ranging from 0 to 1 that tells you how well your data fits to the specified curve A score of 1 tells you that every data point falls exactly on the curve specified by a b c and the curve s equation If you press RCL 04 you should see the value 0 8767 which is RR for the previous example The value for RR just described is dependent upon the number of data points in your sample and upon the number of coefficients a b and c that are estimated for a given curve For this reason RR is not often a good tool for comparing curves However a corrected version of RR one that isn t dependent upon sample size or number of coefficients has been provided stored in register 05 for use when comparing different curves Example 2 Predicting Y at a given X As a met
27. red and Chebyshev approximation and log x blue over the interval 2 4 Vertical divisions are 10 5 Maximum error for the optimal polynomial is 6 07 x 10 5 Right figure Error between optimal polynomial and exp x red and Chebyshev approximation and exp x blue over the interval 1 1 Vertical divisions are 10 4 Maximum error for the optimal polynomial is 5 47 x 10 4 c Angel M Martin Page 110 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 In the graphs above note that the blue error function is sometimes better than inside of the red function but sometimes worse meaning that it is not quite the optimal polynomial Note also that the discrepancy is less serious for the exp function which has an extremely rapidly converging power series than for the log function Chebyshev Approximation CHBCF One can obtain polynomials very close to the optimal one by expanding the given function in terms of Chebyshev polynomials and then cutting off the expansion at the desired degree This is similar to the Fourier analysis of the function using the Chebyshev polynomials instead of the usual trigonometric functions If one calculates the coefficients Ci in the Chebyshev expansion for a function oo f z E anla z 0 and then cuts off the series after the T_N term one gets an Nth degree polynomial approximating f x The reason this polynomial is nearly optimal is that for functio
28. 167 December 2014 SandMath 44 Manual Revision 44_3x3 3 3 4 Log Gamma function LNGM Many times is easier to calculate the Logarithm of the Gamma function instead of the main Gamma value This could be due to numeric range problems remember that the 41 won t support numbers over E100 or due to the poles and singularities of the main definition The SandMath uses the Stirling approximation to compute LogGamma as given by the following expression directly obtained from the formula in page 27 1 1 2n T z n 27 nz z 2m z n sinh 7 2 8102 This approximation is also good to more than 8 decimal digits for z with a real part greater than 8 For smaller values we l use the functional equation to extend it to the region where it s accurate enough and then back calculate the result as appropriate olt l The picture on the left shows the LogGamma function for positive b arguments Interestingly it has a negative results region between 1 and 2 so it isn t always positive Note also the asymptotic behavior near 3 the origin due to the Gamma function pole The implementation on the SandMath uses the analytical continuation to calculate LogGamma for arguments less than 9 ncluding negative values Obvious problems like the poles at negative integer will yield DATA ERROR messages but outside that the approximation should hold since I ztn T z X z i i 1 2 n it follows Ln F z n
29. 2 K x D 2 y Ln x 2 L D Lf fk n x Z 1 gk n x where y is the Euler Mascheroni constant 0 5772 and g n x x 2 n k 1 k 3 k 0 1 2 n 1 f n x x 2 H k H n k k n k amp 0 1 2 and H n is the Aarmonic number defined as H n 1 k k 1 2 n Where Y_ x 1 Y x and K_ x K x note that for x lt 0 Y n x and K n x are complex numbers c Angel M Martin Page 89 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 The graphics below plot the Bessel functions of the second kind Y x and their modified K x for integer orders a 0 1 2 Note that KNBS and YNBS are FOCAL programs that use dedicated MCODE functions specially written for the calculations BS and BS2 Their entries are located in the sub functions FAT thus won t be shown in the main CAT listings in case you wonder about their whereabouts Getting Spherical are we The spherical Bessel functions jn and yn and are very closely related to the ordinary Bessel functions Jn and Yn by r Jaala yla y Yasla 1H In aale Which graphical representation naturally very JBS ish looking is show below Spherical Bessel functions of 1st kind x for n 0 1 o4 Spherical Bessel functions of 2nd kind yf for 0 1 64 Notice that there really isn t any Spherical i n x properly defined but there s one in the Sa
30. 2014 SandMath_44 Manual Revision 3x3 Od HYPERAFULICS 2 3 1 Hyperbolic Functions Yes there are many unanswered questions in the universe but certainly one of them is why oh why didn t HP MotherGoose provide a decent set of MCODE hyperbolic functions in the otherwise pathetic MATH PAC and worse yet adding insult to injury how come that error wasn t corrected in the Advantage ROM For sure we ll never know so it s about time we move on and get on with our lives whilst correcting this forever and ever The first incarnation of these functions came in the AECROM module I believe programmed by Nelson C Crowle a real genius behind such ground breaking module but it was also somehow limited to 10 digit precision The versions in the SandMath all use internally13 digit routines Function Description Author HYP Hyperbolic Launcher Angel Martin H HSIN Hyperbolic Sine ngel Martin H HCOS_ Hyperbolic Cosine ngel Martin i H HTAN _ Hyperbolic Tangent JM Baillard H HASIN Inverse Hyperbolic Sine ngel Martin f H HACOS Inverse Hyperbolic Cosine Angel Martin H HATAN _ Inverse Hyperbolic Tangent JM Baillard The use of function launc
31. 2014 SandMath_ 44 Manual Revision_3x3 Because of the internal structure of the SandMath TAYLOR was split into two distinct parts The first part is a FOCAL program that calculates all the values for f a h and f a h The second is an MCODE version of all the remaining code The main reason to do that was not simply to accelerate the calculation and increase the accuracy with 13 digit OS routines which it certainly does in both accounts but to place the code in the second bank of the lower page which is where the space was available Formulae and Methodology The formulas used are as follows Let a center of the series F f a and A f ath f a h B f a 2h f a 2h C f a 3h f a 3h D f a 4h f a 4h E f a 5h f a 5h G f ath f a h H f a 2h f a 2h I f a 3h f a 3h J f at 4h f a 4h K f a 5h f a 5h then we have h f a 2100A 600 B 150 C 25 D 2 E 2520 h 2 f a 73766 F 42000 G 6000 H 1000 I 125 J 8 K 25200 hA3 f a 70098 A 52428 B 14607 C 2522 D 205 E 30240 hA4 f a 192654 F 140196 G 52428 H 9738 I 1261 J 82 K 15120 hA5 f a 1938 A 1872 B 783 C 152 D 13 E 288 h 6 f a 233244 F 184110 G 88920 H 24795 I 3610 J 247 K 4560 h 7 f a 378A 408 B 207C 52D 5E 24 h 8 f a 462F 378G 204H 69I 13J K 3 h 9 f a 42A 48B 27C 8D E 2 h 10 f a 252F 210G 120H 45I1 10J K To
32. 4 XFS QREM gt X 3 remainder Y 6 quotient Since we used the Alpha Launcher in this example we can take advantage of the LASTF feature to repeat the operation with swapped values 4 ENTER 27 gt X 4 Y 0 c Angel M Martin Page 22 of 167 December 2014 SandMath_44 Manual Revision 3x3 e CBRT calculates the cubic root of a number Note that this could also be done using the mainframe function Y X with Y 1 3 for positive values of X but unfortunately it results in DATA ERROR when X lt 0 and therefore the need for a new function Obviously it follows that CBRT x CBRT x for x gt 0 and are purely shortcut functions which clearly are equivalent to 1 X Y X and to X 2 LASTx respectively but with additional precision due to the 13 digit intermediate calculations Besides it does away with the pesky and totally unjustified issue present with negative numbers as base in Y X Example verify in two different ways that the cubic root of 3 3 is indeed 3 13 CHS x43 CBRT gt 3 000000000 13 CHS x3 3 Y 1 X gt 3 000000000 is used to calculate powers exceeding the numeric range of the calculator simply returning the base in X and the exponent in Y The result is shown in ALPHA in RUN mode For instance calculate 85 69 to obtain is a modified form of the native Y X function with the only difference being its tolerance to the 0 0 case
33. Ber n X Bei n x Ker n x and Kei n x are available in the SandMath implemented as FOCAL programs written by JM Baillard Both values are calculated simultaneously by KLV 2 and left in X Y registers as follows Y n bei nx kei n x Examples 2 SQRT PI XF KLV1 gt ber sqrt 2 z 0 674095951 X lt gt Y gt bei sqrt 2 n 1 597357210 2 SQRT PI XFS KLV2 gt ker sqrt 2 n 0 025901894 X lt gt Y gt kei sqrt 2 x 0 089242867 alternatively sl H K for KLV1 and M81 H SHIFT K for KLV2 c Angel M Martin Page 141 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Kummer Function Kummer s equation has two linearly independent solutions M a b z and U a b Z dw dw z 6 z aw 0 dz dz Kummer s function of the first kind M also called Confluent Hypergeometric function is a generalized hypergeometric series introduced in Kummer 1837 given by a n yn M a 6 z gt T Fila b z ee ae T Where a is the rising factorial defined as a a a 1 a 2 a n 1 The figures below depict two particular cases for a 2 b 3 and a 2 b 3 Re FF 2 3 2 Re Fi 2 3 2 The SandMath implementation is got to be one of the simplest application ot HGF possible which renders acceptable accuracy to the results DAT REGISTERS a R00 b RO1 M a b x Examples Compute M 2 3 7z an
34. EF INPUT gt 1 ENT 4 ENT 7 ENT 2 ENT 3 ENT 6 ENT 5 ENT R S 3 ENTER 4 ENTER PI XEQ HGF gt 3F4 1 4 7 2 3 6 5 7 1 631019643 3 ENTER 4 ENTER PI XEQ HGF gt 3Fy 1 4 7 2 3 6 5 2 0 0002831631328 Example 2 Calculate Fo 1 4 2 5 0 1 1 STOO1 4 STOO2 2 STO 03 5 STO 04 2 ENTER 2 ENTER 0 1 XEQ HGF gt 2F2 1 4 2 5 0 1 0 01289656888 Notes Ifm p 0 HGF returns exp x The function code doesn t check if the series are convergent or not Even when they are convergent execution time may be prohibitive press any key to stop It first checks that for register Rm p existence The SandMath implementation of HGF checks for alpha data Contents of stack register T is preserved and saved in register L LastX c Angel M Martin Page 154 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Regular Coulomb Wave Functions In mathematics a Coulomb wave function is a solution of the Coulomb wave equation named after Charles Augustin de Coulomb They are used to describe the behavior of charged particles in a Coulomb potential and can be written in terms of confluent hypergeometric functions or Whittaker functions of imaginary argument The Coulomb wave equation is show below Pan fu 1 2 HD ua dp p pr where L is usually a non negative integer The solutions are called Coulomb wave functions Putting x 2ip changes the
35. Lommel functions 139 Lerch Trascendent function 140 Kelvin functions 141 Kummer functions 142 Associated Legendre functions 143 Whittaker functions 144 Toronto function 145 c Angel M Martin Page 5 of 167 December 2014 SandMath_44 Manual Revision 3x3 3 6 3 Orphans and Dispossessed Tackle the Simple ones First 146 Polynomial evaluation 1 derivative 147 Decibel Addition 148 Arithmetic Geometric Mean 149 Example Complete Elliptic Integral of 1 Kind 150 Debye Function 151 Dawson Integral 152 Hypergeometric Functions 153 Regular Coulomb Wave function 154 Integrals of Bessel functions 156 Appendix 11 Looking for Zeroes 157 3 7 Solve and Integrate Reloaded 3 7 1 Functions Description _ and Examples 158 3 7 2 MCODE Cathedrals a dissertation 160 Appendix 12 His master s voice 162 END 167 Note Make sure that revision N or higher of the Library 4 module is installed c Angel M Martin Page 6 of 167 December 2014 SandMath_44 Manual Revision 3x3 Preamble What s new in Revisions 3x3 and later Revision N is the eighth generation of the SandMath module Many important architectural changes are added such as a dual bank switched configuration for each of its two pages and thus tripling is initial size to 24k and yet not changing its original 8k footprint The benefits obtained with this layout are easy to see more functions and programs are
36. Martin 2FL PDEG Polynomial Degree JM Baillard XFL LI Logarithmic Integral ngel Martin EFL PSD Poisson Standard Distribution Angel Martin XFL dPL Polynomial first derivative Angel Martin FL DAYS Days between dates MM DDYYVY in X Y HP Co FL JDAY Julian Day number of a Date MM DDYYYY in X Angel Martin FL CDAY Date for a Julian day number day number in X Angel Martin Let s tackle the simpler ones on the list first simply provides the index zero shortcut for FCAT It invokes the sub function CATALOG with Aot keys for individual function launch and general navigation Users of the POWERCL Module will already be familiar with its features as it s exactly the same code which in fact resides in the Library 4 and it s reused by both modules and are auxiliary functions used in the FOCAL programs for the Bessel functions of 2 Kind KBS and YBS They were explained in more detail in the Bessel Functions paragraph Feel free to ignore them as they re not intended for stand alone use e AWL is the Inverse Lambert W function an immediate application of the W definition involving just the exponential but with additional accuracy using the MCODE 13 digit routines in the OS AWL W exp W is the Logarithm Integral also a quick application of the EI function using the formula Li x Ei In x see description for EI earlier in the manual Note how LI starts as a MCODE function
37. Martin Page 72 of 167 December 2014 SandMath_44 Manual Revision 3x3 Appendix 5 Primorials a primordial view NPRML PPRML Welcome to the intersection between factorials and prime numbers In number theory primorial is a function from natural numbers to natural numbers similar to the factorial function but rather than multiplying successive positive integers only successive prime numbers are multiplied The name primorial attributed to Harvey Dubner draws an analogy to primes the same way the name factorial relates to factors There are two conflicting definitions that differ in the interpretation of the argument the first interprets the argument as an ndex into the sequence of prime numbers so that the function is strictly increasing while the second interprets the argument as a bound on the prime numbers to be multiplied so that the function value at any composite number is the same as at its predecessor The figures below plot both definitions comparing their shape and slopes Prnh D a a m m 10 io E 10 yf 10 a 190 x 10 e 10 Pd ne 4 10 a 10 J 4 N 10 e oO 10 Ea 10 N a 10 Ps I gist mE d 1 r 4 s 10 12 14 p as a function of n plotted logarithmically Prime primorial left plot For the nth prime number pn the primorial pn is defined as the product of the first n primes where pk is the kth prime number Ph De k
38. RO8 0 000066832 0 000067418 0 008692041 0 000025386 0 000024802 0 023546488 R09 0 000007608 0 000007491 0 015618742 0 000002726 0 000002756 0 010885341 R10 0 000001031 0 000000749 0 376502003 0 000000003 0 000000276 1 010869565 The exact values for T1 are ak e k and for TO are ak 1 k simple press E in user mode or R S right after the previous steps and input the argument at the prompt X then R S again Repeat as needed Here are the results of our example Exact delta 2 71828183 0 7 3890561 4 06006E 10 20 0855369 2 43359E 06 Eval 2 718281828 7 389056096 20 0855858 2 18281828 7 388834562 20 06646854 2 1828183 O 2 99818E 05 0 000949359 7 3890561 20 0855369 Final Remarks Choosing the increment h between 0 1 and 0 2 is often a good choice The program employs the same h value for all the derivatives but a good choice for f x may be a bad choice for f x and the Same issue appears for all the derivatives See JM Baillard s FOCAL application where h is independently adjusted modified per derivative order which achieves higher accuracy in the results http hp41programs yolasite com taylor php Note that registers ROO thru R09 may be used by the subroutine to program f x c Angel M Martin Page 116 of 167 December 2014 SandMath 44 Manual Revision_3x3 Fourier Series In mathematics a Fourier series decomposes periodic functions or pe
39. SIZE 056 Registers 00 to 07 below are the ones that contain the information pertaining to the curve fit Registers 08 to 55 listed on page 55 contain the accumulated data information required to fit data to the sixteen curves ROO Curve number 0 to 15 RO1 a R02 b RO3 c R04 RR coefficient of determination RO5 RR corrected for comparing curves of different orders RO6 Best RR corrected so far RO7 Rest curve number so far Executing the CURVE program clears all data registers in the HP 41 Example 1 Finding the Best Fit As a genetic engineer you recently completed an experiment dealing with algae growth under varying levels of radiation The experiment yielded nine data pairs which after scaled and rounded to one Significant digit looked like this pRadnt 1 3 3 14 5 5 8 10 11 Growth 5 7 10 9 9 11 12 10 13 Which curve best fits these nine data pairs Solution Assumes FIX 4 Keystrokes Display XEQ CURVE AD FIT Y BST ME 5 ENTER 1 A 1 0000 7 ENTER 3 A 2 0000 10 ENTER 3 A 3 0000 9 ENTER 4 A 4 0000 9 ENTER 5 A 5 0000 11 ENTER 5 A 6 0000 12 ENTER 8 A 7 0000 10 ENTER 10 A 8 0000 13 ENTER 11 A 9 0000 E AD FIT Y BST ME D LINEAR_ RECIPRCL_ HYPERBLA_ CAUCHY LIN EXPN_ c Angel M Martin Page 63 of 167 December 2014 SandMath_44 Manual Revision 3x3 By pressing the D
40. ahead and implemented the Strecok method using the entire 200 coefficients set both in 10 digit and 13 digit formats Without a doubt this was an extravaganza and a bit of an insane task which unfortunately didn t yield satisfactory results despite the resulting huge code stream not to mention the painstakingly error prone programming Adding insult to injury the 13 digit version showed worse accuracy than the 10 digit one which should be explained by a coincidental benefit of the rounding confirming that 13 digits is not enough of an improvement for the region near 1 In case you re interested and want to see by yourself the IERF ROM is available for download on request probably a double record of both the most boring ROM ever produced and the one with fewest functions only eight in an 8k footprint So for a while the only practical alternative was to use an iterative calculation using Halley or Newton methods which would yield acceptable accuracy better than the failed approach above even if the calculation time increases exponentially with the proximity to 1 Further research however uncovered the paper by Michael Giles referring to yet another polynomial approximation but much more tractable and certainly suitable for implementation in the SandMath This is known as the CUDA Library and both a single and a double precision are published in the following references for the paper itself and the source code htto
41. and Y n x H amp k 1 Example calculate H 5 and H 25 Use the main XFL launcher and the LastF functionality ma 5 SHIFT F gt 2 283333333 25 l gt 3 815958178 c ngel M Martin Page 20 of 167 December 2014 SandMath_44 Manual Revision 3x3 Calculates a generalized value of the Faulhaber s formula for integer values of x The few first integer values of x have explicit formulas for the result but that s not the case for a general value which can also be non integer Obviously for x 1 this function returns identical results than the previous one albeit slower due to the additional complexity of the term Example Check the triangular x 1 and pyramidal x 2 formulas for n 10 which are particular cases of the Faulhaber s Formula involving Binomial coefficients and Bernoulli s numbers See the link below for details http en wikipedia org wiki Faulhaber 27s formula 10 ENTER 1 Stal SHIFT A gt 55 00000000 10 ENTER 2 dat gt 385 0000000 ee T k 14243 4 4n 2434 4n 5 k eget ME _ oe eee Pa 2B amp 1 And using the convention B 1 0 5 the formula is _ 33 1 m 1 iS Om n f Jp n aie T k 0 Which can be programmed using a few of the SandMath functions albeit it will be considerably slower due to the impact of the Zeta algorithms part of Bernoulli s kicking in for n gt 4 e CHSY
42. as now the Matrix and Polynomial functions have an independent life of their own in separate modules like the SandMatrix more on that to come c Angel M Martin Page 9 of 167 December 2014 SandMath_44 Manual Revision 3x3 Conventions used in this manual This manual is a more or less concise document that only covers the normal use of the functions All throughout this manual the following convention will be used in the function tables to denote the availability of each function in the different function launchers hait assigned to the keyboard by MKEYS is direct execution from the main launcher XFL H executed from the hyperbolic launcher HYP F executed from the fractions launcher FRC R executed from the RCL launcher RCL CR executed from the Carlson Launcher no separate function exists HK executed from the Hankel launcher no separate function exists xx executed from the Statistics Menu ST PRB sub function in the secondary FAT XFS MKEYS prompts for the asiign de assign action use the Y N keys or back arrow to cancel There are a total of 25 functions assigned refer to the SandMath overlay for details Note that MKEYS is programmable as well Xtra Bonus High Rollers Game There is an Easter egg included in the SandMath 3x3 hidden somewhere there s a rendition of the High Rollers game so you can relax in between hard thinking sessions of math really There was simpl
43. counterparts The transform elements are expected to be stored in data registers before DHT is executed Their existence is checked but there s no check for Alpha Data which will trigger a DATA ERROR condition The transform results will be stored in a block of registers same size of the input data and located right following the last element of the initial elements Input parameters for DHT are the dimension of the vector matrix in ROO the data elements stored in registers RO1 to Rm n J and the index of the result element in X use zero for all as a convenient shortcut A few auxiliary programs are also provided for the data entry and review of the results which can be a tedious process for relatively large size vectors or matrices These are as follows c Angel M Martin Page 120 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 IN and OUT to sequentially enter or review a block of registers e Enter the initial register index for IN then proceed with all required entries and terminate with a blank R S to end the sequence Control word bbb eee is in X upon termination e Input the control word in X in the form bbb eee and OUT will display all registers sequentially Use flag 21 to control the display prompt set or not clear INPUT LIST to enter a set of coefficients in a List using the ALPHA register e Simply type the control word in X and XF INPUT Use E
44. for N PV PMT and FV all use a direct formula based on the values for the other four variables TVM uses 13 digit math routines for extended precision thus the accuracy should in theory be better than the FOCAL programs used elsewhere like the Advantage s own TVM The calculation for the interest rate uses an iterative method to solve the non explicit equation This is done applying Newton s formula for the successive estimations of the solution starting with the following initial value i0 abs PV n PMT FV 1 The function s derivative for Newton s formula is calculated using the expression f i PMT i i 1 t n PMT 1 ip i _ FV d During the calculation the display shows a blinking message shortly followed by the calculated result iz wH et eo USER and then Data Registers The usage of data registers in TVM is compatible with the other FOCAL programs in the Advantage Pac TVM and in the PPC ROM FI This is convenient if you want to use them interchangeably to compare the speed and accuracy of the different implementations You can see the current contents with the RCL function and the top row keys as arguments from 01 to 05 as follows N RO1 RO2 PV R03 PMT R04 FV RO5 When you call TVM it first makes a copy of the contents of these data registers into the stack and uses those values for the calculations Upon completion the obtained result is
45. forms described before Nevertheless the SandMath also includes LEI1 LEI2 and LEIS three FOCAL programs based on the Carlson formulas to calculate them Here are the definitions again The incomplete elliptic integral of the first kind is defined as i 1 F k 0 1 k sin t San calculated with LEI1 or with ELIPF the second kind as i a 1 Elo k f J1 k2 sin t dt j calculated with LEI2 And the third kind as H 1 II n k aes bo eee E 1 nsin t L k sin t calculated with LEIS Note also that the respective complete elliptic integrals are easily obtained by setting the value of the amplitude the upper limit of the integrals to 7 2 The formulas used to calculate them are as follows E sin Rr cos 1 m sin 1 m 3 sin Ry cos 1 m sin 1 P sin Rr cos 1 m sin 1 n 3 sin Ry cos 1 m sin 1 1 n sin Stack input of the three are the amplitude q in Y and the argument in degrees in X and LEIS also expects the characteristic n in Z The result is always returned to X Examples in DEG mode calculate F 0 7 84 E 0 7 84 and P 0 9 0 7 84 0 7 ENTER 84 XF LEI1 gt F 84 0 7 1 884976271 0 7 ENTER 84 XF LEI2 gt E 84 0 7 1 184070048 0 9 ENTER 0 7 ENTER 84 F LEI3 gt P 0 9 84 0 7 1 336853616 Obviously we could have used ELIP
46. functions is included in the SandMath including the four basic operations plus a fraction viewer and two test functions FCR Fractions Launcher Angel Martin F D gt F Calculates a fraction that gives the number in X _ Frans de Vries F F Fraction addition Angel Martin F F Fraction subtraction ngel Martin F F Fraction multiplication Angel Martin F F Fraction division Angel Martin F FRC Is X a fractional number Angel Martin F INT Is X an integer number Angel Martin is the key function within this group Shows in the display the smallest possible fraction that results in the decimal number in X for the current display precision set Change the display precision as appropriate to adjust the accuracy of the results This means the fraction obtained may be different depending on the settings returning different results For example the following approximations are found for n n 104348 33215 in FIX 9 FIX 8 and FIX 7 mv 355 113 in FIX 6 FIX 5 and FIX 4 m 333 106 in FIX 3 n 22 7 in FIX 2 FIX 1 and FIX 0 This function was written by Frans de Vries and published in DataFile DF V9N7 p8 It uses the same algorithm as the PPC ROM DF routine As per the fraction arithmetic functions there s not much to say about them apart from the fact that they use the four stack levels to enter both fractions compone
47. in the function set but back in 1979 when the 41 was designed things were a little different So here there are finally and for the record calculates Permutations defined as the number of possible different arrangements of N different items taken in quantities of R items at a time No item occurs more than once in an arrangement and different orders of the same R items in an arrangement are counted separately The formula is l Th n k calculates Combinations defined as the number of possible sets or N different items taken in quantities or R items at a time No item occurs more than once in a set and different orders of the same R items is a set are not counted separately The formula is m kl n k The general operation include the following enhanced features Gets the integer part of the input values forcing them to be positive Checks that neither one is Zero and that n gt r Uses the minimum of r n r to expedite the calculation time Checks the Out of Range condition at every multiplication so if it occurs its determined as soon as possible The chain of multiplication proceeds right to left with the largest quotients first e The algorithm works within the numeric range of the 41 Example nCr 335 167 is calculated without problems e It doesn t perform any rounding on the results Partial divisions are done to calculate NCR as opposed to calculating first NPR and dividing it by r Pro
48. l h amp 2 Yan x k 1 Jn i X Jax 2n x Jax The execution time is substantially longer than the direct approach but as an additional benefit JNX1 will also calculate J 0 x in addition leaving this value in the Y register upon completion Example J3 100 7 628420178 10 Pa osd i Order k Argume rt c Angel M Martin Page 93 of 167 December 2014 SandMath_44 Manual Revision 44 3x3 Appendix 7 FOCAL program used to calculate the Bessel Functions of the second kind As you can see it s just a simple driver of the MCODE functions with the additional task of orchestrating the logic for the different cases Note the usage of the sub functions from the auxiliary FAT as well as other SandMath functions 01 02 03 04 05 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 bo S n a FO O 37 38 39 40 41 42 43 44 45 46 47 48 49 50 F F F F F F F F F F F F F F LBL KBS e e m LBL YBS CF 01 single case x 0 HALFX x 2 swap things Multi Function Launcher Recurrence Sum BS J n x save J n x here used by Hankel is KBS partial result n is YBS ee J HALFX EEE Salad ce ala a 1 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 F F F F F F F F F F F F F F F F orden argument swapped default cas
49. made one month after the money Is loaned This requires End mode Cash Fiows Example 1 I 14 25 PV 4 a N 30x 12 1 2 3 4 5 ove 356 357 358 359 360 PMT i 1 e i j 650 650 Using the TVM solver we ll input the known variables first and then use the unknown function key to obtain the result Input Keys Result 650 CHS TVM D PMT 650 14 25 ENTER 12 TVM B T 1 1875 30 ENTER 12 TVMS A N 180 0 TVM E FV 0 TVM C PV 53 955 91959 Example _2 How much money must be set aside in a savings account each quarter in order to accumulate 4 000 in 3 years The account earns 11 interest compounded quarterly and deposits begin immediately Cash Flows Example 2 14 4 000 I r Fy 4 N 3x 4 1 2 3 4 t0 11 12 PMT f p t f TOR Input Keys Result TVM J BEGIN MODE sets flag 00 11 ENTER 4 TVM B T 2 7500 3 ENTER 4 TVMS A N 12 4000 TVM E FV 4 000 0 TMVS C PV 0 TMV D PMT 278 223688 Notice that when you press a key after keying in a value the calculator stores that value in the indicated variable equivalent to STO into the register However when you press it without first keying in a value the calculator computes a value for the indicated variable c Angel M Martin Page 49 of 167 December 2014 SandMath_44 Manual Revision 3x3 Programming Information The calculation
50. not complex The small program in the next page shows the FOCAL code to just drive the execution of both JBS and YBS piercing them together via ZOUT or ZAWIEW in the 41Z module Getting Spherical are we SHK1 SHK2 Finally there are also spherical analogues of the Hankel functions as follows AM x jn x iyn J z a 5 hy i jn x tyn x The FOCAL programs below list the simple code snippets to program the three pairs of functions just covered as follows 1 Hankel functions HK1 and HK2 2 Spherical Bessel functions SJBS and SYBS 3 Spherical Hankel functions SHK1 and SHkK2 Note the symmetry in the code for the spherical programs making good use of the stack efficiency derived from the utilization of the MCODE JBS function The plots on the left show the Spherical Hankel 1 function for orders 1 and 2 for a short range of the argument x Explicitly the first few are real part imaginary part fy te l hg iz i1 is i fy z g E 4 gt a i 1 at tiiz d ha g ie ix from 4 to 4 Fa hy z 6iz 152 15i real part hg a imaginary part zZ c ngel M Martin Page 136 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 i XEQ SJBS STO 04 STO 04 g RCL Z RCL 00 l 10 RCL Z i 10 RCLO 10 ne ie 11 ST x xe 11 SI xX 11 FO 03 12 YBS 12 AEQ SYBES 12 CHS n ie 13 FC C 03 13 FO C 04 13 Meo 14 CHS 14 CHS 14 JES 15
51. oeople maths ox ac uk gilesm files gems _erfinv pdf htto goucomputing net g node 1828 The final SandMath implementation is entirely a MCODE function very fast that uses the single precision approximation for the central region and the double precision for the upper end region determined by the condition Ln 1 x 2 lt 6 25 that is x 2 gt 1 exp 6 25 0 998069546 Examples Using ERF and IERF complete the table below Note the relative error column Delta indicating the more than reasonable accuracy of both functions combined both in the central and extreme regions equally X 0 000100000 0 001000000 0 010000000 0 100000000 0 200000000 0 300000000 0 400000000 0 500000000 0 900000000 0 995000000 0 999500000 0 999950000 0 999995000 0 999999500 0 999999950 0 999999995 c ngel M Martin ierf 0 000088623 0 000886227 0 008862501 0 088855991 0 179143455 0 272462715 0 370807159 0 476936276 1 163087154 1 984872613 2 461266226 2 867761312 3 227792264 3 554139637 3 854657923 4 134484326 erf 0 000100000 0 001000000 0 010000000 0 100000000 0 200000000 0 300000000 0 400000000 0 500000000 0 900000000 0 994999999 0 999500001 0 999950001 0 999995000 0 999999501 0 999999951 0 999999994 Page 106 of 167 Delta 0 000000000E 00 0 000000000E 00 0 000000000E 00 0 000000000E 00 0 000000000E 00 0 000000000E 00 0 000000000E 00 0 000000000E 00 0 000000000E 00 1 005025097E 09 1 000500222E 09
52. of 167 December 2014 SandMath 44 Manual Revision 44_3x3 3 3 10 Lambert W function The last function deals with the implementation of the Lambert W function Oddly enough its definition is typically given as the inverse of another function aS opposed to having a direct expression This makes it a bit backwards looking initially but in fact it is significantly easier to implement than the Riemann Zeta seen before The Lambert W function named after Johann Heinrich Lambert also called the Omega function or product log is the inverse function of f w w exp w where exp w is the natural exponential function and w is any complex number The function is denoted here by W eC TE For every complex number z a z Wize 0 5 aa The Lambert W function cannot be expressed in _ terms of elementary functions It is useful in _ combinatory for instance in the enumeration of neue 7 i trees P 0 5 It can be used to solve various equations involving exponentials and also occurs in the a Solution of delay differential equations The graph of W x for l es x s4 h The Taylor series of WO around 0 can be found using the Lagrange inversion theorem and is given by where n is the factorial However this series oscillates between ever larger positive and negative values for real z gt 0 4 and so cannot be used for practical numerical computation The W function may be approximated using Newton s
53. of 167 December 2014 SandMath_44 Manual Revision 3x3 2 1 3 First Second and Third degree Equations A MCODE implementation of these offers no doubt the ultimate solution even if it doesn t involve any high level math or sophisticated technique The Stack is used for the coefficients as input and for the roots as output No data registers are used n Des ription Author Functio STLINE Calculates straight line coefficients from two data points Angel Martin F QROOT Calculates the two roots of the equation Angel Martin QROUT Displays the roots in X and Y Angel Martin CROOT Calculates the three roots of the equation Angel Martin 7 CVIETA Driver program for CROOT Angel Martin e STLINE is a simple function to calculate the straight line coefficients from two of its data points P x1 y1 and P2 x2 y2 The formulas used are Y ax b with a y2 y X2 X1 and b y1 a X It is trivial to obtain the root once a and b are known using X9 b a Example Get the equation of the line passing through the points 1 2 and 1 3 3 ENTER 1 ENTER 2 ENTER 1 STLINE gt Y 2 500 X 0 500 and its root is left in register Z RDN RDN gt 5 000 e QROOT The general forms of the Quadratic Equation is 2 ax bxr c 0 witha o Given the quadratic equation above QROOT calculates its two solutio
54. of real numbers The N real numbers xO XN 1 are transformed into the N real numbers HO HN 1 according to the formula N 1 9 9 A y Ta cos Tnk sin nk n 0 k 1 2 N 1 The combination Cos z sin z is sometimes denoted Cas z with the well known expression based on the double angle formula l N cos z sin z V2cos z 7 The transform can be interpreted as the multiplication of the vector x0 xN 1 by an NXN matrix therefore the discrete Hartley transform is a linear operator The matrix is invertible the inverse transformation which allows one to recover the xn from the Hk is simply the DHT of Hk multiplied by 1 N That is the DHT is its own inverse involutary up to an overall scale factor The DHT can be used to compute the DFT and vice versa For real inputs xn the DFT output Xk has a real part Hk HN k 2 and an imaginary part HN k Hk 2 Conversely the DHT is equivalent to computing the DFT of xn multiplied by 1 i then taking the real part of the result Implementation details The SandMath includes DHT written by JM Baillard to calculate the transform for both vectors in R as well as for matrices of order nxm The transformation is strictly symmetrical thus all coefficients are divided by sqrt n m so DHT DHT A A but for small round off errors as usual is an all MCODE function with the considerable speed advantage over equivalent FOCAL
55. one shown above It has the clear advantage of not having to calculate Gamma for each term in the sum contributing to a much faster and robust algorithm The iterative relationships are as follows J n x X U k k 1 2 x 2 T n 1 where U k U k 1 x 2 42 k k n with U 0 1 c Angel M Martin Page 88 of 167 December 2014 SandMath_ 44 Manual Revision 44_3x3 The graphics below plot the Bessel functions of the first kind J x and their modified I x for integer orders a 0 1 2 Note that for large values of the argument the order or both these algorithms will return incorrect results for J n x This is due to the alternating character of the series which fools the convergence criteria at premature times and fouls the intermediate results Unfortunately there isn t an absolute criteria for validity but a practical rule of thumb is to doubt the result if n x it greater than 20 Bessel functions of the Second kind K n x and Y n x The formulae used are as follows J x cosan Jal i nials Ials sin a7 Kala 2 sin a7r Yal These expressions are valid for any real number as order with the same issues as the first kind functions above when the order is integer To avoid the singularities and to reduce the calculation time the following expressions are used for integer orders T Ya x 2 y Ln x 2 Jn x Z 1 fk n x Z gK n x
56. q K as Function is available in the auxiliary FAT It is a FOCAL program like the one listed below which calculates the perimeter from the semi axis values input in Y and X stack registers a sweet and short application of the Elliptic Integrals at work Note how the pesky input conventions are observed the parameter k is squared 1 LBL ELIPER 2 X lt gt Y 3 STO 05 a 4 b a 5 XA2 b a 2 6 CHS 7 8 1 b a 2 9 90 E 10 DEG 11 LEIZ uses ROO R04 qa 120 ROOS OOO aaan 13 ST X 2a R l 4 Major Axis Ma 15 ST X 4a E n 2 e a 16 END Minor Axis Example calculate the perimeter for a 3 and b 2 3 ENTER 2 XF ELP gt 15 86543959 A related magnitud appearing in formulas related to ellipses is the ratio a b a b sometimes Squared There s no proper name for this parameter unlike eccentricity but regardless the sub funcion appropriately also without a proper name in the Auxiliary FAT the very last one in the catalog is available to compute it using the values in Y and X registers Example for Y 1 and X 3 returns 0 5 c Angel M Martin Page 128 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Jacobi Elliptic functions JEF AJF In mathematics the Jacobi elliptic functions are a set of basic elliptic functions and auxiliary theta functions that are of historical importance Many of their feature
57. roots of the equation x 2 x 1 0 1 ENTER ENTER QROOT gt Re z 0 500000000 RDN gt Im z 0 866025404 e CROOT The general forms of the Cubic Equation is 3 2 J ax br cr d 0 with a o Given the cubic equation above CROOT calculates the three solutions or roots You need to input the four coefficients in the stack registers T Z Y X using a ENTER b ENTER c ENTER d ENTER CROOT uses the well known Cardano Vieta formulas to obtain the roots The highest order coefficient doesn t need to be equal to 1 but errors will occur if the first term is zero for obvious reasons The SandMath implementation does reasonably well with multiple roots but sure enough you can find corner cases that will make it fail yet not more so than an equivalent FOCAL program Appendix 2 lists the code as well as an equivalent FOCAL program to compare the sizes much shorter but surely much slower and with data registers requirements Both functions can return real or complex roots If the roots are complex the functions will flag it in the following manners 1 QROOT will clear the Z register indicating that X and Y contain the real and imaginary parts of the two solutions Conversely if Z 0 then X and Y contain the two real roots 2 CROOT will leave the calculator in RAD mode indicating that X and Y contain the real and imaginary parts of the second and third roots The real root will alway
58. short MCODE routine originally written by Jean Marc Baillard which calculates the series terms and adds them until they don t have a contribution to the final result It is a slow converging series and therefore the execution time can be rather long at normal CPU speeds Data input follows the usual conventions for the stack registers entering x as the last parameter in register X despite the written form Examples PI ENTER 0 6 ENTER 0 7 F LERCH gt 0 7 7 0 6 5 170601130 3 ENTER 4 6 ENTER 0 8 F LERCH gt 0 8 3 4 6 3 152827048 Alternatively M81 H R using the main launcher instead c Angel M Martin Page 140 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Kelvin Functions i In applied mathematics the Kelvin functions of the first kind Berv x and Beiv x and of the Second kind Kerv x and Keiv x are the real and imaginary parts respectively of Sari 4 mifa Jy xe for the 1st Kind K ze for the 2nd Kind These functions are named after William Thomson ist Baron Kelvin For integers n Bern x and Bein x have the following series expansion penta 2 ose Dal 27 2 lt kE n k 1 4 and Bei x Z sin 7 3 7 2 1 ai a a hoe 2 lt KP n k 1 4 The figure below shows Ber n x and Ker n x for the first 4 integer orders and real arguments 10 10 g 2 4 a a 10 g
59. simple to program but its precision suffers for small values of the argument A version suitable for calculators is as follows 2 7 riz Valid for Re z gt 0 and with reasonable precision when Re z gt 8 c Angel M Martin Page 78 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 For smaller values than that it s possible to use the recurrence functional equation taking it to the safe region and back calculating the result with the appropriate adjusting factor C z 1 zI z Incidentally this method can be used for any approximation method not only for Stirling s The method used on the SandMath is the Lanczos approximation which lends itself better to its implementation and can be adjusted to have better precision with careful selection of the number of coefficients used For complex numbers on the positive semi plane Re z gt 0 the formula used is as follows l 75122 6331530 g 80916 6278952 a2 I 363082951477 T z 4 5 spe g 8687 24529705 e n Jo qe 1168 92649479 g 83 8676043424 q 2 5066282 Although the formula as stated here is only valid for arguments in the right complex half plane it can be extended to the entire complex plane by the reflection formula l 1 z T z snr An excellent reference source is found under http www rskey org gamma htm written by Viktor T Toth Let s mention that this method yields good en
60. solver or you can use XEQ SOLVER A i e TRIA TRIA helps you solve all the characteristics of any triangle given three defining quantities for that triangle Here s a picture of an arbitrary triangle with its angles and sides labeled 3 The sides are labeled in a counterclockwise order around the triangle and the angles are numbered according to the opposite side This is the way that TRIA expects a triangle to be oriented To call up the TRIA menu execute the function SOLVER then A Remember the calculator must be in USER mode You will see the menu on the left USER RAD The first three keys in the left side menu represent the three sides of the triangle AR shows that using the D key you can solve for the AREA of a triangle and the M shows you that the E key brings up the menu at any time Now press e You will see the menu on the right side This is the shifted menu of TRIA This menu shows you that by using the shifted top row keys a b and c you can input or solve for any of the three angles of a triangle Plus the S selection executes SARR described before So remember the TRIA function has two menus The E key calls up the unshifted menu and e calls the shifted menu TRIA has fairly specific rules for inputting the three knowns that define a triangle Once the triangle is oriented similar to the previous diagram sides labeled counterclockwise the k
61. some of the properties of a circle that can be calculated using CIRC Important The radius Rd the sector angle AN lt and the X distance Xd are the only allowed inputs You can also calculate the circumference CI of the circle the area AR of the circle the segment area SG and the sector area ST Three menus are available for CIRC To view each menu execute the function XEQ SOLVER B and press the E e and J keys s l i DI ARC CI CHR ME The menu that comes up when you press the E keis key tells you the meanings of the top row keys the menu on the Jie key applies to the shifted top row and the J menu tells you the meanings of the wal pe E i keys in the second row Example 1 Calculate the diameter area and circumference of a circle with a radius of 10 0 Also calculate the arc length chord length chord rise sector area and segment area of a sector in that circle with a central angle of 30 degrees Solution With USER mode on and the display set to FIX 4 press SOLVER B i e CIRC E 10 A 30 B Then To calculate Press Result diameter I a 20 0000 area F 314 1593 circumference c 62 8319 arc length b 5 2360 chord length C 5 1764 chord rise lid 0 3407 sector area H 26 1799 segment area G 1 1799 c ngel M Martin Page 44 of 167 December 2014 SandMath_44 Manual Revision 3x3 Example 2 Calculate the rise YR at X
62. split the result in mantissa and exponent same as it was described for the extended factorials sections seen earlier in the manual The user instructions are simply to input the index n in X and call APNB with the sub function launcher XF The result will by placed in stack registers Y exponent and X mantissa as well as shown as an ALPHA message in RUN mode Examples 68 XFS APNB gt Ag7 5 08229 E100 Shown as follows in RUN mode Other Examples 41 EF APNB gt A41 4 944386782 E59 100 EF APNB gt Agog 2 824655679 E149 329 F APNB gt A329 1 990511251 E499 c Angel M Martin Page 75 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 Synergy in action A glimpse of what s ahead Relationship between common special functions Taken from John Cook s web site http www johndcook com special_ function diagram html Note Make sure that revision N or higher of the Library 4 module is installed c Angel M Martin Page 76 of 167 December 2014 SandMath_ 44 Manual Revision 44_3x3 A word about the approach The Hyper Geometric Function as a generic function generator or the case of the chameleon function in disguise Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis functional analysis physics or other applications fre
63. the SandMath has been resolved with an iterative calculation approach making use of the Digamma function directly in the Newton method Let r x Val the value for which a suitable argument x is sought Thus the function to find a root is f x T x Val and applying Newton s method to calculate the successive approximations Cni Z En Flan f ia but in this case f x f x 1 x which simplifies considerably the calculation The only remaining aspect is that of the initial approximation x0 We have used that formula provided by D Cantrell which involves the Lambert W function as well Approx Inv Gamma or AIG x L x W L x e 2 Letting L x In x c Sqrt 27 with c 0 036534 See reference htto mathforum org kb thread jspa message lD 342551 Even if this initial calculation takes longer than say using the Logarithm or a polynomial approximation of Gamma DataFit the benefits of a more accurate initial value are fewer number of iterations and therefore shorter total execution times See below a tabulated comparison of the execution times using the two initial approaches X Direct David Cantrell DataFit Gerson Barbosa 1 0 2 370024 2 9339976 1 5 15 4800000 17 6000040 2 0 17 96998 17 219989 2 5 11 85998 17 469972 3 0 10 98 17 66 3 5 10 36008 15 289992 4 0 10 47996 14 72004 4 5 10 179972 15 17004 5 0 10 110024 14 7900024 10 9 34992 14 230008 15 8 740008 13 86 20 9 36 14 349996 Natu
64. the auxiliary FAT of the upper page Hope you agree it was a small price to pay for such a rewarding addition definitely worth the extra effort c Angel M Martin Page 45 of 167 December 2014 SandMath_44 Manual Revision 3x3 2 5 2 The Time Value of Money solver Putting a second yellow ribbon around the box from revision M the SandMath also includes the new TVMS solver functionality taken from the just released TVM ROM This is an all MCODE implementation of the classic functions that rivals with the HP 12C implementation in speed and accuracy use it to solve for any of the five money variables with the other four known N I PV PMT and FV First of all accessing the TMV solver is also possible using a dedicated entry in the launcher just press the USER key directly at its prompt thus no need to go through the SOLVER function even if for consistency reasons it s also included there The direct way saves keystroke pressings therefore it s the recommended approach Also important to know is that to input the data TVM expects the value already in X before calling the corresponding menu choice This is reversed from the geometric solvers which first present the prompt with the menu choices for informational purposes not a launcher The same option key is used to either input the variable value or to calculate it based on the other four This duality is possible by relying on the status of the
65. the calculations calls the MCODE function after doing the change x 1 x which obviously is gt 1 perx gt 27 yl gin Ehr l r per x gl c ngel M Martin Page 95 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 Really the direct method isn t very useful at all and it s more of an anecdotal implementation with academic value but not practical The Borwein algorithm provides an iterative alternative to the direct method with a much faster convergence even as a FOCAL program and more comfortable treatment It is implemented in the SandMath as a courtesy of JM Baillard in the function ZETAX For example using ZETAX to calculate Z 1 001 returns the correct solution 1 005 577289 in a few seconds See the appendices for a FOCAL listing of the program if interested Examples Complete the table below for C x using both the direct method and the Borwein algorithm Use the result in WolframAlfa as reference to also determine their respective errors x C x Direct error Borwein error 5 0 0039682539682 0 003968254 8 0136E 09 0 003968254 8 0136E 09 5 1 036927755 1 03692775 4 96019E 09 1 036927755 1 38255E 10 3 1 202056903 1 20205676 1 19096E 07 1 202056903 1 32764E 10 2 1 6449340668482 1 644934066 5 15644E 10 1 1 10 58444846 n a n a 10 58444847 4 77115E 10 We see that not only is the Borwein algorithm faster and more capable in range but also their results are more
66. top row of keys you need to clear those assignments Example The slope of a line is 776 What is the angle between the line and level Solution In USER mode press XEQ SOLVER C i e SARR then 0 776 A B 37 8115 When you execute SARR and switch to USER mode the keys in the top row take on new meanings To see these new meanings press the E key in the top row at any time The calculator shows you the menu c Angel M Martin Page 40 of 167 December 2014 SandMath_44 Manual Revision 3x3 The first four keys in the top row represent the values SLope ANgle RIse and RuN and the fifth key calls the menu Pressing one of the first four keys can mean either take the value in the X register as an input or calculate a value depending on when you press it When you key in a value and press one of the first four keys the HP 41 takes it as an input But immediately following an input pressing one of the first four keys means compute this value In the above example the only known value was the slope which is all you need to know to solve for the angle You simply keyed in the slope 0 776 A and solved for the angle B To solve for anyone of the four unknowns you need to input knowns according to the following table To Solve for You need to input Slope Run and Rise or Angle Angle Run and Rise or Slope Rise Run and Angle or Slope Run Rise and Angle or Slope IMPORTANT RULE Always key
67. ups are available toggled by the SHIFT key xx Default Lineup Linear Regression xx Shifted Lineup Probability Note the inclusion of the mainframe functions MEAN and SDEV in the menus for a more rounded coverage of the statistical scope With the menus up you just select the functions by pressing the key under the function abbreviated name Use SHIFT to toggle back and forth between both lineups and the back arrow key to cancel out to the OS Obviously the data pairs must be already in the X REG registers for these functions to operate meaningfully c Angel M Martin Page 51 of 167 December 2014 SandMath_44 Manual Revision 3x3 Alea jacta est i It s a little known fact that the SandMath module also uses a buffer to store the current seed used for random number generation The buffer id is 9 and it is automatically created by SEEDT or RAND the first time any of them is executed and subsequently upon start up by the Module during the initialization steps using the polling points e SEEDT will take the fractional part of the number in X as seed for RNG storing it into the buffer If x 0 then a new seed will taken using the Time Module really the only real random source within the complete system will compute a RNG using the current seed using the same popular algorithm described in the PPC ROM and incidentally also used in the CCD module s function RNG Both functions were written by
68. user flag 22 the data entry flag to determine whether it s an input or a calculation action UF 22 set means input whereas UF 22 clear means calculation Remember that to actually set the satus of flag 22 you need to press a key on the numeric pad i e the digits 0 9 the Radix or EEX keys Any other key will not activate it in particular RCL CHS and ENTER so you need to work around those cases as appropriate when a new value is to be entered TVM will clear UF 22 upon completion of the command either inputting or calculating this enables a repeat calculation of different values just by pressing each menu choice in sequence After the input or calculation is done a message will show the result value for the variable chosen If the value is an integer number then decimal settings in the calculator will be ignored for further clarity Not shown in the main menu are the following actions e B E key J use it to toggle between BEGIN END modes A message is displayed to inform of the selected mode and it also toggles UF 00 annunciator in the display as a reminder of the currently selected mode e SHOW keys F to I use it to sequentially review the current values of each of the Money variables N I PV PMT and FV For additional consistency with the data entering approach both B E and SHOW will also clear UF 22 upon completion Rather than attempt to explain the usage and complete functionality let s bo
69. 00 27 16 XEQ 00 SIN 17 SF 00 1g DEG The screenshot below shows the 9 Fy ILPER output of the process 20 0 21 RCL 00 Example fix x2 22 INTEG Printer 23 RCL 00 1 LBL X2 EROM FOURN 24 2 MAD F_NAME 25 ST X 3 END x2 RUN 26 g T 27 FS 00 PI 28 b Example f x x 3 141593 29 jj 2 000000 30 RCLO2 1 LBL XK G Z83185 31 AINT 2 END RUN j jja PREC 7 pee 000000 RUN 34 PROMPT N om 7 000000 RUN 36 RTN 37 GTO E Using this program we ll calculate the coeffcients for the 7 and 9 terms for f x x 2 a7 0 081633 b7 1 795196 and a9 0 049383 b9 1 396263 c ngel M Martin Page 119 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Discrete Hartley Transform A discrete Hartley transform DHT is a Fourier related transform of discrete periodic data similar to the discrete Fourier transform DFT with analogous applications in signal processing and related fields Its main distinction from the DFT is that it transforms real inputs to real outputs with no intrinsic involvement of complex numbers Just as the DFT is the discrete analogue of the continuous Fourier transform the DHT is the discrete analogue of the continuous Hartley transform introduced by R V L Hartley in 1942 Definition and Properties Formally the discrete Hartley transform is a linear invertible function H R gt R where R denotes the set
70. 1 8 Solution 1 8 D I 0 1774 Notice that the only values you can input are the radius RA on the A key the central angle AN on the B key and the X distance XDIS on the D key These values must be knowns if you wish to calculate any properties that depend upon them Geometric Solvers Implementation Details This section has some comments on the integration to the SandMath Ignore it altogether if you re not interested in what s going on under the hood Adding the AECROM geometric solvers to the SandMath has been an exercise of discovery and patience The first due to the appreciation of the ingenuity used by the original developers to integrate the MCODE accuracy and speed to a basically FOCAL driven data input process which relies on the user flag 22 DATA Entry flag as trigger for known unknown elements As mentioned at the beginning of this section there are three FOCAL programs which account for all the possible menu selections made in the three launchers that is a total of 30 choices The uncanny thing about those programs is that for every one and each of them the same instruction is always executed and that instruction is nothing less that the AECROM header function itself How then does the function know which option is called up for The answer lies in the actual program pointer position of the calling step thus the relative location of the code was of utmost importance which accounts for the pat
71. 12N5 p44 STSORT sorts in descending order the contents of the four stack registers X Y Z and T Obviously no input parameters are required This function was written by David Phillips and published in PPCCJ V12N2 p13 finds the maximum within a block of consecutive registers which will be placed in X returning also the register number to Y The register block is defined with the control word in X as input with the same format as before bbb eee is a handy and super fast way to calculate the sum of the data registers specified by the control word bbb eee in X It was written by Jean Marc Baillard exchanges the contents of five statistical registers and the stack including L Use it as a convenient method to review their values when knowing their actual location is not required And EVEN are simple tests to see is the number in X is odd or even The answer is YES NO and in program mode the following line is skipped it the test is false The implementation is based on the MOD function using MOD x 2 0 as criteria for evenness c Angel M Martin Page 55 of 167 December 2014 SandMath_44 Manual Revision 3x3 Normal Probability Distribution Function In probability theory the normal or Gaussian distribution is a continuous probability distribution that has a bell shaped probability density function known as the Gaussian function or informally as the bell curve 1 f a p 0 e a
72. 2p 17 1 amp 8 f 18 19 IN 19 194 20 6 ENTER 20 21 ENTER 21 Z 22 JE 22 McY 23 EAX 23 Mc gt y 24 f 2 STO 25 WHLO 26 f 27 S 28 28 STO initial guess 29 STO initial Guess 29 LBL loop here 30 LBL _ loop here 30 RCLOL argument 31 RCLOL argument 31 RCL CUTENt guess 32 RCL current guess 32 GAMMA 33 GAMMA a f 34 i 34 CHS 35 CHS 35 F a6 JE 36 37 ae RCL 38 RCL 38 PSI 39 PSI 39 f 40 F 40 ST adjust result 41 ST adjust result 41 VIEW show current 42 VIEW 00 show current a2 ABS 43 ABS 43 E tolerance 44 E S tolerance NY less than it 45 X lt Y less than it 45 GTO no next pass 46 GTO 00 no next pass 46 RCL yes Recall result 47 RCL yes Recall result 47 END done 4g END done c ngel M Martin Page 86 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 3 3 7 Euler s Beta function BETA The beta function also called the Euler integral of the first kind is a special function defined by 1 B y ey tat Rela Rely gt 0 af The beta function was studied by Euler and Legendre and was given its name by Jacques Binet The most common way to formulate it refers to its relation to the Gamma function as follows P x Ty Bis y T z y As a graphical example the picture below shows B X 0 5 for values of x between 4 and 4 As it s expected the same Gamma problem points are inherited by the Beta function The implementation on the Sand
73. 4 SandMath_ 44 Manual Revision_3x3 Realistic estimates greatly facilitate the speedy and accurate determination of a root If the variable x has a limited range in which it is meaningful and realistic as a solution it is reasonable to choose initial estimates within this range Nega tive roots for instance are often unrealistic for physical problems e FROOT requires thirteen unused program registers If enough spare program registers are not available FROOT will not run and the error NO ROOM results Execute PACK in Program mode to see how many program registers are available e Before running FROOT you must have a program stored in program memory or a plug in module that evaluates your function f x at zero This program must be named with a global label FROOT then iteratively calls your program to calculate successively more accurate estimates of x Your program can take advantage of the fact that FROOT fills the stack with its current estimate of x each time it calls your program e You then enter two initial estimates for the root a and b into the X and Y registers Lastly put the name of your program that evaluates the function into the Alpha register and then XEQ FROOT When the program stops and the calculator displays a number the contents of the stack are Z the value of the function at x root this value should be zero Y the previous estimate of the root should be close to the resulting root
74. 4 20 The parameter u is the mean or expectation location of the peak and o 2 is the variance o is known as the standard deviation The distribution with u 0 and o 2 1 is called the standard normal distribution or the unit normal distribution y expects the mean and standard deviation in the Z and Y stack registers as well as the argument x in the X register Upon completion x will be saved in LASTx and f 1 0 x will be placed in X It has an all MCODE implementation using 13 digit routines for increased accuracy PDF is a function borrowed from the Curve Fitting Module which contains others for different distribution types With the Normal distribution being the most common one it was the logical choice to include in the SandMath Probability density function Cumulative distribution function The red curve its the standard nommel distnbution The figures above show both the density functions as well as the cumulative probability function for several cases The Error function ERF in the SandMath can be used to calculate the CPF no need to apply brute force and use PDF in an INTEG like scenario much longer to obtain or course The relation to use is 1 x H F z u o 2 1 ert E TER Example program The routine below calculates CPF Enter u o and x values in the stack 01 LBL CPF 08 02 RCL Z 09 ERF 03 11 INCX 04 X lt gt Y 12 2 05 13 06 2 14 END 07 SQRT c ng
75. 9 HK1 10 HK2 11 HNX 12 ITI 13 ITJ 14 JNX1 15 KLV 16 KLV2 17 KUMR 18 LERCH 19 LI 20 LNX 21 LOMS1 22 LOMS2 23 RCWF 24 RHGF 25 SHK1 26 SHK2 27 SIBS Cat header does FCAT Aux routine All Bessel Aux routine 2nd Order Integers Airy Functions Ai x amp Bi x ngel Martin ngel Martin ngel Martin JM Baillard Associated Legendre function 1st kind Pnm x JM Baillard Inverse Lambert W Dawson integral Debye functions Hypergeometric function Hankel1 Function Hankel2 Function Struve H Function Integral if IBS Integral of JBS Bessel Jn for large arguments Ber amp Bei functions Ker amp Kei functions Kummer Function Lerch Transcendent function Logarythmic Integral Struve Ln Function Lommel s1 function Lommel S2 Regular Coulomb Wave Function Regularized hypergeometric function Spherical Hankel1 Spherical Hankel2 Spherical IBS ngel Martin JM Baillard JM Baillard JM Baillard ngel Martin ngel Martin JM Baillard ngel Martin ngel Martin Keith Jarret JM Baillard JM Baillard ngel Martin JM Baillard ngel Martin JM Baillard JM Baillard JM Baillard JM Baillard JM Baillard ngel Martin ngel Martin ngel Martin c ngel M Martin Page 17 of 167 December 2014 SandMath_44 Manual Revision 3x3 28 TMNR 29 WEBAN 30 WHIM 31 WL 32 ZOUT 33 ELLIPTIC 34 AJF 35 BRHM 36 CLAUS 37 CRE 38 CRG 39 CRJ 40 CSX 41 ELIPF
76. A is a driver program for CROOT including the prompts for the equation coefficients The results are placed in the stack following the same conventions explained above See the program listing below showcasing the use of a few SandMath functions 01 LBL CVIETA 16 RCL 02 in stack as required LBL for new starts 17 RCL 01 coeff index 18 RCL 00 19 CROOT 05 a 0 y always real D6 AINT fa ARCL Z 1 D7 2 PROMPT show first root TE PROMPT 23 FS 43 is RAD mode on To STO IND 2 24 GTO 02 yes complex roots 10 RDN 35 M lt gt 21 no clear the Z register 1 DSEX 3 6 CLX indicates real case 2 NOP 7 M lt gt Z 1 restore stack 13 k O did them all 14 GTO 01 no next coeff 29 OROUT show other roots s5 RCL 03 yes prepare data 30 GTO 00 new start 34 END c Angel M Martin Page 29 of 167 December 2014 SandMath_44 Manual Revision 3x3 Appendix 2 CVIETA equivalent FOCAL program replaced now with an all MCODE implementation 01 LBL CVIETA 64 7 02 65 imaginary part 03 RA 66 RCL 1 cbrt x R3 2 04 ST T 0 q 2 a 3 in T 67 RCLO03 cbrt x R3 2 05 ST Z 1 a 1 a 3 inZ r 68 o 69 2 07 STO 00 a0 a 0 a 3 7 08 RDN 71 CHS 09 STO 01 a1 a l a 3 F 72 RCL 02 a2 3 F 40 RDN a2 a2 aq3 3 eee real part OOOO u 3 a a2 V 75 STO T 0 flag it as Complex i F 13 STO 02 a2 3 T 76 RDN Z 0 indicate
77. CL 03 cbrt x R3 2 122 120 60 123 ST 03 6 3 124 1sG 05 62 SQRT 125 GTO 08 E 63 126 ED c Angel M Martin Page 30 of 167 December 2014 SandMath_44 Manual Revision 3x3 2 1 4 Additional Tests Rounded and otherwise Ending the first section we have the following additional test functions Function Description Author X 1 Is X exactly equal to 1 Nelson C Crowle E X gt Y Is X equal to or greater than Y Ken Emery X gt 0 Is X equal to or greater than zero Angel Martin E X YR Rounded Comparison Angel Martin F FRC Is X a fractional number ngel Martin F INT Is X an integer number ngel Martin EVEN IsXan even integer ngel Martin 7 ODD Is X an odd integer Angel Martin E ZOUT Combines the values in Y and X into a complex result ngel Martin They follow the general rule returning YES NO in RUN mode and skipping a program line if false in a program Their criteria are self explanatory for the first three These functions come very handy to reduce program steps and improve the legibility of the FOCAL programs compares the values in the X and Y registers skipping one line if false compares with zero the value in the X register skipping one line if false These functions are arguably missing on the mainframe set a fact partiall
78. C_OS X module has a general purpose prompt lengthener activated by pressing the ON key while the function prompt is up c Angel M Martin Page 37 of 167 December 2014 SandMath_44 Manual Revision 3x3 Pressing ALPHA at the RCL prompt will invoke function AIRCL _ _ This will in turn prompt for a data register number and once filled it ll append the integer part of the value stored in that register to the ALPHA register thus equivalent to what AINT does with the x register Note that AIRCL _ _ is fully programmable When entered in a program you d ignore the prompts and the program step following it will be used to hold the register number to be used by ARCLI when the program runs This technique is known as non mergead functions to work around the limitation of the OS Too bad we can t use the Byte Table locations wasted by eGObEEP and W instead This method is used in several functions of the SandMath module like the RCL math functions just described Appendix 3 A trip down to Memory Lane From the HP 41 User s Handbook Automatic Primary Extended Memory Data Storage Data Storage Stack Registers Registers T Roo Reo Z ALPHA iT The standard Raon Y Rao HP 410 has Rag You can add up to four memory modules X LAST X of 63 EEs bringing the total te 100 primary and 219 If slimmed ones primary storage extended storage registers allocated to storage registers Res peau ne each additional module would R
79. Chebyshev s Coefficients JM Baillard 85 CRVF Curve Fitting AECROM Nelson F Crowle 86 D Difference Percent Angel Martin 87 DAYS Days between Dates HP Co 88 DHT Discrete Hartley transform JM Baillard 89 dPL First derivative of Polynomial Angel Martin 90 IN Input Data in Registers Angel Martin 91 INPUT Data input as ALPHA Lists Angel Martin 92 JDAY Julian Day Number Angel Martin 93 OUT Output Data from Registers Angel Martin 94 PDEG Polyn degree from control word JM Baillard 95 PL Polynomial Evaluation Angel Martin 96 Calculates Y X Y X Angel Martin 97 dB Decibel Addition Angel Martin 98 REV Shows Module revision Angel Martin 99 FCAT Function Catalog ngel Martin The best way to access FCAT is through the main launcher PAR then pressing SHIFT ENTERN CN provides usability enhancements for admin and housekeeping It invokes the sub function CATALOG with Aot keys for individual function launch and general navigation Users of the POWERCL Module will already be familiar with its features as it s exactly the same code which in fact resides in the Library 4 and it s reused by both modules and the SandMatrix as well The hot keys and their actions are listed below R S halts the enumeration SST BST moves the listing one function up down SHIFT sets the direction of the listing forwards backwards XEQ direct execution of the listed function or entered in a program line ENTER moves to th
80. Conjectandi of 1713 Ada Lovelace s note G on the analytical engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage s machine As a result the Bernoulli numbers have the distinction of being the subject of the first computer program i 01 LBL BN2 h DE n k 03 GTOO1 04 x 0 05 INK 06 X Re 07 RTN D 0 ODD Ct Sn ge ens ae 09 CLX 10 x 0 LD i 11 RTW H i 19 12 2 C 2 13 X lt Y 14 GTOO00 15 j6 There are several or rather many algorithms and approaches to 16 1 X the calculation of Bn In this particular example we ll use the 17 RTN expression based on the Riemann s Zeta function according to 1g LBL OO a which the values of the Riemann zeta function satisfy 19 Si X 20 XcY 21 GTO 00 n 1 n Bn te l 23 1 X for all integers n20 The expression n 1 n for n 0 is to be o Wal understood as the limit of x 1 x when x gt 0 lame oe Li 26 XoY l l 27 LBLOL The FOCAL program on the right shows the implemented SandMath 3E STOM code As you can see it is a super short application of the ZETA 39 CHS function even if it s used for negative arguments Obviously we ve 30 INC single cased the troublesome points to avoid execution times 31 ETA unreasonably long but apart from that it s quite generic in its 39 RCM approach It also uses a few others SandMath functions as additional 33 CHS bonus 34 35 END c ngel M Martin Page 97
81. Coulomb wave equation into the Whittaker equation so Coulomb wave functions can be expressed in terms of Whittaker functions with imaginary arguments Two special solutions called the regular and irregular Coulomb wave functions are denoted by FL n p and GL n p and defined in terms of the confluent hypergeometric function by the friendly expression below he E 14 in 2L 2 where Mk u Whittaker s function of the ist kind which is included in the SandMath but without Support for complex numbers and therefore can t be used for this purpose Fr p poe M L 1 in 2L 2 2ip The formulas used instead are as follows as per JM Baillard s implementation as usual F n r CL rh 5 Aw n pe for k gt L and integer with CLM 1 P 2L42 24 e I rA 1 n and Ara 1 Are n L 1 k L k L 1 Ak 2n Aki Ago k gt L 2 further we avoid using gamma for complex arguments by replacing the last modulus calculation with the following expressions IPC 1 y r y sinh ny and Ir 1 y L y L Hy J oe 1 y ny sinh ny The resulting FOCAL program is RCWF which takes as inputs the values for L n anr r in the stack registers Z Y and X respectively returning the result into X Example calculate F 2 0 7 1 8 2 ENTER 0 7 ENTER 1 8 EF RCWF gt F2 0 7 1 8 0 141767746 or alternatively MA1 H SHIFT R using the main launcher instead Note the res
82. ENTER 2 EF LastF gt 1 389795845 Final remarks on the Jacobi Elliptic functions Note the interesting role of the parameter m as it moves from 0 to 1 The condition m O causes the functions to become the same as the trigonometric sin and cos whereas in the other extreme for m 1 they become the hyperbolic tanh and sech In more proper terms these functions are doubly periodic generalizations of the trigonometric functions satisfying sn v 0 1 1 sn v sech v sinv cn v 0 cosv and dn v 0 tanhv cn v 0 sechv and dn v 1 The figures in next page represent three intermediate stages observe the tendency as the elliptic modulus k varies towards both ends of the range Quite a remarkable behavior showing how the interrelationships amongst seemingly unrelated topics appear c Angel M Martin Page 130 of 167 December 2014 SandMath 44 Manual Revision _3x3 7A OS OO Pc E 0 4 3K Sx 5 JK K 16399 JIQ VEL VO wis 0 99 3K x 3K_K 3 3566 Figure 22 3 4 k 0 999999 3K x 3K K 7 9474 c ngel M Martin Page 131 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Airy Functions AIRY For real values of x the Airy function of the first kind is defined by the improper integral Ai z cos 3 at dt T Jo which converges because the positive and negative parts of the rapid oscillations tend to cancel one another out as can b
83. F 65 LOGMF 66 MANTXP 67 MFCT 68 NPRML 69 POCH 70 PRML 71 PSD 72 TNL 73 SFCT 74 XFCT Description Section Header Arithmetic Geometric Mean Apery Numbers Bernouilly Numbers Cumulative probability uo Display Digits setting Generalized Error Function Falling Factorial Inverse Cumulative Prob Logarithm Hyper Factorial Logarithm Multi Factorial Mantissa Multi Factorial Number Primorials Pochhammer symbol Prime Primorials Poisson Standard Distribution Quantile Standard Normal ICP Super Factorial Extended Factorial Author n a ngel Martin JM Baillard ngel Martin ngel Martin ngel Martin JM Baillard ngel Martin ngel Martin ngel Martin JM Baillard David Yerka JM Baillard ngel Martin ngel Martin ngel Martin ngel Martin ngel Martin JM Baillard ngel Martin c ngel M Martin Page 18 of 167 December 2014 SandMath_44 Manual Revision 3x3 And the last section takes us into the Transforms and Approximation theory very loosely speaking index Name Description Author 75 TRANSFORM Section Header n a 76 ALIST Input Data in List Angel Martin 77 ANUMDL ANUM with Deletion HP Co 78 b e Array size from control word Angel Martin 79 b lt gt e index swapping Angel Martin 80 CDAY Calendar Day Angel Martin 81 CdT Aux for CHBAP JM Baillard 82 CHB Chebyshev Polinomials 1st Kind JM Baillard 83 CHB2 Chebyshev Polinomials 2nd Kind JM Baillard 84 CHBCF
84. F for the first case which has a slightly faster execution and yields the same result c Angel M Martin Page 126 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Application Examples j r Function Description Author SAE Surface Area of an ellipsoid Angel Martin ELP Perimeter of a ellipse Angel Martin Calculates Y X Y X Angel Martin The following two examples should illustrate the applicability of these special functions in the geometry subjects related to ellipses and ellipsoids and therefore provide some context to their origins and development Example 1 Surface Area of an Ellipsoid is a direct application of the Carlson Symmetrical Integral of second kind CRG used to calculate the surface aerea of an escalene ellipsoid i e not of revolution ye y x4 gtp aTh which formula is o Area 4rn Rg ab f ac f bc withc lt b lt a Example a 2 b 4 c 9 gt A 283 4273843 c ngel M Martin Page 127 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Example 2 Perimeter of the Ellipse For an ellipse with semi major axis a and semi minor axis b and eccentricity e the complete elliptic integral of the second kind is equal to one quarter of the perimeter C of the ellipse measured in units of the semi major axis In other words c 4aE e i o A Pa or more compactly in terms of the incomplete integral of the second kind E
85. F operation is also supported when excuting a sub function from within the FCAT enumeration using the XEQ hot key which is very handy for those with elusive spelling Another new enhancement is the display of the sub function names when using the index based launcher XF which provides visual feedback that the chosen function is the intended one or not This feature is active in RUN mode when entering it into a program and when single stepping a program execution but obviously not so during the standard program runs LASTF Operating Instructions No separate function exists The Last Function feature is triggered by pressing the radix key decimal point the same key used by LastX while the launcher prompts are up This is consistently implemented across all launchers supporting the functionality in the three modules SandMath SandMatrix and PowerCL they all work the same way When this feature is invoked it first briefly shows LASTF in the display quickly followed by the last function name Keeping the key depressed for a while shows NULL and cancels the action In RUN mode the function is executed and in PRGM mode it s added as a program step if programmable or directly executed if not programmable The functionality is a two step process a first one to capture the function id and a second that retrieves it shows the function name and finally parses it All launchers have been enhanced to store the approp
86. Fraction Divide 47 LCM Least Common Multiple 48 ERC is X fractional 48 LR Linear Regression c Angel M Martin Page 16 of 167 December 2014 SandMath_44 Manual Revision 3x3 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 Name INT HACOS HASIN HATAN HCOS HSIN HTAN AIRCL RCL RCL RCL RCL RCL Description Is X Integer 49 50 Hypebolic ACOS 51 Hyperbolic ASIN 52 Hyperbolic ATAN 53 Hyperbolic COS 54 Hyperbolic SIN 55 Hyperbolic TAN 56 57 ARCL Integer Part 58 Recall Power 59 Recall Add 60 Recall Subtract 61 Recall Multiply 62 Recall Divide 63 Name LR Y value Description Combinations Permutations Is X Odd Probability Distribution Function Prime Factorization in Alpha Is X Prime Random Number Block Maximun Register Sort Register Sum Stores Seed for RNDM Exchange ST amp XREG Stack Sort Time Value of Money Launcher Functions in blue are all in MCODE Functions in black are MCODE entries to FOCAL programs Pink background denote new in revisions 3x3 3x3 and 3x3 And now the sub functions within the Special Functions Group deeply indebted to Jean Marc s contribution and not the only section in the module Note there are two sections within this auxiliary FAT you can use the FCAT hot keys to navigate the groups Description Author index 0 1 BS 2 BS2 3 AIRY 4 ALF 5 AWL 6 DAW 7 DBY 8 HGF
87. H kan Th rngren an old hand 41 programmer and MCODE expert and published in PPC V13N4 p20 Determines whether the number in the X register is Prime i e only divisible by itself and one If not t returns the smallest divisor found and stores the original number into the LASTX register PRIME Also acts as a test YES or NO are shown depending of the result in RUN mode When in a program the execution will skip one step if the result is false i e not a prime number enabling so the conditional branching options This gem of a function was written by Jason DeLooze and published in PPCCJ V11N7 p30 Example program The following routine shows the prime numbers starting with 3 and using diverse Sandbox Math functions 01 LBL PRIMES 05 PRIME 09 INCX 02 3 06 VIEW X lt yes gt 10 GTO 00 03 LBL 00 07 X Y lt no gt 11 END 04 RPLX 08 LASTX See other examples later in the manual relative to prime factorization programs in the secondary FAT returns in X the number of decimal places currently set in the display mode 0 regardless whether it s FIX SCI or END Little more than a curiosity it can be used to restore the initial settings after changing them for displaying or formatting purposes c Angel M Martin Page 52 of 167 December 2014 SandMath_44 Manual Revision 3x3 Combinations and Permutations two must have classics Nowadays would be unconceivable to release a calculator without this pair
88. HT is a FOCAL program despite its MCODE appearance in the FAT The execution may be stopped and resumed in single step mode if so desired The program listing is shown below 01 LEL DHST 02 DS 7 sample type 03 PROMPT dimension o4 STO saved in ROO 05 INT matrix 06 GTO 00 no skip 07 b e yes get dimension 08 LBL 00 oo E3 E normalize 10 INPUT enter elements 11 0 all set 12 DHST Hartiey transform 13 RCL 00 dimension 14 INT matrix 15 6TO 00 no skip 16 b e yes get dimension 17 LBL 00 i 18 ENTER 19 STX Skip on set 20 E3 E normalize 21 beginning reg 22 SF 21 pause between 23 OUT output elements 24 END done Note how the auxiliary functions need to be used after the INT conditional tests due to their multi line structure The program has the 4 byte GTO jumps pre compiled so there are no LBL 00 steps The sub function b e is also available in the auxiliary FAT I simply calculates the product of the integer part of a number by its fractional part normalized to three decimal digits It is therefore the matrix dimension in this case Note that both b e and b lt gt e which swaps the begin end formats have lower case letters in their names but despite that fact you should use upper letters when spelling them at the XF prompt c Angel M Martin Page 122 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 3 5 More Special Functions in the Secondary FAT This section
89. I The order of entry is unimportant If you use only four variables then the fifth must equal zero All variables are set to zero when you first run TVM or clear the financial data so you do not have to enter a zero in these cases e You should clear the financial data B before beginning a completely new calculation otherwise previous data that is not overwritten will be used i e for the fourth unused variable Running the program anew also clears the financial data e Remember to specify cash inflows arrow up as positive values and cash outflows arrow down as negative values The results are also given as positive ox negative indicating inflow or outflow e Check that the payment mode is what you want If you see the flag 00 annunciator a small 0 below the main display line then Begin mode is set If not End mode is set To change the mode press J a toggle The display will then show what you have just set BEGIN MODE or END MODE The default is End mode flag 00 clear e Remember that the interest rate must be consistent with the number of compounding periods An annual percentage rate is appropriate only if the number of compounding periods also equals the number of years e You might want to set the display format for two or three decimal places FIX 2 3 This menu will show you which key corresponds to which function in TVM Press to recall this menu to the display at any time This will not disturb the prog
90. Math Betaly 5 makes no attempt to discover new approaches or utilize any numeric equivalence it simple applies the definition formula using the Gamma subroutine Obvious disadvantages include the reduced numeric range aggravated by the gt multiplication of gamma values in the a numerator Execution time corresponds to three times that of the Gamma function plus the small overhead to perform the Alpha Data checks and the arithmetic operations y between the three gamma values 3 3 8 Incomplete Beta Function ICBT The incomplete beta function a generalization of the beta function is defined as a a 1 1 1 4 dt wv For x 1 the incomplete beta function coincides with the complete beta function The relationship between the two functions is like that between the gamma function and its generalization the incomplete gamma function And it s also given in terms of the Hypergeometric function the expression by Bizi a b oF a 1 b a 1 z z CE Examples Calculate B 0 7 n e and B 0 4 21 40 Type PI 1 E X 0 7 XEQ ICBT gt 0 029623046 21 ENTER 40 ENTER 0 4 XEQ ICBT gt 4 8989756 18 c Angel M Martin Page 87 of 167 December 2014 SandMath_ 44 Manual Revision 44_3x3 3 3 9 Bessel functions and Modified The next logical group comprises the Bessel functions and Spherical variants Function Description Author cA IBS Bessel I n x
91. NTER 0 ENTER 3 CHS R S This leaves the control word in X thus we just enter the evaluation point and call the appropriate functions as shown below 0 003 ENTER F PL gt 21 0000 RDN 2 EF DPL gt 44 0000 Note how the function name is spelled using upper case letters The FOCAL programs shown below were written by JM Baillard They perfrom the same tasks and are provided for your sheer enjoyment and as an example of how efficient FOCAL can be specially with a 4 stack register pile and the capability to use indirect addressing c Angel M Martin Page 147 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Consider that the minimalistic programs below have an equivalence of abour 150 bytes in MCODE by the time you re done with the error handling and math syntax to use the OS routines However the speed advantage and the ability to locate the code in a secondary bank are well worth the effort g D1 LBL dPL GTO 00 02 RCLY T TET D3 PDEG A w STO M 5 21 LBL 01 wo xX RCL 2 LBL O02 RCL 2 me CO 09 RCLINDZ 1 T0 e gt M5 11 229 ST M 5 12 gt M5 13 14 SG 2 1 15 NOP 16 DSE M 5 17 s GTO 02 Decibel Addition pE 74 RCL IND 2 1 a5 26 ISG 2 1 j GTO 01 29 KY 30 SIGN 31 RDN 32 END The decibel dB is a logarithmic unit used to express the ratio between
92. NTER to separate the list entries while you re in data entry mode terminating with R S e Entries can be negative or positive integer or fractional the only limitation is no E character for exponents is possible in this mode use IN instead e Remember also the maximum length is limited to 24 characters including the blank spaces in between the entries Use it repeated times with smaller range if this limit is expected to be insufficient for the complete list Note that INPUT is a FOCAL program that drives its MCODE heart i e LIST originally written for the Polynomial Data entry in the Polynomial ROM and later modified for Matrix Input as well INPUT also uses ANUMDL under the hood to read the numeric values from the ALPHA string deleting them in a loop repeated as many times as elements on the list All these functions reside in the Library 4 ROM so only FAT pointers are added to the SandMath Let s see a couple of examples from JM s web page ttp hp41programs yolasite com hartley php Example1 One dimensional data Let A be the vector A 124 7 here n 4 amp me 1 Input the data elements using INPUT ideally suited to this type of integer data and review the results using OUT 1 004 XF INPUT gt A_ 1 ENTER 2 ENTERS 4 ENTER 7 R S 4 STO_00 0 XF DHT gt 7 0000 value of b1 5 008 XFS OUT gt R05 RO6 R07 RO8 7 4 2 1 listed sequentially Ex
93. Press XEQ CURVE and the top row of keys takes on the following meanings A AD Accumulate an x y Data pair B FIT Fit the Data to the curve specified in register 00 curve number 0 15 C y Calculate a y value on the current curve for an input x D BST Find the BeST fit of the sixteen available curves for the current data E ME Bring up this Menu SHIFT a Remove an x y Data pair for error corrections The Sixteen curves The sixteen curves available and their equations are listed below according to curve number LINEAR y a bx RECIPRCL reciprocal of linear y 1 a bx HYPERBLA hyperbola y a b x RECIP HYP reciprocal of hyperbola y x ax b POWER y ax b MOD PWER modified power y a b x ROOT y a b i x EXPONENL exponential y a e bx eae E a c ngel M Martin Page 62 of 167 December 2014 SandMath_44 Manual Revision 3x3 8 LOGRTHMC logarithmic y a b x LNx 9 LIN HYP linear hyperbolic y a bx c x 10 2 ORD HY second order hyperbolic y a b x c x 2 11 PARABOLA y a bx C x 2 12 LIN EXPN linear exponential y ax b x 13 NORMAL y a e b 2 c 14 LOG NORM log normal y a e b LNx 2 c 15 CAUCHY y I a x b 2 c Data Register usage In order to run the CURVE program or use the CRVF function you need to have 56 registers available for data storage XEQ
94. Primary Data Storage Registers Extended Data Storage Registers When allocated data storage registers numbered Ryj99 through Ry319 are extended Data Storage Registers Program Memory All registers that are not allocated as Primary of Extended data storage registers are part of program memory When allocated as program memory the standard HP 41C registers provide space for 200 400 fully merged lines of programs When allocated as program memory each memory module adds 200 400 lines The total can be 1000 2000 lines when alle 319 registers are allocated as program memory Variations in storage capacity depend on the kinds of functions stored in program memory Gi ee ee ee ee ee ee ee ee Note Make sure that revision N or higher of the Library 4 module is installed c Angel M Martin Page 39 of 167 December 2014 SandMath_44 Manual Revision 3x3 24 SULVERS What good is a math related module without a Triangle solver application Never too basic to be unimportant especially if it can be tucked away in an auxiliary bank and doesn t take the customary set of FAT entries FAT space is always at a premium Revision 3x3 of the SandMath includes the AECROM TRIA which can arguably be considered the best Triangle solver ever written for the 41 And it is incorporated into a new single function that acts as consolidated launcher for it and the other two geometric solvers All improved with 13 digit math rou
95. RCL O4 15 RCL O4 15 FC O03 16 ZAVIEW 16 ZAVIEW 16 G amp GTO00 17 END 17 END 17 RCL OO H 13 INCX H Plot of H 1H 1 x 19 CHSYX I fart aS x from 10 to 10 real part imaginary part af Examples Calculate H1 H2 SH1 and SH2 for the following values in the table pon x Z 0 440 J0 781 Z 0 440 J0 781 Z 0 301 J1 382 Z 0 301 J1 382 Oi f dt DATA ERROR DATA ERROR 1 1 Z 0 671 J0 431 Z 0 671 J0 431 Z 0 552 J0 979 Z 0 552 J0 979 Where we see that for negative arguments integer and non integer orders both the result of the Bessel function of the second kind is itself a complex number therefore the DATA ERROR message Note also the symmetric nature of the values for each of the function pairs H1 with H2 and SH1 with SH2 c Angel M Martin Page 137 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Struve functions LHX Struve functions are solutions y x of the non homogenous Bessel s differential equation d A x 2 er PU gD 4 aay varT a 3 Struve functions H n x and Modified Struve Functions L n x have the following power series forms OSO pays Ho 2 D rong Dron rari 2 i O0 gy utl 1 xy 2k d 3 Leppresean a 4 a 4 2 comme ta ate Me a aT The figure below shows a few Struve functions or integer order n 1 to 5 for 10 lt x lt 10 we L x y n 3 H x E 100 ak y 10 10 Struve functions of any order c
96. Remarks 91 Appendix 7 FOCAL program for Yn x Kn x 92 c Angel M Martin Page 4 of 167 December 2014 SandMath_44 Manual Revision 3x3 Riemann Zeta Function 93 Appendix 8 Putting Zeta to work Bernoulli numbers 95 Lambert W Function 96 3 4 Remaining Special Functions in Main FAT The unsung Hero 100 Exponential Integral and associates 101 Generalized Exponential Integrals 103 Errare humanum est 104 Generalized Error Functions 104 Appendix 9 Inverse Error function coefficients galore 105 Appendix 9b IERF using the CUDA Library approach 106 How many logarithms did you say 107 Clausen and Lobachevsky functions 108 3 5 Approximations and Transforms 3 5 1 The basics Approximation theory 110 Chebyshev s Approximation 111 Chebyshev Polynomials 113 Taylor Coefficients and Approximation 114 Fourier Series 117 Appendix 10 Fourier Coefficients by brute force 119 Discrete Hartley symmetrical Transform 120 3 6 More Special Functions in Secondary FAT 3 6 1 Carlson Integrals and associates The Elliptic Integrals 124 Carlson Symmetric Form 125 Complete and Incomplete Legendre Forms 126 Example Perimeter of an Ellipse 128 Jacobi Elliptic Functions 129 JacobianTheta Functions 130 Airy Functions 132 Fresnel integrals 133 Weber and Anger Functions 134 3 6 2 Hankel Struve and others A Lambert relapse 135 Hankel functions yet a Bessel 3 Kind 136 Getting Spherical are we 136 Struve Functions 138
97. Ry is the same for any permutation of its first three arguments The Carlson Symmetric Elliptic integral of the 2 Kind is defined as 1 fe t T t z A Jo J t alt y t z t ra try tz in the SandMath is calculated using the following expression involving CRF and CRJ 1 2 2 Re x3y3Z Z Re X3y3Z x Z y z 3 Rp x3y3z x y z with Rp x y z Ry x3y z z Examples Calculate Rp 2 3 4 and Ro 2 3 4 4 ENTER 3 ENTER 2 EF CRF gt R 2 3 4 0 584082842 4 ENTER 3 ENTER 2 EF CRG gt R 2 3 4 1 725503028 Examples Calculate Rj 1 2 3 4 and Rj 1 2 4 7 4 ENTER 3 ENTER 2 ENTER 1 EF CRJ gt R 1 2 3 4 0 239848100 7 ENTER 4 ENTER 2 ENTER 1 al gt R 1 2 4 7 0 147854445 Where the second call was made using the last function shortcut c Angel M Martin Page 125 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Complete and Incomplete Legendre Forms j In mathematics the Legendre forms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced Legendre chose the name elliptic integrals because the second kind gives the arc length of an ellipse of unit semi minor axis and eccentricity k the ellipse being defined parametrically by s 4 1 k cos t y sin t In modern times the Legendre forms have largely been supplanted by an alternative canonical set the Carlson symmetric
98. SandMath_44 Manual Revision 3x3 SHNIMATHOYY Axdtt Moth Ertaensions for hae HF H User s Manual and Quick Reference Guide Written and programmed by Angel M Martin December 2014 c ngel M Martin Page 1 of 167 December 2014 SandMath_44 Manual Revision 3x3 This compilation revision 5 77 77 Copyright 2012 2014 Angel M Martin p ii e aae AA f f l W z A EEN ay TO D DY ya PE af Be fa ye Acknowledgments Documentation wise this manual begs steals and borrows from many other sources in particular Jean Marc Baillard s program collection on the web Really both the SandMath and this manual would be a much lesser product without Jean Marc s contributions There are multiple graphics and figures taken from Wikipedia and Wolfram Alpha notably when it comes to the Special Functions sections I m not aware of any copyright infringement but should that be the case I ll of course remove them and create new ones using the SandMath function definition and PRPLOT Just kidding An important contribution comes from the AECROM Geometric Solvers and Curve Fitting and the HP 41 Advantage Pac FROOT and FINTEG Original authors retain all copyrights and should be mentioned in writing by any party utilizing this material No commercial usage of any kind is allowed Screen captures taken from V41 Windows based emulator developed by Warren Furlow Its
99. THETA Theta functions JM Baillard K KLv2 Kelvin Functions 2 kind JM Baillard Here we finally find both branches of the Lambert W function WLO and WL1 described previously in the manual as well as a nice selection of other related sub functions So your several choices in terms of launchers are as follows a Function in main FAT XEQ WLO the ordinary method ae M shortcut using the main launcher xF WLO since xF also finds functions in the main FAT Dae ALPHA WLO b Functions and in secondary FAT MA1 H W M H SHIFT W F 031 ZF 032 ZF WLO EF WLI PAA ALPHA WLO DAA ALPHA WL1 Now that s what I d call both a digression and multiple ways to skin this cat c Angel M Martin Page 135 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Hankel functions yet a Bessel third kind HK1 HK2_ Another important formulation of the two linearly independent solutions to Bessel s equation are the Hankel functions Ha 1 x and Ha 2 x defined by H 2 Jo a a 2 H Ja a iYa 2 where i is the imaginary unit These linear combinations are also known as Bessel functions of the third kind and it s just an association of the previous two kinds together This definition allows for relatively simple programming only using the real domain Bessel programs assuming the individual results for J and Y are
100. Tom Bruns XF MANTXP Mantissa and Exponent of number David Yerka P gt R Modified Polar to Rectangular lt in 0 360 Tom Bruns R gt P Modified Rectangular to Polar lt in 0 360 Tom Bruns R gt S Rectangular to Spherical Angel Martin S gt R Spherical to Rectangular Angel Martin DAs VMANT Shows full precision 10 digit mantissa Ken Emery is a small divertimento useful in pseudo random numbers generation It simply returns the sum of the mantissa digits of the argument at light blazing speed using just a few MCODE instructions More about random numbers will be covered in the Probability Stats section later on Example calculate the sum of all digits of the HP 41 s rendition of PI PI XEQ XDGT gt 40 000000000 also in the secondary FAT returns in X the number of decimal places currently set in the display mode 0 regardless whether it s FIX SCI or END Little more than a curiosity it can be used to restore the initial settings under program control after changing them for displaying or formatting purposes elegantly solves the classic dilemma to append an index value to ALPHA without its radix and decimal part eliminating the need for FIX 0 and CF 29 instructions taking extra steps and losing the original calculator settings Note that HP added AIP to the Advantage module and the CCD has ARCLI to do exactly the same and are related functions that dea
101. X 10 GTO 01 10 PRIME 11 X lt gt L 11 GTO 01 2 GTO 03 12 X lt gt L gl GTO 00 l 17 GTO 00 18 RCLZ 19 END Both routines only use the stack no data registers or user flags are used Clearly the numeric range will again be the weakest link reaching it for n 54 for PPRML and n 251 for NPRML c Angel M Martin Page 74 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 Apery Numbers In mathematics Ap ry s numbers were defined by Roger Ap ry in his proof of irrationality of the Apery s constant 3 and are defined by the following sums of binomial coefficients DAIO in KN ie A k amp RLY tn A There s an expression based on the Generalized Hypergeometric Function will be covered later in the manual which is the one used in the SandMath albeit in an independent MCODE function thus not calling HFG Said formula is An 4Fi n n nt lat lili ly A short FOCAL program using this formula is listed below 01 LBL APNB 07 STO 06 13 4 02 CHS 08 STO 07 14 PI 03 STO 01 09 X lt gt Y 15 INT 04 STO 02 10 16 1 05 1 11 STO 03 17 HGF 06 STO 05 12 STO 04 18 END They are also given by the recurrence equation 34 m 51 n 27 n 5 Ani n 1P Ana Ag a 2 The fisrt few are 1 5 73 1445 33001 see Sloane s A005259 Their values grow very large quickly therefore exceeding the 41 numeric range for n gt 67 The technique used has been to
102. X is related to the same subject and in general relevant to the summation of alternating series It can be regarded as an extension of CHS but dependent of the number in X Its expression is CHS y x y 1 x and thus changing the sign of Y when the number in X is odd e ATAN2 is the two argument variant of arctangent Its expression is given by the following definitions arctan r gt m arctan 4 y gt 0r lt 0 T arctan y lt O 2 lt 0 atan2 y r 4 _ j 5 y gt Peres Ul E y lt 0z undefined y 0z 0 Example Calculate ATAN2 z 27 using the main XFL launcher PI PI 2 MA SHIF SHIFT SST gt 1 107148718 Those amongst you with a penchant for complex variable would no dobut recognize this as the principal value of the argument of the logarithm of a complex number c Angel M Martin Page 21 of 167 December 2014 SandMath_44 Manual Revision 3x3 e E3 E does just what its name implies adds one to the result of dividing the argument in x by one thousand Extensively used throughout this module and in countless matrix programs to prepare the element indexes e FLOOR and CEIL The floor and ceiling functions map a real number to the largest previous or the smallest following integer respectively More precisely floor x x is the largest integer not greater than x and ceiling x x is the smallest integer not less than x The SandMath implementation uses t
103. accurate than the direct approach MCODE or not 13 digit internal subroutines notwithstanding Note The following links to the MAA and the now defunct Zetagrid make fascinating reading on the Zeta zeros current trends and historic perspective make sure you don t miss them http www maa org editorial mathgames mathgames_10_18_04 html http www zetagrid net c Angel M Martin Page 96 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 Appendix 8 Putting Zeta to work Bernoulli numbers In mathematics the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory The values of the first few Bernoulli numbers are BO 1 B1 1 2 B2 1 6 B3 0 B4 1 30 B5 0 B6 1 42 B7 0 B8 1 30 If the convention B1 1 2 is used this sequence is also Known as the first Bernoulli numbers with the convention B1 1 2 is known as the second Bernoulli numbers Except for this one difference the first and second Bernoulli numbers agree Since Bn 0 for all odd n gt 1 and many formulas only involve even index Bernoulli numbers some authors write Bn instead of B2n The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jakob Bernoulli after whom they are named and independently by Japanese mathematician Seki Kowa Seki s discovery was posthumously published in 1712 in his work Katsuyo Sampo Bernoulli s also posthumously in his Ars
104. al programmers we are told adapted the already existing code form the HP 15C but they had to overcome a couple of real challenges to port it smoothly into the 41 platform Possibly the 15C also required similar trickery but I don t know its internal architecture so I can t say The first striking thing is of course being able to return to an MCODE code stream after executing a user code program FOCAL which calculates f x That alone can leave you thinking as it did to me suspecting the hows and the abouts being explained somehow by SIRTN and SILOOP the two auxiliary functions written for this purpose Yet how did they exactly work We need to understand the buffer 14 paradigm before we can answer this Yes there s the question of Buffer 14 the dedicated buffer in the Advantage that exhibits a rather idiosyncratic temperament contrary to all other modules the Advantage seems to be on a search and destroy mission with the apparent aim to kill any previous existence of the buffer judging by the polling events CALC_OFF and IO_SRVC Equally intriguing is the location of said buffer which is situated while it s allowed to exist below the Key assignments area and not above it as it s the normal way This fact conflicts with the OS routines that manage the I O area like PKIOAS and others and would create real havoc if it weren t because the Advantage manages the buffer dynamically creating it on the fly just
105. allurgist you are testing a new additive to an alloy This additive influences the strength of the alloy and this influence varies according to the percentage of the additive in the alloy In tensile strength experiments you measured failure points in wires of different additive percentages The following table of scaled data was produced Additive Failure Scale Additive Failure Scale 0 1 00 4 0 4 165 0 5 1 131 4 5 4 629 1 0 1 079 5 0 4 811 1 5 1 354 5 5 5 577 2 0 1 382 6 0 5 391 2 5 2 350 6 5 4 735 3 0 3 767 7 0 4 618 3 5 3 945 Input the data and find the best fit Use the failure variables as the values of y and the percentages as the x s Then based on this best fit curve find the scaled failure point for a wire with an additive percentage of 4 3 Solution The best fit is the NORMAL curve or NORMAL distribution the equation is y a e x b 2 c The values for a b and c are 5 173 6 234 and 19 175 respectively Once you have determined this to be the curve to get the y value a t x 4 3 press 4 3 E C That value is 4 256 c Angel M Martin Page 64 of 167 December 2014 SandMath_44 Manual Revision 3x3 A few more Geometry Functions Function Description Author PP2 2D Distance between 2 points Angel Martin VMOD 3D Vector Module Angel Martin VXA 3D Cross Product Angel Martin V A 3D Dot Product Angel Martin HERON Area of a Triangle JM Baillard
106. ammer symbol by cn D x The usual factorial n Is related to the rising factorial by n he Whereas for the falling factorial the expression is n n p Examples Calculate the rising factorial for n 7 x 4 and the falling factorial for n 7 x 7 7 ENTER 4 XF POCH gt 604 800 0000 7 ENTER 7 CHS 2F POCH 7 XEQ CHSYX gt 5 040 000000 c Angel M Martin Page 69 of 167 December 2014 SandMath_44 Manual Revision 3x3 Multifactorial Superfactorial and Hyperfactorial MFCT SFCT HFCT This section covers the main extensions and generalizations of the factorial There are different ways to expand the definition depending on the actual sequences of numbers used in the calculation The double factorial of a positive integer n is a generalization of the usual factorial n defined by ne an 2 5 3 1 va gt Oodd nlledn n 2 6 4 2 n gt OQOeven d n 0 Even though the formulas for the odd and even double factorials can be easily combined into nl n 2z i ien The double factorial is a special case of the multifactorial which uses the same formula but with different steps subtracting k instead of 2 from the original number thus n n n 1 a 2 2 n n n 2 n 4 nl t on n 3 m 6 where the products run through positive integers Obviously for k 1 we have the standard FACT O
107. ample2 Two dimensional data Let M be the 2x3 matrix defined by 1 2 4 3 5 6 Repeating the same process as above Note that for two dimensional cases the elements are introduced in column order 1 006 XFS INPUTS gt 4_ 1 ENTER 3 ENTER 2 ENTER 5 ENTER 4 ENTER 6 R S 2 003 STO_00 0 F DHT gt 8 573214097 value of b1 7 012 XF OUT gt RO7 to R12 listed sequentially as show below B 8 5732 2 8978 0 7765 rounded to 4 decimals 2 8577 0 1494 0 5577 If you copy R07 R12 to RO1 R06 and press 0 XFS DHT again you ll get the elements of the original matrix M with a mean error of about 3 E 9 c Angel M Martin Page 121 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 A driver program for DHT DSHT Revision N of the SandMath includes many small enhancements and improvements in several areas as well as DSHT an all new driver program for DHT which has been moved to the secondary FAT With DSHT the data entry is automated with prompts under program control so the user needs not to remember the parametes before hand The dimension can be either an integer number or a 2 column matrix There s no need to use a 1 for the one dimensional case It s however important to remember that for 2 dimensional data the element entry and output are made in COLUM order as opposed to other matrix applications DS
108. an be expressed in terms of the Generalized Hypergeometric function 1F2 which is not the Gauss Hypergeometric function 2F1 This is the expression used in the SandMath implementation E z 22412 V2nr a 3 2 in other words referred to the Rationalized Generalized Hypergeometric function which with such a long name it definitely must be a formidable function but it s just the same divided by Gamma H z 1 Fo 1 3 2 a 3 2 z 4 H x x 2 F 1 3 2 n 3 2 x14 L x 3 2 1 F 1 3 2 n 3 2 x 4 Examples Compute H 1 2 3 4 and L 1 2 3 4 1 2 ENTER 3 4 F HNX gt H 1 2 3 4 1 113372657 1 2 ENTER 3 4 F LNX gt L 1 2 3 4 4 649129471 Alternatively M81 H H for HNX and lal H SHIFT H for LNX c ngel M Martin Page 138 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Lommel functions The Lommel differential equation is an inhomogeneous form of the Bessel differential equation Feu Two solutions are given by the Lommel functions su v z and Su v z introduced by Eugen von Lommel 1880 Sulz Zr Y 2 z I z dz JL z a ale dz Syu 2 Suu 2 A OMG c08 u 2 2 2 od y dy 2 p y 2PH where Jv z is a Bessel function of the first kind and Yv z a Bessel function of the second kind Using the Generalized Hypergeometric function the expressions for s1 m n x is sP a
109. and the function MCODE are included in their entirety With them you can make fast and convenient curve fitting calculations to 16 different curve types choosing the best fit amongst them based on the correlation coefficients obtained The following paragraphs are extracted from the AECROM Users manual by itself an excellent work perfect complement to the world class programming that went into the Module They should provide enough information to get you going It s only after some consideration that I decided to include them both begging forgiveness and asking permission you re encouraged to consult the original manual available at Atto www hp41 org LibvView cfm Command View amp ItemID 581 The AECROM Curve Fitter With the AECROM program CURVE you can fit an unlimited number of data pairs x y to sixteen different curves CURVE will automatically determine which of the sixteen curves best fits the supplied data or you can specify the curve to fit Once the data has been fit to a curve CURVE will return predicted y values for x values you supply A menu driven program The program CURVE is menu driven that is it redefines the meanings of the top row of keys and those new meanings can be shown in the display above the keys In order to use CURVE you must set your calculator to USER mode press USER to turn on the word USER in the display and you must clear any global assignments on the top row of keys
110. as follows 0i LBL FF 05 PI 02 ENTER 06 03 X 2 07 1 X 04 08 END Store FF in ALPHA 2 in R11 and 5 in R12 interval begin and end points Type 10 XF CHBCF gt 18 028 after 5 minutes on a normal speed 41 This is the control word that indicates that the coefficients are stored as follows R18 c0 0 061130486 R24 c6 0 000001210 R19 cl 0 038060320 R25 c7 0 000000410 R20 c2 0 008422922 R26 c8 0 000000170 R21 c3 0 001522665 R27 c9 0 000000042 R22 c4 0 000227407 R28 c10 0 000000008 R23 c5 0 000025750 Note you can use the program OUT also included in the secondary FAT to review and output those results Use SF 21 if you want the display to halt after each value resuming with R S Example Let s now evaluate f 3 amp f 3 using CHBAP and flag 01 to select the case First we set flag 06 to bypass the data entry prompts then we store the control word bbb eee in R13 CF 01 3 XEQ CHBAP gt 0 066043252 SF 01 3 XEQ CHBAP gt 0 030531990 If you re missing automation you ll be glad to know there is some Rather than a different driver program CHBAP doubles as one when flag 06 is clear triggering the data entry prompts which drive the data entry At the prompt a b N enter the three values separated by ENTER then R S After a long time the coefficients are calculated and the program prompts X your chance to input
111. avy lifting is thus performed by CRG which together with CRJ do all the hard work in the calculation for the Elliptic Integrals of first second and third kinds The figure below shows the first and third kinds in comparison Elliptic Of the First Kind Elliptic Of the Third Kind n O oe t 25 n 0 5 1 25 neO 25 wer Hk pei r f i Er d t ohi elit Nk r eint This is a perhaps a good moment to define the Carlson symmetric forms The Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced They are a modern alternative to the Legendre forms The Legendre forms may be expressed in terms of the Carlson forms and vice versa c Angel M Martin Page 124 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 The Carlson Symmetric Elliptic integrals of the First and Third kinds are defined as dt E ay t y 2 Yp Ra e y 2 p t p t a t y t4 z and are the SandMath functions that calculate their values located in the auxiliary FAT Their arguments x y z are expected to be in the corresponding stack registers and the result will be placed in X Reg upon completion Re x y 2 The term symmetric refers to the fact that in contrast to the Legendre forms these functions are unchanged by the exchange of certain of their arguments The value of Rr is the same for any permutation of its arguments and the value of
112. breakpoints capability and MCODE trace console are a godsend to programmers See www hp41 org SandMath Overlays 2009 2014 Lujan Garcia Published under the GNU software licence agreement c Angel M Martin Page 2 of 167 December 2014 SandMath_44 Manual Revision 3x3 Tab e of Lontents Revision dAXdtt 0 Preamble to Revision 3x3 7 Configuring the SandMath 3x3 revision N 8 1 Introduction Function Launchers and Mass key assignments 9 Used Conventions and disclaimers 10 Getting Started Accessing the functions 11 Main and Dedicated Launchers the Overlay 12 Appendix 0 The Hyper shift keyboard 13 Appendix 1 Last Function and Launcher Maps 15 Function index at a glance 16 2 Lower Page Functions in Detail 2 1 SandMath44 Group Elementary Math functions 20 Number Displaying and Coordinate conversions 24 Base Conversions 26 First Second and Third degree Equations 27 Appendix 2 FOCAL program listing 30 Additional Test Functions rounded and otherwise 31 2 2 Fractions Calculator Fraction Arithmetic and displaying 32 2 3 Hyperbolic Functions Direct and Indirect Hyperbolics 34 Errors and Examples 35 2 4 Recall Math Individual RCL Math functions 36 RCL Launcher the Total Rekall 37 Appendix 3 A trip down memory lane 38 2 5 Geometric and TVM Solvers Introduction yet a new Launcher 40 Triangles Circles and Slopes 41 Implementation Details 45 The Time Value of Money So
113. d M 2 3 7 2 ENTER 3 ENTER PI CHS F KUMR gt M 2 3 x 0 166374562 2 ENTER 3 ENTER PI FL gt M 2 3 n 10 24518011 Alternatively M81 H SHIFT K using the main launcher instead c Angel M Martin Page 142 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Associated Legendre Functions In mathematics the Legendre functions P A Q A and associated Legendre functions Pu A and Qu A are generalizations of Legendre polynomials to non integer degree Associated Legendre functions are solutions of the Legendre equation 1 2 y 2ar AA y 7 where the complex numbers A and u are called the degree and order of the associated Legendre functions respectively Legendre polynomials are the associated Legendre functions of order y 0 These functions may actually be defined for general complex parameters and argument 1 fee al l z p E ee Eep P Mp i z oF A A 1 1 u 5 for l z lt 2 The figures below give a couple of graphical plots for the Legendre Polynomials ast egerdre Pornervals a ooa _ 2 M a amm o D _ ba pl s ha pael wa pag P x jpe a teow ee Pi v i i j i i 4 t ah A a E t i n a i j Pix Px P x 4 3 b N gt C N p pe p ia ee ee ae EN D 9 LO i REGISTERS ROO thru RO5 FLAGS
114. d positive integer n 2k 1 k gt 1 itis k 2k 1 2i 1 i l A common related notation is to use multiple exclamation points to denote a multifactorial the product of integers in steps of two n three n or more The double factorial is the most commonly used variant but one can similarly define the triple factorial n and so on One can define the k th factorial denoted by n recursively for non negative integers as wie as ifO lt n lt k E n n k I ifn gt k The figures below show the plots for X right a comparison with log abs gamma red versus log abs doublegamma green left eS ee ee H 14 i 1 t t 4 n Using the Logarithm is helpful to deal with large arguments as these functions go beyond the calculator numeric range very quickly Also ran out of space in the module to have more than one function on this subject thus LOGMF was chosen given its more general purpose character The implementation is thru an all MCODE function yet execution times may be large depending on the arguments LOGMF may also be used to compute factorials use n 1 and then E X on the result Obviously the accuracy won t be the greatest but it s a reasonable compromise Output a a LGMF x Examples 2 ENTER 100 F LOGMF gt Log 100 79 53457468 999 ENTER 123456 XFL gt Log 123456 578 0564932 c Angel M
115. d the second kind terms This assumes that the convergence occurs at comparable number of terms which fortunately is the case given their relative fast convergence Note that in order to obtain similar expressions for both Yn and Kn and so getting simpler program code we can re write Kn as follows 1 2 K x 2 fy Ln x 2 I x Df f n x D L C1 gx n x Dedicated MCODE Functions To further decrease the execution time of the programs two dedicated functions have been written implemented as MCODE routines as follows Function Flag 00 Clear Flag 00 Set BS x U n x K 0 1 2 where Uk Ux x 2 42 k k n BS2 DV f n x k 0 1 2 or 2 n x K 0 1 n 1 The first function BS is used equally in the calculation of the first kind and the second kind of non integer orders Integer Non integer JBS BS IBS ie 2x BS2 KBS As it was said before the summation will continue until the contribution of the newer term is negligible to the total sum value All calculations are done using the full 13 digit precision of the calculator No rounding is made until the final comparison which is done on 10 digit values From the definition above it s clear that BS coincides with either Jn x or In x depending on the status of the CPU flag 9 and for positive orders The functions JBS and IBS are just MCODE extensions of BS that set up the specific settings prior to invoking it and dependi
116. dule that evaluates your function f x This program must be named with a global label Your program can take advantage of the fact that FINTG fills the stack with its current estimate of x each time it calls your program e You then enter the two limits a and b into the X and Y registers Lastly put the name of your program that evaluates the funct ion into the Alpha register and then XEQ FINTG c Angel M Martin Page 164 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 When the program stops and the calculator displays the integral the contents of the stack are T the lower limit of the integrat ion a Z the upper limit of the integration b Y the uncertainty of the approximation of the integral X the approximation of the integral this is what is shown in the display Accuracy of FINTG Since the calculator cannot compute the value of an integral exactly it approximates it The accuracy of this approximation depends on the accuracy of the integrand s function itself as calculated by your program While in tegrals of functions with certain characteristics such as spikes or rapid oscillations might be calculated inaccurately these functions are rare This is affected by round off error in the calculator and the accuracy of empirical constants To specify the accuracy of the func tion set the display format FIX n SCI n or ENG n so that n is no greater than the number of decimal digits tha
117. dule FAT if the sub function name is not found within the multi function group so the user needn t remember where a specific function sought for was located In fact XF will also find a function from any other plugged in module in the system even outside of the SandMath module Main Launcher and Dedicated Secondary Launchers The Module s main launcher is DAAI Think of it as the trunk from which all the other launchers stem providing the branches for the different functions in more or less direct number of keystrokes With a well thought out logic in the functions arrangement then it s much easier to remember a particular function placement even if its exact name or spelling isn t know without having to type it or being assigned to any key Despite its unassuming character the prompt provides direct access to many functions Just press the appropriate key to launch them using the SandMath Overly as visual guide the individual functions are printed in BLUE with their names set aside of the corresponding key They become active when the XF _ prompt is in the display r a USER O14 4 USER Hig 4 ALFHA d or Besides providing direct access to the most common Special Functions XFL will also trigger the dedicated function launchers for other groups Think of these groupings as secondary menus and you ll have a good idea of their intended use The following keys activate the secondary menus A act
118. e 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 o s a a O O O O O O O O O O O O O O O 100 101 X lt 0 is it negative SF 02 negative order ABS remember this fact ____ STO 01 abs n STO 00 reset counter STO 02 and partial sum RDN Ses skip if n 0_ Lnn IGTO 06 E C ooo o ooo selects B2 sts SPF CH X gk n x k 0 1 n 1 BS2 CHS STO 02 RCL 01 abs n RCL 03 x 2 selects B1 L fk n x k 0 1 2 partial result showing off ST X 3 ST 02 partial result RCL 02 FS 01 is it YBS SL fon a SS a ae ee ae ee GTO 04 yes cut the chase meee eee i ee l c Angel M Martin Page 94 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 3 3 9 Riemann Zeta function Perhaps one of the most studied functions in mathematics it owes its popularity to its deep rooted connections with prime numbers theory Not having an easy approximation to work with its implementation on the 41 will be a bit of a challenge mainly due to the very slow convergence of the series representation used to program it Be assured that this numeric calculation won t help you prove the Riemann hypothesis and collect the 1M prize so adjust your expectations accordingly The Riemann zeta function is a function of complex argument s that analytically continues the sum of the infinite series A mo ygt 5 gt L Rs g
119. e checked by integration by parts The Airy function of the second kind denoted Bi x is defined as the solution with the same amplitude of oscillation as Ai x as x goes to co which differs in phase by n 2 Bi a lexp t at sin t zt dt The expressions used to program them are again based on HGF as follows Ai x 3 T 2 3 oF 1 2 3 x 9 x 37 T1 3 oF 1 4 3 x 9 Bi x 37 T 2 3 oF y 2 3 x 9 x 3 8 T73 oF 1 4 3 7 9 The figure below shows Ai and Bi plotted for 15 lt x lt 5 1 00 0 50 REGISTERS ROO thru R04 FLAGS mene Stack Example 0 4 EF AIRY gt Ai 0 4 0 254742355 or M81 O Y X lt gt Y gt Bi 0 4 0 801773001 c Angel M Martin Page 132 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Fresnel Integrals CSX Fresnel integrals S x and C x are two transcendental functions named after Augustin Jean Fresnel that are used in optics They arise in the description of near field Fresnel diffraction phenomena and are defined through the following integral representations al a S x J sin t dt C x J cos t2 dt E The function CSX will calculate both S x and C x for the argument in X returning the results in Y and X respectively It is a short FOCAL program that uses yes you guessed it the Generalized Hyper geometric function acc
120. e definition range of the function It has been implemented on the SandMath using the formulas derived from the called Gauss digamma theorem although further simplified in the following algorithm 1 4 1 1 W x log z 5 fara t To0gt 2528 P programmed as u 2 u 2 20 1 21 u 2 1 10Ju 2 1 12 Lnu u 2 The implementation also makes use of the analytic continuation to take it to arguments greater than 9 same as it s done for LogGamma using the following recurrence relation to relate it to smaller values which logically can be applied for negative arguments as well as required Wir 1 V x 7 Examples PI XEQ PSI gt Psi x 0 977213308 1 XEQ PSI gt Psi 1 0 577215665 opposite of Euler s constant 7 28 XEQ PSI gt Psi 7 28 4 651194216 1234 5 XEQ PSI gt Psi 1234 5 7 118826276 c Angel M Martin Page 83 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 The Polygamma Function PSIN In mathematics the polygamma function of order m is a meromorphic function on C and defined as the m 1 th derivative of the logarithm of the gamma function are a j co ye z a z InI z 7 dom qzmtrl For m o the expression holds where w 0 z is the digamma function and z is the gamma function They are holomorph on C NO At all the negative integers these polygamma functions have a pole of order m 1 The func
121. e in Y George Eldridge D gt H Decimal to Hex William Graham H gt D Hex to Decimal William Graham T gt BS__ Base Ten to Base prompting version Ken Emery e The first two are FOCAL programs taken from the PPC ROM They are the generic base b to from Decimal conversions The Direct conversion expects the base in Y and the decimal number in X returning the base b result in Alpha The inverse function uses the string in Alpha and the base in X as arguments You can chain them to end with the same decimal number after the two executions Ten to Base is the MCODE equivalent to D gt BS much faster and more elegant due to its prompt where in RUN mode you input the destination base The result is shown in the display and also left in ALPHA so it could be also used by BS gt D once the base is in X Note that the original argument decimal value is left in X unaltered so you can use T gt BS repeated times changing the base to see the results in multiple bases without having to re enter the decimal value T gt BS is programmable In PRGM mode the prompt is ignored and the base is expected to be in the Y register much the same as its FOCAL counterpart D gt BS Obviously using zero or one for the base will result in DATA ERROR The maximum base allowed is 36 and the custom error message BASE gt 36 will be shown it that s exceeded note that larger bases would require characters beyond Z The maximum decimal val
122. e next previous section depending on SHIFT status lt back arrow cancels the catalog One limitation of the sub functions scheme that you ll soon realize is that contrary to the standard functions they cannot be assigned to a key for the USER keyboard Typing the full name or entering its index at the XFL prompt is always required This can become annoying if you want to repeatedly execute a given sub function The LAST Function implementation certainly reduces this issue for repeat executions of the last sub function called without a dedicated key assignment required Another work around consists of writing a micro FOCAL program with just the sub function as a single pair of program lines and then assign it to the key of choice Not perfect but it works c Angel M Martin Page 19 of 167 December 2014 SandMath_44 Manual Revision 3x3 2 Lower Page Functions in detail The following sections of this document describe the usage and utilization of the functions included in the SandMath_44 Module While some are very intuitive to use others require a little elaboration as to their input parameters or control options which should be covered here Reference to the original author or publication is always given for additional information that can and should also be consulted 2 4 SANIMATH GROUP 2 1 1 Elementary Math functions Even the most complex project has its basis simple enough but reliable so that
123. ea account for the following register F addresses iia The Function Rej O5 The Indirect Address Register Ros 10 0000 The Desired Register Recalled into the reqister Rio 2 3400 Storage Register Arithmetic Arithmetic can be performed upon the contents of all storage registers by executing followed by the arithmetic function followed in tum by the register address For exarnple Opertion Result O1 Number tn X register is added to the contents of register Eoi and the sum is placed inte Roy The display execution form of this is sT 02 Number in A register is subtracted from the contents of register Roo and the difference is placed inte Egos The display execution form of this is st 1E Number in resister is multiplhed by the contents of register Kos and the product is placed inte Ros The display execution form of this is BT x 04 Number in Rog is dmided by the number in the M register and the quotient is placed into Rog The display execution form of this is st c Angel M Martin Page 38 of 167 December 2014 SandMath_44 Manual Revision 3x3 Primary Data Storage Registers The standard HP 41C has 63 registers that can be allocated to data storage or program memmory in axy combination As you add HP memory Modules up to four the total number of registers can increase to 319 64 registers for each memory module When allocated data storage registers numbered Ro through Roo are
124. el M Martin Page 56 of 167 December 2014 SandMath_44 Manual Revision 3x3 Cumulative Probability Function and its Inverse P Since revision 2x2 the SandMath includes a few functions to calculate the cumulative probability and its inverse both for the standard and general cases any standard deviation and mean of a Normal Distribution The direct function is CDF which basically employs the Error function erf with the appropriate adjustment factors as described in the previous example The inverse function is ICPF which benefits from a native implementation of the inverse Error Function ierf more about this later 01 LBL ICPE The expression used is conversely 02 ST x l 3 E x u o sqr 2 ierf 2 P x u 0 1 IERF where x ICPF and P x u o CPF programmed as shown in the listing at the left a very simple FOCAL program that directly relies on IERF to do all the work Note that the stack is expected to contained the three parameters defining the distribution 04 05 06 07 SORT 0E 09 10 11 END Both CPF and ICPF require the mean in Z the standard deviation in Y and the argument in X You can use the fact that they are inverse from each other to verify the results The third function is QNTL which basically is a particular case for ICPF for the Standard Normal with o 1 and u 0 It is calculated with an iterative approach using the Halley method to converge to the r
125. en FROOT modifies the secant method to ensure that c bl lt 100 la bl This is especially important because it also reduces the tendency for the secant method to go astray when rounding error becomes significant near a root If FROOT has already found values a and b such that f a and f b have opposite signs it modifies the secant method to ensure that c always lies within the interval containing the sign change This guarantees that the search interval decreases with each iteration eventually finding a root If this does not yield a root FROOT fits a parabola through the function values at a b and c and finds the value d at the parabola s maximum or minimum The search continues using the secant method replacing a with d If three successive parabolic fits yield no root or d b the calculator displays NO In the X and Z registers remain b and f b respectively with a or c in the Y register At this point you could resume the search where it left off direct the search elsewhere decide that f b is negligible so that x bisa root transform the equation into another equation easier to solve or conclude that no root exists Instructions In calculating roots FROOT repeatedly calls up and executes a program that you write for evaluating f x You must also provide FROOT with two initial estimates for x providing a range for it to begin its sea rch for the root c Angel M Martin Page 162 of 167 December 201
126. en if the M code length more than doubles the equivalent FOCAL program there s a lot to say about how efficient FOCAL code is See JM Baillard s page for the FOCAL code at http hp41programs yolasite com lobachevsky php Because the expression programmed is truncated to 8 terms the Bernoulli numbers have been hard coded in the code so there s no need to use ZETA as subroutine The accuracy of this approach appears to be good enough within the 9 decimal digits resolution of the machine CLAUS uses a more general approach actually calculating as many terms as needed until their contribution to the partial sum is negligible It is however limited to arguments in the interval 0 27 Note that CLAUS will read the input in the set angular mode but it will change it to DEG The code is also taken from JM Baillard s page posted at http hp41programs yolasite com Causen php Examples Calculate both Clausen and Lobachevsky s functions for the three arguments given in the table below and compare their relative results Use the LOBACH result as reference obtaining the adjusted value for Cl2 2x 2 using CLAUS i e Cl2 2x 2 A x x 2X 2x 2 Lobach x 0 33831387 0 338313869 delta 2 95584E 09 0 330783505 0 330783505 out of range 0 0 445441712 n a As you can see for small arguments the results are identical this is because for those cases calculating ZETA is not required for CLAUS either thus b
127. ero then that series is also called a Maclaurin series named after the Scottish mathematician Colin Maclaurin who made extensive use of this special case of Taylor series in the 18th century It is common practice to approximate a function 7 by using a finite number of terms of its Taylor a series Taylor s theorem gives quantitative estimates on the error in this approximation Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial The Taylor series of a function is the limit of that 1 function s Taylor polynomials provided that the limit exists A function may not be equal to its Taylor series even if its Taylor series converges 2 at every point A function that is equal to its Taylor series in an open interval or a disc in the complex plane is Known as an analytic function The Taylor series of a real or complex valued s function f x that is infinitely differentiable in a 10 neighborhood of a real or complex number a is o 2 6 2 02 4 the power series As the degree of the Taylor polynomial rises it j _ fn approaches the correct function This image shows sin T f o a vl and Taylor approximations polynomials of degree 1 2 7 ni x _ 9 11 and n 0 where n denotes the factorial of n and f a denotes the n th derivative of f evaluated at the point a The derivative of order zero f is defined to be f itself and x a 0 and O are bot
128. es EI 7 cI ISI j The first sub section covers the Exponential Logarithmic Trigonometric and Hyperbolic integrals They re all calculated using their expressions using the Generalized Hyper geometric function in a clear demonstration of the usefulness or the adopted approach For real nonzero values of x the exponential integral Ei x is defined as E t lt dt co ft Ei x Integrating the Taylor series for exp t and extracting the logarithmic singularity we can derive the following series representation for real values ak a Fife y ln a2 zr 0 k k k 1 where we substitute the series by its Hyper Geometric representation L x k kk x F 1 1 2 2 x The logarithmic integral has an integral representation defined for all positive real numbers by the definite integral z d o Int li x The function li x is related to the exponential integral Ei x via the equation li a Ei lne which is the one used to program it in the SandMath module Examples 1 4 XEQ EI gt Ei 1 4 3 007207463 or bg R 1 4 XFL LI gt Li 1 4 0 144991005 c Angel M Martin Page 101 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 LI is the Logarithm Integral also a quick application of the EI function using the formula Li x Ei Un x Note how LI starts as a MCODE functions that transfers into the FOCAL code calculating EI so strictly speaki
129. ess to 10 functions using the keys in the top row directly below the function symbol The Fractions functions are encircled by a red line on the overlay at the bottom and left rows of the keyboard They include the fraction math plus a fraction Viewer and fraction Integer tests The Hankel and Carlson aunchers present their choices in their prompts and will be covered in a dedicated section later in the manual c Angel M Martin Page 12 of 167 December 2014 SandMath_44 Manual Revision 3x3 Appendix 0 The Hyper SHIFT keyboard HYP The available room in the auxiliary banks has proven useful to extend the HYP launcher beyond the strictly hyperbolic functions Presing the SHIFT key twice activates the hyper SHIFT mode and then repeat pressings of SHIFT will toggle between the normal and hyper Shift modes The hyper SHIFT extensions are mainly about adding a SHIFTED HYP mode with a full keyboard of assignments like those for functions assigned by MKEYS to the HYP prompt choices The picture below shows the function map for the HYP and SHIFT HYP launchers HYP As it s now customary the SHIFT key will toggle between these two and the back arrow will return to the main XFL launcher Note that this arrangement includes both main and sub functions in the same second layer keyboard This is a very convenient way to circumvent the inability to directly assign sub functions to keys Later on i
130. esult Obviously the results should be equivalent to ICPF with the standard parameters inputted Halley s method uses the following expression to calculate the successive approximations to the root ooo e f tn tanti n T arer wt rr ref A 2f an Flan f En where our function in this case is f x CPF x Value thus we take advantage of the fact f x PDF and f x k f x thus the above expression gets simplified considerably Examples Which argument yields a probability of 75 for a Standard Normal distribution a Using ICPF 0 ENTER 1 ENTER 0 75 EF ICPF gt 0 674489750 b Using QNTL 0 75 EF QNTL gt 0 674489750 What is the cumulative probability for the argument obtained in the previous example Type 0 ENTER 1 RCL Z XFS CPF gt 0 750000000 The accuracy is quite good also holding up well across the entire range of values for both ICPF and QNTL thanks to the thorough implementation of IERF and to the iterative Halley approach employed Execution speed is much faster for ICPF than for QNTL but this one is more accurate for arguments in the vicinity of 1 c Angel M Martin Page 57 of 167 December 2014 SandMath_44 Manual Revision 3x3 Poisson Standard Distribution is another Statistical function which calculates the Poisson Standard Distribution In probability theory and statistics the Poisson distribution is a discrete pr
131. et TE Af n l k 7 A Nothing short of magical if you ask me what I d call going out with a bang A few examples note the convenient usage of the LASTF feature for repeat executions of the same function 1 4 ENTER 3 ZF ITJ gt 0 3 J 1 4 x dx 1 049262785 1 4 ENTER 3 ZF ITI gt 0 3 1 1 4 x dx 2 918753200 1 ENTER 3 ZF ITJ gt 0 3 J 1 x dx 1 260051955 0 ENTER 10 FL gt 0 10 J 0 x dx 1 067011304 50 ENTER 30 ZFL gt 0 30 J 50 x dx 1 478729947 E 8 c Angel M Martin Page 156 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Appendix 11 Looking for Zeros Once again we re just connecting the dots here s a brute force crude implementation of a root finder for Bessel functions made possible once the major task i e calculating the function value is reduced to a single MCODE function The following trivial looking program it really can t get any simpler uses SOLVE within the Advantage Pack or FROOT in the SandMath no less Starting with zero obvious guess values are the previous root and the root incremented by one Successive repetitions will unearth all those roots just make sure you have the turbo mode enabled on V41 or equivalent emulator Enjoy The first few roots j n k of the Bessel function Jn x are given in the following table for small nonnegative integer values of n and k See also h
132. ettings determine the accuracy of the conversions even if it s not obviously seen This has the advantage that the result is always reduced to the best possible fit For instance when calculating 2 4 plus 18 24 in program mode with the four values in the stack the result will be 120 in Y and 96 in X thus 120 96 However on RUN mode or SST ing the program will show the reduced fraction A good way to check that the result is expressed in irreducible form is pressing GCD verifying that the result is indeed 1 try it out if you re curious If you want to see the reduced result from a program execution you ll need to add program steps to perform the division and add a conversion to fraction after the fraction math operation step The code Snippet below describes this see lines 10 and 11 01 LBL TEST 02 2 03 ENTER 04 4 05 ENTER 06 18 07 ENTER 08 24 09 F D gt F 12 END and are two more test functions which criteria is the integer or fractional nature of the number in X Having them available comes very handy for decision branching in FOCAL programs The Fractions section of the module is the natural placement for them The answer is YES NO depending on whether the condition is true or false In program mode the following line is skipped it the test is false Note Make sure that revision N or higher of the Library 4 module is installed c Angel M Martin Page 33 of 167 December
133. f you don t care about the High Level Math Special Functions and Statistics sections Think however that the FAT entries for the Function launchers are in the upper page so they ll be gone as well if you use the reduced foot print version effective 4k only of the SandMath l Function Launchers HP Advantage Solve High Level Math Stats Curve Fitting 8 Integrate Lower SandMath_44 FRC HYP RCL Math TVM AECROM Geometry Solvers Note that it is not possible to do it the other way around that is plugging only the upper page of the module will be dysfunctional for the most part and likely to freeze the calculator do not attempt c Angel M Martin Page 8 of 167 December 2014 SandMath_44 Manual Revision 3x3 SandMath_44 Module Revision 3x3 Math Extensions for the HP 41 System 1 Introduction Simply put here s the ultimate compilation of MCODE Math functions and FOCAL applications to extend the native function set of the HP 41 system At this point in time way over 30 years after the machine s launch it s more than likely not realistic to expect them to be profusely employed in FOCAL programs anymore yet they ve been included for either intrinsic interest read challenging MCODE or difficult to realize or because of their inherent value for those math oriented folks This module is a record breaking 24k implementation arranged in a dual bank switched configuration The lower pages include more ge
134. function Consider that the OS routine ASRCH is used to locate them both and it s a sequential search first RAM for LBL f x then ROM and there may be several plugged in You no doubt have noticed that in the SandMath there are only three functions related to this FROOT not so fruity FINTG and FLOOP the fluppy one which sure correspond to SOLVE INTEG and SILOOP But what about the whereabouts of SIRTN No it s not one of the section headers already used for other purposes and nor is it in the secondary FAT that d be impossible to pull off fortunately this is one of the added pluses of going bank switched SIRTN is in the FAT of bank 3 all by itself so it ll be found while ASRCH is called from the MCODE all that extra work payed off and so we saved a precious FAT entry in the main FAT All in all a stroke of genious with all the ingredients of a work of art if you ask me So I feel especially glad to finally have cracked this nut and managed to include it in the SandMath the yellow ribbon around the box Hope this dissertation wasn t too boring and that you enjoy it at least as much as I did working on it a ANE as created by FROOT ifs eee oes ee oS USER FAD PRGM as created by FINTG To see this by yourself insert function BFCAT in the LBL f x then stop the enumeration c Angel M Martin Page 161 of 167 December 2014 SandMath_ 44 Manual Revision_3x3
135. functions used are related to the harmonic analysis though Here s an nteresting one the Christmas Tree function and its Fourier representation for different number of terms c Angel M Martin Page 118 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Appendix 10 Fourier Coefficients by brute force Since the coefficients are basically integrals of the functions combined with trigonometric functions nothing besides common sense stops us from using INTEG to calculate them This brute force approach is just a work around considering the time requirements for the execution but it can be useful to calcuate a single term randomly as opposed to the sequential approach used by FFOUR So here the idea is to calculate the n th Coefficient independently which responds to the following definig equation laaa a ah Notice that the module SIROM a Linai diea Solve and Integrate ROM contains a ekdi not only FROOT and FINTG but ii also the program FOURN in its 2 an APPLIED section so you can use g k that 4k rom instead of the Advantage Pamer en that ll also save you from having to z l I type in the program 10 FIX 4 4 ST X Simply enter the information asked at the prompts including the precision desired number of decimal digits 11 IBLE 1 INDEX Pl 13 PROMPT FC 00 1d STO 02 COS function name and its chosen period 15 CF 00 F5
136. h defined to be 1 In the case that a 0 the series is also called a Maclaurin series Establishing an analogy with the Chebyshev approximation one would notice that here the approximation is made using certain coefficients affecting the Taylor polynomials terms which are simpler versions than Chebyshev s basically x n for the McLaurin case Thus it s intuitively understandable that a similarly good approximation i e with small enough error will require a larger number of Taylor terms to accomplish it Numerically however we re faced with the problem to calculate all the function derivatives of a given function This is approached using the Taylor Expansion using the notion of small increments of both the function and the argument to estimate the derivatives Let h be that small increment then The Taylor expansion of a function f in a point near the center is f ath f a hf a h 2 f a 2 h n f a n Given a function f x we seek approximations of a1 f a a2 f a 2 an f a n The SandMath implementation is a direct application of JM Baillard s method using a 10 degree polynomial to approximate the derivatives This theoretically provides perfect accuracy for polynomials of degree lt 10 but in practice due to roundoff errors the precision decreases as k increases and the estimation of a degree gt 10 is often very doubtful c Angel M Martin Page 114 of 167 December
137. has leaving five returns for a program that calls FINTG Note that FINTG cannot be used recursively calling itself If it is the program stops and displays RECURSION You can use FINTG with FROOT A routine that combines both FINTG and FROOT requires 32 available program registers to operate c Angel M Martin Page 165 of 167 December 2014 SandMath 44 Manual Revision _3x3 c Angel M Martin Page 166 of 167 December 2014 SandMath 44 Manual Revision _3x3 This completes this manual Don t forget to check Jean Marc Baillard extensive and authoritative references on the web despite its unassuming web site name located at http hp41programs yolasite com A treasure chest awaits you enjoy the ride c Angel M Martin Page 167 of 167 December 2014
138. he 41 CL make sure that the six ROM images are stored in the appropriate block locations in memory either SRAM or Flash and that you use the MAX control string in ALPHA for the execution of the PLUG command Hint the same conventions used for the Advantage Pack are applicable here place the 3 lower banks in the first three blocks within a sector and the 3 upper banks in the 5 6 and 7 blocks of the same sector thus leaving a gap of one block in between which can be used to store other modules without a conflict There are only a few new functions in revision 3x3 not included in the 2x2 but they alone account for two additional banks one on each page lower and upper The difference is therefore substantial despite the apparent sameness between revisions You may of course choose which one to use depending on which one is more convenient for your hardware The optimal setup is the 3x3 revision gathering the most benefits from the bank switching imlpementation on line code that doesn t take additional footprint Note that the LASTF features are only available in the 3x3 verison of the module Note for Advanced Users Even if the SandMath is a 24k module it is possible to configure only the first lower page as an independent bank switched 4k ROM This may be helpful when you need the upper port to become available for other modules like mapping the CL s MMU to another module temporarily or permanently i
139. he fast execution due to the Fc MCODE speed F 6 7 F g See the appendix in the next pages with both the actual code for PFCT Fg in the SandMath and for PRMF a more capable implementation using F 40 the Matrix functions from the HP Advantage to store the prime factor eT PRIME and their repetition indexes really the best way to present the results F a 12 SF 00 r 43 For that second case the function PF gt X restores the original argument F i from the matrix values Also function TOTNT is but a simple extension 15 using the same approach F 16 17 18 49 20 21 22 3 4 35 c ngel M Martin Page 59 of 167 December 2014 SandMath_44 Manual Revision 3x3 Appendix 4 Enhanced Prime factor decomposition The FOCAL programs listed below are for PFCT included in the SandMath and PRMF a more capable implementation that uses the Matrix functions from the HP Advantage or the SandMatrix ROM For sure a matrix is a much better place than the ALPHA register to hold that information as is done in PFCT The drawback is of course the execution speed much faster in PFCT e PRMEF stores all the different prime factors and their repetition indices in a n x 2 matrix The matrix is re dimensioned dynamically each time a new prime factor is found and the repetition index is incremented each time the same prime factor shows up e PF gt xX is the reverse function that restores the original number fro
140. he native MOD function through the expressions CEIL x x MOD x 1 and FLOOR x x MOD x 1 GEU is a new constant added to the HP 41 the Euler Mascheroni constant defined as the limiting difference between the harmonic series and the natural logarithm Th y lm gt gt In n moo k l The numerical value of this constant to 10 decimal places is y 0 5772156649 The stack lift is enabled allowing for normal RPN style calculations It appears in formulas to calculate the Y Psi function Digamma and the Bessel functions of 2 Kind amongst others e LOGYX is the base b Logarithm defined by the expression below where the base b is expected to be in register Y and the argument in register X l T los x log 2 Ogio _ og x log 19 0 logib Example verify that 5 55 Log 2 2 5 55 using 2 X 1 and LOGXY 5 55 2 X 1 1 2 x lt gt Y LOGYX gt 5 55000000 e Calculates the Remainder R and the Quotient Q of the Euclidean division between the numbers in the Y dividend and X divisor registers Q is returned to the Y registers and R is placed in the X register The general equation is Y Q X R where both Q and R are integers Note that if the dividend is smaller than the divisor the function will return zero for the quotient and the remainder will be the divisor itself Example calculate the remainder and quotient of dividing 27 over 4 27 ENTER
141. hers permits convenient access to these six functions without having to assign them to any key in USER mode Efficient usage of the keyboard which can double up for other launchers or the standard USER mode assignment if that s also required Combining the XFL and the SHIFT keys does the trick in a clean and logical way The formulas used are well known and don t require any special consideration to program 10 l T pT 5 sinh t 2 et A j cosh t a g H 2 2T et tanhzs gE Ei 1 5 H The SINH code is also used as a subroutine for the Digamma function 10 c ngel M Martin Page 34 of 167 December 2014 SandMath_44 Manual Revision 3x3 The direct functions are basically exponentials l o 3 whilst the inverses are basically logarithms arsinh x In x VE 1 Both cases are well covered with the mainframe arcoshz ln z 3 fr y gt 1 internal math routines without any need to worry on about singularities or special error handling 1 r artanha ln fel a 2 l gz For all hyperbolic functions the input value is expected in X and the return value will also be left in X The original argument is saved in LASTx No data registers are used Examples Complete the table below calculating the inverses of the results to compare them with the original arguments Use FIX 9 to see the complete decimal range 4 Z as Se TA hia S W 1 Jo E 1 bs IA SIN H COS TAN
142. herwise unchanged The initial x coordinate is saved in LastX so you can restore the original vector using X lt gt L Example Calculate the magnitude of vector V 1 3 4 Type 4 ENTER 3 ENTER 1 2F VMOD gt 5 099019514 Note the reversed order in the data entry sequence c Angel M Martin Page 65 of 167 December 2014 SandMath_44 Manual Revision 3x3 3D Dot and Cross products VXA Here the first vector is stored in stack registers X Y Z and the second in ALPHA registers M N O Obviously having an auxiliary function like ST lt gt A will come handy such is available in the AMC_OS X module You can also use STO M STO N and STO O for each coordinate VXA returns the result coordinates in the Stack replacing the initial values ALPHA is unchanged V A returns the result value in X the stack and ALPHA are unchanged Examples Calculate the cross and dot products between Vi 1 2 3 andV2 4 5 6 Type 6 ENTER 5 ENTER 4 ENTER and 3 ENTER 2 ENTER 1 ENTER F V A gt 32 00000000 Then use X lt gt L to restore the initial value and XF VXA gt 3 000000000 Use RDN twice to see the result vector is V1 x V2 3 6 3 Remember also that the cross product is not commutative thus V1xV2 V2xV1 Here s a way to check your results in WolframAlpha Attp www wolframalpha com input 7i cross product 1 2 3 id G G
143. i e 64 functions Now they have been pushed even further off to the secondary FAT used for the sub functions group Bessel Function Summed Functions by BS2 Flag 00 Flag 01 g gK n x Set rag C1 g n x CAF fdn aa Note also that for integer orders there are two infinite summations involved for the Bessel functions of the second kind as calculating the 1 kind function is also required This is done simultaneously within BS2 when user flag 02 is set as both series converge in very similar conditions i e with the Same number of terms Main functions IBS JBS KBS and YBS The first kind pair IBS and JBS are entirely written in MCODE including exception handling and special cases This is the only version known to the author of a full MCODE implementation on the 41 platform and it is however a good example of the capabilities of this machine No data registers are used but both the stack and the Alpha registers are used The number of terms required for the convergence is stored in register N upon termination The second kind pair KBS and YBS is implemented using a FOCAL driver program for the auxiliary functions BS and BS2 in the secondary FAT Notably more demanding than the previous two their expressions require additional calculations that exceed the reasonable MCODE capabilities Although they re not normally supposed to be used outside of the Bessel program BS and BS2 could be called indepe
144. icients to calculate in X After it s done the control word for the coefficients is returned to X in the form bbb eee and the coefficients are stored in the corresponding registers The execution time will be very long recommended to use TURBO mode on V41 or the CL CHBAP obtains the approximation of the function using these coefficients calculated by CHBCF It uses the status of flag 01 to control whether the function or its first derivative will be approximated with all the data stored in R11 R12 the coefficients and the argument x in X Obviously CHBAP requires that the coefficients have been calculated previously but repeated estimations can be calculated using the same coefficients with no further need to re calculate them every time Setting user flag 06 will allow you to call CHBAP directly which will do the coefficients calculations invoking CHBCF internally saving you the additional step c Angel M Martin Page 111 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 CHBCF is located in the secondary FAT thus you need to use XFL or alternatively XFL 080 You need to make sure that enough number of registers are available to store the results setting SIZE 19 n for n coefficients is a MCODE auxiliary function to expedite CHBAP execution It is also in the secondary FAT but typically you ll have no need to call it separately Example f x 1 x2 x zx and a b 2 5 Which is easily programmed
145. ience part as I had to move and shift large sections of code to accommodate for the demanding requirements of the FOCAL newcomers So there were just about 180 words in the main bank in total but what a tricky thing to adjust for on an already packed module with interdependencies across three banks and two pages Fortunately the bulk of the code is the MCODE for SOLVER itself which has been conveniently located in bank 3 of the lower page briefly coming up to the main one for the partial key sequence prompts and every time the execution ends to the FOCAL program I got partial vindication by consolidating the three FOCAL drivers into a single launcher which furthermore allowed the removal of the three FAT entries TRIA SARR and CIRC a definitive plus given that the FAT was already full You can explore those programs switching to PRGM mode during their execution in between entries standard procedure You probably have noticed that I changed the text presented by the different menus to a less busy version of the same Perhaps more importantly I also swapped the D and ld actions in the TRIA solver so now the unshifted D calculates the Area This provides consistency to the d key as the gate to interconnect SARR and TRIA on both cases Subtle differences probably just a matter of taste As a side effect of the modification only one function MANTXP was removed from the main FAT it has been placed into
146. ime Let s add to this mounting MCODE nightmare the requirement that both SOLVE and INTEG would work in a nested way which is something that the code will only discover having already created the buffer for the first function so the buffer would potentially have to be resized on the fly not losing any previous information already contained c Angel M Martin Page 160 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 And adding insult to injury welcome to the parallel dimension of bank swithching imagine now attempting to do all that from within an auxiliary bank say bank 3 which when activated would not know a thing about the main one little details like the FAT etc so it s there to live and die by its own sword Case in point being what if the function f x to solve or integrate contains functions available within the same module how then could they be found Well at least this one has an easy answer runs thus obviously the main FAT is also there and all will work out So provided we can identify the exact points in the code where the execution is transferred to the FOCAL label we d be home free or would we But beware because then not only the MCODE execution needs to be resumed how it does it is still pending clarification but itll also have to re activate bank 3 as the very first thing it does Buffer 14 comes to the rescue Say there are two auxiliary functions one of them SIRTN is sought for d
147. imizes speed and accuracy thanks again to internal usage of 13 digit OS routines The reuse made of it more than pays off for its lengthy code A few examples will illustrate this 27 erf x T iF 4 3 2 it aan oFi a 1 2 H z GO iiil Fa 1 3 2 3 2 z 4 V2nT a 3 2 Naturally this is not the case for any special function and even when there s such an expression it may be more appropriate to use the direct definition instead or an alternative one for the implementation This is the case of the Bessel functions which use the series expansion approach in the SandMath the Gamma function using the Lanczos formula etc With that said let s delve into the individual functions comprising the High Level Math group First off come those more frequently used so that they have gained their place in the ROM s main FAT Looking at the authorship you ll see the tight collaboration between JM and the author as stated in the opening statements of this manual c Angel M Martin Page 77 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 3 3 1 Gamma function and associates Let s further separate these by logical groups depending on their similarities and applicability The first one is the GAMMA and related functions 1 GM PSI PSIN LNGM ICGM BETA and ICBT all of them a Quantum leap from the previous functions described in the manual both in terms of the mathematical definition and a
148. in j JEF Jacobi Elliptic Integrals JM Baillard C CLAUS Clausen integral JM Baillard L LOBACH Lobachesvki function ngel Martin W WHIM Whittaker M function _ JM Baillard Y DBY Deby function JM Baillard c Angel M Martin Page 123 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 The Elliptic Integrals In integral calculus elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse They were first studied by Giulio Fagnano and Leonhard Euler Modern mathematics defines an elliptic integral as any function f which can be expressed in the form f z R t VPE at where R is a rational function of its two arguments P is a polynomial of degree 3 or 4 with no repeated roots and c is a constant The most common ones are the incomplete Elliptic Integrals of the first second and third kinds Besides the Legendre form given below the elliptic integrals may also be expressed in Carlson symmetric form which has been the basis for the implementation in the SandMath completely based on the JMB_MATH ROM The incomplete elliptic integral of the first kind F is defined as P d F p k F p k F sing k Woman which in terms of the Carlson Symmetric form Rr it results F k sin dR cos 1 k sin 1 ELIPF is implemented as a MCODE function which simply calls CRG with the appropriate input parameters All the he
149. in your knowns from right to eft in the menu Example The center riser on a triangular roof truss is four feet high and the length from one end of the truss to the midpoint is 22 feet What is the slope of the roof Solution This solution assumes you have already pressed XEQ SOLVER C Now press E to view the menu and 22 D 4 C A The answer is 0 1818 Example With a theodolite you measured the angle of an imaginary line going from the top of a tree to the ground at a distance of 5 749924998 m from the base of the tree to be 57 degrees How tall is the tree Assuming you just completed the previous example press remember you need to be in USER mode and also in DEG mode for this example Then 5 749924998 D 57 B C will give you the answer 8 854108050 m In the above example you are solving for the Rise given the Run and the angle 57 degrees Notice that when you key in a number before you press a key in the top row the calculator takes it as an input But when you press one of the top row keys without first keying in a number the HP 41 calculates that value based on the numbers you ve just keyed in fN we 18 10 3 8 Converted from the original feet value in the manual c Angel M Martin Page 41 of 167 December 2014 SandMath_44 Manual Revision 3x3 2 lt gt TRIA THE TRIANGLE SOLVER From SARR you can press SHIFT d to execute the TRIA
150. internal routines to accommodate this case so stack prompts are excluded Note that even if the Stack arguments are not directly allowed controlled by the prompting bits it is unfortunately possible to use the decimal key in an indirect register sequence that is after pressing the SHIFT key This won t work properly in the current design so must be avoided 2 4 2 RCL Launcher the Total Rekall The basic idea of a launcher is a function capable of calling a set of other functions The grouping in this case will be for the five RCL Math functions described above plus logically the standard RCL operation inclusive its indirect registers addressing Other enhancements include the prompt lengthener to three fields for registers over 99 albeit this is de facto limited to 128 as we ll see later on The keyboard mapping for Yell is as follows Numeric keypad or Top rows to perform the standard RCL SHIFT for Indirect register addresses EEX for the prompt lengthener to three places RCL to revert to the standard RCL function see note below Math keys and to invoke the RCL Match functions Back arrow to cancel out to the OS Note that RCL is not programmable This is done intentionally by design so that t can be used in a program to enter any of the RCL Math functions directly as a program line ignoring the corresponding prompt Using the RCL hot key reverts to the standard RCL operation This nift
151. ion of functions by generalized Fourier series that is approximations based upon summation of a series of terms based upon orthogonal polynomials One problem of particular interest is that of approximating a function in a computer mathematical library using operations that can be performed on the computer or calculator e g addition and multiplication such that the result is as close to the actual function as possible This is typically done with polynomial or rational ratio of polynomials approximations The objective is to make the approximation as close as possible to the actual function typically with an accuracy close to that of the underlying computer s floating point arithmetic This is accomplished by using a polynomial of high degree and or narrowing the domain over which the polynomial has to approximate the function Narrowing the domain can often be done through the use of various addition or scaling formulas for the function being approximated Modern mathematical libraries often reduce the domain into many tiny segments and use a low degree polynomial for each segment Optimal polynomials Once the domain and degree of the polynomial are defined the polynomial itself is chosen in such a way as to minimize the worst case error That is the goal is to minimize the maximum value of P x f x where P x is the approximating polynomial and f x is the actual function Left figure Error between optimal polynomial and log x
152. it can be used as solid foundation for the more complex parts The following functions extend the HP 41 Math function set and many of them will be used either as MCODE subroutines or directly in FOCAL programs Function Description Author 24X 1 Self descriptive faster and better precision than FOCAL Angel Martin 1 N Harmonic Number H n Angel Martin NAX Geometric Sums Angel Martin ATAN2 Two argument arctangent complex argument Angel Martin CBRT Cubic root main branch Angel Martin ied CEIL Ceiling function of a number Angel Martin CHSYX Multiple CHS by Y Angel Martin E3 E Index builder Angel Martin FLOOR Floor function of a number Angel Martin GEU Euler Mascheroni constant Angel Martin LOGYX Base Y Natural logarithm of X Angel Martin QREM Quotient Remainder Ken Emery XA3 Cube power of X Angel Martin Y 1 X x th root of Y ngel Martin YAAX Very large powers of X result gt 1E100 Angel Martin YX Modified Y X does 0 0 1 JM Baillard provides a more accurate result for smaller arguments than the FOCAL equivalents It will be used in the ZETAX program to calculate the Zeta function using the Borwein algorithm e 1 N calculates the Harmonic number of the argument in X that is the sum of the reciprocals of the natural numbers which excludes zero lower and equal to n It will be used in the calculation of the Kelvin functions and the Bessel functions of the second kind K n x
153. ivates the STAT PRB menus H and O activate the Hankel and Carlson groups launchers respectively 0 activates the FRC Fractions launcher Radix activates the LastFunction SHIFT switches into the hyperbolic choices pressing it twice enables the second overlay ALPHA and PRGM activate the XF and XF sub functions launchers respectively USER activates the TVM launcher latest addition to the module lt back arrow cancels it or returns to it from a secondary menu c Angel M Martin Page 11 of 167 December 2014 SandMath_44 Manual Revision 3x3 As it occurs with standard functions the name of the launched function will be shown on the display while you hold the corresponding key and NULLED if kept pressed This provides visual feedback on the action for additional assurance This is a good moment to familiarize yourself with the EFU launcher Go ahead and try it using it also in PRGM mode to enter the functions as program lines Note that when activating XF youll NOT need to press ALPHA a second time to spell the sub function name unlike standard functions like COPY or XEQ This saves keystrokes as you can start spelling the function name directly You still need to press ALPHA to terminate the sequence Direct access function keys A Stat Prob MENUS B Euler s Beta Function C Digamma PSI D Rieman s Zeta Function E Gamma Natural log F One over Gamma G Eule
154. l with the mantissa and exponent parts of a number MANTXP places the mantissa in X and the exponent in Y whereas VMANT shows the full mantissa for a few instants before returning to the normal display form or permanently if any key is pressed and held during such time interval similar to the HP 42S implementation of SHOW and are modified versions of the mainframe functions R P and P R The difference lies in the convention used for the arguments in Polar form which here varies between 0 and 360 as opposed to the 180 180 convention in the mainframe Example convert the point 1 1 to the modified polar coordinates and back to rectangular DEG 1 CHS ENTER R gt P gt 1 414213562 X lt gt Y gt 225 0000000 and not 135 X lt gt Y gt original point c Angel M Martin Page 24 of 167 December 2014 SandMath_44 Manual Revision 3x3 and contine with the coordinate conversion theme This pair of functions can be used to change between rectangular and spherical coordinates Example convert the rectangular point 1 2 rectangular 3 ENTER 2 ENTER 1 RDN RDN RDN RDN The convention used is shown in the figure below defining the origin and direction of the azimuth and polar angles as referred to the rectangular axis r phi theta lt gt X Y Z The SandMath implementation makes use of the fact that with the appropriate selection of origins dual P R conversio
155. lar Gamma function y gammai x z p Taylor series expansion around 0 gives T z 2 12 Plot of 1 x along the real axis 4 The SandMath however uses the expression based in continuous fractions according to which r x x x 1 2 sqrt 2m exp x 1 12 x 1 30 x 53 210 x 195 371 x Comparing the results obtained by GAMMA using Lanczos and continuous fractions it appears that the precision ts generally better in the Lanczos case which also happens to be faster due to its polynomial like form and the absence of loops to adjust the result for smaller arguments Note the special case for x 0 which is not a pole for this function but it is a singularity for all the others that used the common subroutines therefore the dedicated check in the routine listing 3 3 3 Lower Incomplete Gamma function ICGM In mathematics the upper and the lower incomplete gamma functions are respectively as follow C s 2 J t e dt aag J Et e dt x Connection with Kummer s confluent hypergeometric function when the real part of z is positive which is the expression used to program it in the SandMath y s z s tze M 1 s 1 z The Upper incomplete Gamma function can be easily obtained from the relationship s Ts T s Examples 3 ENTER 4 XEQ ICGM gt 1 523793389 1 2 ENTER 1 7 XEQ ICGM gt 0 697290898 c ngel M Martin Page 81 of
156. lve and Integrate Chances are that if you re reading this you re already familiar with SOLVE and INTEG from the Advantage Pac module needless to say this is about the same functions so we won t get into a lengthy discussion on the functions methodology and attributes both are assumed to be already known to you Function Description Comments FROOT Calculates roots of f x in an interval Same as SOLVE FINTG Calculates the integral of f x between limits Same as INTEG FLOOP Auxiliary function for control Does nothing by itself SIRTN Auxiliary function for control In hidden FAT bank 3 FROOT will attempt to obtain a real root for the function in an interval defined by the values in Y XJ and FINTG will numerically calculate the definite integral of a function f x between the integration limits defined in registers Y lower limit and X upper limit In both cases the function needs to be programmed in a FOCAL program and its global LBL name needs to be in ALPHA when FROOT or FINTG are executed in program mode Note that this means it won t work for mainframe or MCODE functions from plug in ROMS which will need a dummy user code program to host them Also note that contrary to the original SOLVE and INTEG on the SandMath implementation these functions will when executed in RUN mode ALPHA will be turned on automatically for convenience Let s see a couple of examples The first one should be a repeat of the exercise from
157. lver 46 c Angel M Martin Page 3 of 167 December 2014 SandMath_44 Manual Revision 3x3 3 Upper Page Functions in Detail 3 1 a Statistics Probability Statistical Menu Another type of Launcher 51 Alea jacta est 52 Combinations and Permutations 53 Linear Regression Let s not digress 54 Ratios Sorting and Register Maxima 55 Probability Distribution Function 56 Cumulative Probability and Inverse 57 Poisson Standard Distribution 58 And what about Prime Factorization 59 Appendix 4 Prime Factors decomposition 60 Curve Fitting The AECROM Fitter 62 3 1 b A few Geometry Functions 3D vectors and 2D distance 65 More Triangles and tetrahedrons 67 3 2 Factorials A timid foray into Number Theory 68 Pochhammer symbol rising and falling empires 69 Multifactorial Superfactorial and Hyperfactorial 70 Logarithm Multi Factorial 72 Appendix 5 Primorials a primordial view 73 Apery Numbers 75 3 3 High Level Math The case of the Chameleon function in disguise 77 Gamma Function and associates 78 Lanczos Formula 79 Appendix 6 Comparison of Gamma results 80 Reciprocal Gamma function 81 Incomplete Gamma function lower 81 Logarithm Gamma function 82 Digamma and Polygamma functions 83 Inverse Gamma Function 85 Euler s Beta function 87 Incomplete Beta function 87 Bessel Functions and Modified 88 Bessel functions of the 1 Kind 88 Bessel functions of the 2 Kind 89 Getting Spherical are we 90 Programming
158. m the values stored in the matrix e TOTNT Totient function is but a simple extension also shown in the listings below PRMF PC gt X and TOTNT are included in the Advanced MATRIX ROM Below is the program listing for PFCT as implemented in the SandMath 1 LBL PRIME 32 skip if Prime 2 CF 00 33 3 INT 34 reduced number 4 ABS 39 5 CLA 36 Prime found 6 AINT 37 RCL 00 7 ss 380 Kel 8 PRIME 9 SF 00 10 AINT first prime factor 11 K 17 12 GTO 01 13 FS C if prime we re done 14 GTO 01 15 STO 01 save for grouping 16 ST L 17 LASTX reduced number 18 19 E 20 STO 00 reset counter 21 RDN 23 RCL 01 previous prime factor 24 KY 25 PRIME 26 SF 00 27 KHY different PF 28 GTO 02 YES 29 Isc 00 increase counter 60 AVIEW 30 NOP SAME pf 61 ANUM 31 FS 7C 00 Was it prime 62 END c Angel M Martin Page 60 of 167 December 2014 SandMath_44 Manual Revision 3x3 Below is the Enhanced version allowing for any number of different prime factors and repetition indices all stored in a n x 2 matrix file in extended memory PRMF Note how the program structure is basically the same despite the addition of the matrix handling Since the Advantage module is required we ve used AIP instead of AINT totally interchangeable as they re basically the same function LBL PRMF 46 RCL 00 2 PRMF 47 MSRt 3 48 RDN 4 49 ST L 5 50 LASTX
159. method with successive approximations to w W z so z w e being wje i 7 1 W C 1 j a 4 wer The implementation in the SandMath uses this iterative method to solve for W z the roots of its functional equation given the functions argument z An important consideration is the selection of the initial estimations For that the general practice is to start with Ln x as lower limit and 1 Ln x as upper value Another aspect of the W function is the existence of two branches The second branch is defined for arguments between 1 e and 0 with function values between 1 and infinite The lower branch is also available in the SandMath as the function WL1 In fact the MCODE algorithm is the same one with just different initial estimations depending on the branch to calculate c Angel M Martin Page 98 of 167 December 2014 SandMath_ 44 Manual Revision 44 _3x3 Example 1 calculate W for x 5 5 WLO gt RUNNING followed by 1 326724665 We can use the inverse Lambert function AWL to check the accuracy of the results simply executing it after WLO and comparing with the original argument Note the AWL will be seen later on in the Secondary FAT Sub functions group This it requires XF to call it not XEQ 5 WLO F AWL gt 4 999999998 an error of err 4 E 10 where EF can be called using the main launcher A41 ALPHA Example 2 calculate the Omega constant W 1
160. n is based on the Gauss transformation with the formulas used being With m 1 m let be u 1 sqrt m 1 sqrt m and v u 1 sqrt u we have sn ulm 1 4sqrt u sn vlu 1 4sqrt u sn viv cn ulm cn vlu dn vlu 14sqrt u sn vip dn ulm 1 sqrt u sn vlu 1 sqrt u sn viv These formulas are applied recursively until u is small enough to use The program calculates the three functions simultaneously returning the result in the stack registers X sn Y cn and Z dn The input parameters are the amplitude m and the argument u expected in Y and X respectively before calling JEF Two functions are included in the SandMath JEF and AJF The main program is JEF which can be used to calculate the results for any value of the amplitude m AJF is a MCODE funtion used to speed up the calculations applicable when the amplitude lies between 0 and 1 You could use AJF directly in this case since JEF does nothing but calling it in that circumstance If m lt 9999999999 the program can give wrong results c Angel M Martin Page 129 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Example 1 Evaluate sn 0 7 0 3 ocn 0 7 0 3 dn 0 7 0 3 0 3 ENTER 0 7 XFS JEF gt sn 0 7 0 3 0 632304776 RDN gt cn 0 7 0 3 0 774719736 RDN gt dn 0 7 0 3 0 938113640 Example 2 Likewise for x 0 7 and amplitudes 1 2 3 sn 0 7 1 0 604367777 sn
161. n the manual we ll see dedicated launchers for other subfunctions in the CARLSON and HANKEL sections completing the round A Prime Factors B Discrete Hartley Transform C Curve Fitting D Rieman s Zeta Borwein E Poly Logarithm F Fourier Series G Inverse Gamma H Inverse Hyp SINE I Inverse Hyp COS J Inverse Hyp TAN Toggles Hyp Launchers K Days between Dates L Cubic Equation Roots M Shortcut to Launcher SST ATAN2 Complex argument N INPUT data in registers O Taylor Series P Arithmetic Geometric Mean E8 Cancels out to Agi Q Probability Distribution Function R Generalized Exponential Integral S Generalized Error Function T Inverse Error Function U Cumulative Probability Function V Hyperbolic Sine Integral W Whittakert Function M X Lobachesvsky function Y Inverse Cumulative Probability Z Hyperbolic Sine Integral Clausen function Straight Line Equation Reg Maximum Spc Register Sort Stack Sort R S Ceiling function c ngel M Martin Page 13 of 167 December 2014 SandMath_44 Manual Revision 3x3 This implementation effectively supersedes the MKEYS approach respecting the default keyboard no need to toggle USER mode and without the extra KA registers consumption Note also that the HYP keyboard is compatible with the SandMath Overlay of which finally real life units were made Perhaps it s time fo
162. nchers and additional functionality described in the introduction of this manual at STHT PROH The following functions are in this general group Some of them are plain catch up with the aim to complete the set of basic functions Some others are a little more advanced reaching into the high level math as well Function Description Author T _ Compound Percent of x over y Angel Martin EVEN Tests whether x is an even number Angel Martin GCD Greatest Common Divider ngel Martin LCM Least Common Multiple ngel Martin NCR Combinations of N elements taken in groups of R Angel Martin NPR Permutations of N elements taken in groups of R Angel Martin ODD Tests whether x in an odd number Angel Martin _PDF Normal Probability Density Function Angel Martin lied PFCT Prime Factorization Angel Martin PRIME Primality Test finds one factor Jason DeLooze RAND Random Number from Seed in buffer Hakan Thorgren ial RGMAX Maximum in a register block JM Baillard ial RGSORT Sorts a block of registers Hajo David RGSUM Sums a block of registers JM Baillard SEEDT SEED with Timer Hakan Th reren ST lt gt d XREG exchange with Stack Nelson C Crowle fg STSORT Stack Sort David Phillips Statistical Menu Another type of Launcher Pressing MA1 twice will present the STAT PROB functions menu allowing access to 10 functions using the top row keys A J Two line
163. ndMath just the same using the same relationship as for j n x and y n x Once again remember than as n x increases the accuracy of the results decreases specially for J n x Y n x and the spherical counterparts where the returned value can be completely incorrect if n x gt 20 a practical rule not an absolute criterion c Angel M Martin Page 90 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 Programming Remarks The basic algorithms use the summation definition of the functions calculating the successive values of the sum until there s convergence for the maximum precision 10 decimal places on the display Therefore the execution time can take a little long a fact that becomes a non issue on the CL Or when using 41 emulator programs like V41 setting the turbo mode on There are different algorithms depending on whether the order is integer or not This speeds up the calculations and avoids running into singularities as mentioned before Note that for integer indexes the gamma function changes to a factorial calculation which benefits from faster execution on the calculator Non integer orders utilize the special MCODE function GAMMA with shorter execution times than equivalent FOCAL programs but still longer than FACT when integers Besides that for integer orders the execution time is further reduced by calculating simultaneously the two infinite sums involved in the first kind an
164. ndently Both use the same input parameters index in Y and half of the argument in X Pay close attention to the status of user flags 00 and 01 as they directly influence their result c Angel M Martin Page 92 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 Examples J 1 1 0 440050586 I 1 1 0 565159104 J 1 1 0 440050586 I 1 1 0 565159104 J 0 5 0 5 0 540973790 1 0 5 0 5 0 587993086 J 0 5 0 5 0 990245881 I 0 5 0 5 1 272389647 Y 1 1 0 781212821 K 1 1 0 601907230 Y 1 2 0 107032431 K 1 2 0 139865882 Y 0 5 0 5 0 990245881 K 0 5 0 5 1 075047604 Y 0 5 0 5 0 540973790 K 0 5 0 5 1 075047604 Error Messages Note that the functions will return a DATA ERROR message when the solution is a complex number like J 0 5 0 5 or I 0 5 0 5 There s no way around that save in some particular cases of the order You can always use the versions available in the 41Z Module for full complex range coverage OUT OF RANGE occurs when the calculator numeric range is exceeded This typically occurs for large indexes during the power exponentiation step ALPHA DATA indicates alphabetic data in registers X or Y May also trigger DATA ERROR Iterative Method for large arguments JINX The FOCAL program JNX1 is also available in the secondary FAT for cases involving large values of the arguments and integer orders It uses the relations
165. ne can define the k th factorial denoted by ni recursively for non negative integers as yh h ifO lt n lt k lV n n k I ifn gt k Another extension to the factorial is the Superfactorial It doesn t use any step size as variant rather it follows a similar formula but using the factorial of the numbers instead of the numbers themselves sf n k KH 17 27 3 n 1 n k k 1 ie Both the multifactorial and specially the superfactorial will exceed the calculator numeric range rather quickly so the SandMath functions use a separate mantissa and exponent approach using registers X and Y respectively tte ion lune Nevertheless the functions will put up a io i consolidated combined representation l Ps in the display using the letter E to wal alll separate both amounts Make sure to adjust the FIX settings as approriate 484889 3928 USER RAD c Angel M Martin Page 70 of 167 December 2014 SandMath_44 Manual Revision 3x3 Examples Calculate the multi and superfactorials given below 2345111 type 6 ENTER 2345 MFCT gt 1234 type 5 ENTER 1234 MFCT gt Sf 41 type 41 EF SFCT gt Sf 100 type 100 EFL gt To complete this trinity of factorials Occasionally the hyperfactorial of n is considered It is written as H n and defined by H n J ke 11 22 33 n 1 1 n The figures below sh
166. neral purpose functions re visiting the usual themes Fractions Base conversion Hyperbolic functions RCL Math extensions Triangles and Circles as well as simple but neat little gems to round off the page In sum all the usual suspects for a nice ride time The upper pages delve into deeper territory touching upon the special functions approximation theory and Probability Statistics Some functions are plain catch up for the 41 system sorely lacking in its native incarnation whilst others are a divertimento into a tad more complex math realms All in all a mixed and matched collection that hopefully adds some value to the legacy of this superb machine for many of us the best one ever I am especially thankful for the essential contributions from Jean Marc Baillard more than 3 4ths of this module are directly attributable to his original programs one way or another Wherever possible the 13 digit OS routines have been used throughout the module ensuring the optimal use of the available resources to the MCODE programmer This prevents accuracy loss in intermediate calculations and thus more exact results For a limited precision CPU certainly per today s standards the Coconut chip still delivers a superb performance when treated nicely The module uses routines from the Page 4 Library a k a Library 4 Many routines in the library are general purpose system extensions but some of them are strictly math related a
167. nfluent hypergeometric function defined by Heatley 1943 as oT 5m 3 Fin 1 Which to untrained eyes just appears to be a twisted cocktail of the Kummer function adding the exponential to the mix and scaling it with Gamma TF Se Oe iF m zn l r DATA REGISTERS ROO thru R04 Output Flags none EE ae Example 2 SQRT 3 SQRT PI XFS TMNR gt T sqrt 2 sqrt 3 n 0 963524225 Alternatively ial H T using the main launcher instead Gausszexpl e ypeak vs xpeak 100 Nay a d P n D 50 f J Pei bt 0 50 100 150 200 250 xpeak l T T gL e e Gaussexpl residuals 4 a 5 aieia of Bah male x z gt fof a hi ee Jy Fo 2 H 4l l i L l j 0 50 100 150 200 250 xpeak c ngel M Martin Page 145 of 167 December 2014 SandMath_44 Manual Revision_3x3 3 5 3 Orphans and dispossessed The last group of sub functions include those not belonging to any particular launcher for no other particular reason that there s no more available space in the ROM Keep in mind that the only dual way to execute them is using the XFL or XFL launchers Function Description Author EFL SP FNC Section header does FCAT Angel Martin EFL BS Aux routine All Bessel Angel Martin FL BS2 Aux routine 2nd kind Integer orders Angel Martin EFL AMG Arithmetic geometric Mean Angel Martin EFL AWL Inverse Lambert ngel
168. ng it s a sort of hybrid natured function The different trigonometric and hyperbolic integral definitions and their relations with the Hyper Geometric funcion for the relevant integral in the definition are as follows z sinht Shi x J dt x F 1 2 3 2 3 2 x14 cosht 1 Chi r n f at 0 x 4 5F3 1 1 2 2 3 2 x 4 x 4 F 1 1 2 2 3 2 x14 Examples 1 4 XEQ SI gt Si 1 4 1 256226733 or 5 Z 1 4 XEQ CI gt Ci 1 4 0 462006585 or az V 1 4 XEQ HSI gt Shi 1 4 1 561713390 or 5 1 4 XEQ HSI gt Chi 1 4 1 445494076 or D 1 Z V The figure below shows the function plots for Si and Ci for 0 lt X lt 15 oY FT N EE Tees 22s 1 NP s oH 5 10 15 uN es X H Nota also that even if support for complex arguments is not covered by the SandMath the following relation between the Exponential and Trigonometric Integrals is available E ix i 50 Si ax Cia x gt 0 c Angel M Martin Page 102 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 Generalized Exponential Integrals ENX The exponential integral may also be generalized to wg ag Et E J C at pm which can be written as a special case of the upper incomplete gamma function Enlt 2 T 1 n We also have EO x 1 x exp x and En 0 1 n 1 if n g
169. ng of the period Begin mode Begin mode sets flag 00 Ending payments are common in mortgages and direct reduction loans beginning payments are common in leasing Equation l ti n i Where iis the periodic interest rate as a fraction i 1 100 p 1 in Begin mode or 0 in End made Valid Input Values for Data Use a cash flow diagram to determine what your cash flow inputs are and whether to specify them as positive or negative The cash ftow diagram is just a time line divided into time periods Cash flows transactions are indicated by vertical arrows an upward arrow is positive for cash received while a downward arrow is negafive for cash paid out For example the six period time line on the left shows a 20 cash outflow initially and a 50 cash inflow at the end of the fourth period Begn mode cannot be used in calculating PV or FV The five period time line on the right shows a 1 000 cash outflow initially and a 100 inflow at the end of each period ending with an additional 1 000 inflow at the end of the fifth period 1 000 i kP Cash 50 vas Cash i Inflow 100 100 100 100 100 inflow a i i i g Dash Cash Outflow 20 Outfiaw 1 000 c Angel M Martin Page 47 of 167 December 2014 SandMath_ 44 Manual Revision 3x3 Instructions e The program TVM will solve for any one of the variables N I PV PMT or FV given the other three or four which must include either N or
170. ng on the signs of the orders and the arguments possibly adjust the result after it s completed c Angel M Martin Page 91 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 The second function BS2 is only used for second kind functions with integer orders It s a finite sum and not an infinite summation Its contribution to the final result grows as the function order increases Its main goal was to reduce execution time as much as possible derived from the speed gains of MCODE versus FOCAL The definition of fk n x is as follows f n x x 2 k n k PACs H n k k 0 1 2 The definition of gk n x is as follows g n x x 2 n k 1 k 3 k 0 1 n 1 Despite GAMMA s execution time being reasonably fast it is noticeably longer than that of the Factorial for integer indexes therefore BS2 will use FACT instead for integer orders The Harmonic Numbers H n are obtained using another SandMath function as subroutine 21 N You see that the internals of BS2 perform quite an involved procedure utilizing multiple resources within the SandMath module Furthermore BS2 is called twice within the FOCAL program to calculate KBS or YBS once for the first infinite summation and a second time for the second finite sum The status of User Flag 00 controls the calculation made That was done to save one FAT entry when the limiting factor was the maximumm nuber of functions per page
171. now available However double storage space doesn t mean duplicating the number of functions for several reasons 1 Bank switched pages are not available simultaneously thus the code must be structured taking into account this limitation as well as the different requirements imposed by the OS For technical reasons FOCAL code can only reside in the primary bank thus the usage of secondary banks is limited to MCODE only Furthermore all the menu launchers use the partial data entry technique less demanding on battery consumption than keystroke pressing detection which is also restricted to the main bank as the OS will always switch back to the main bank when the CPU goes to light sleep Some of the new functions are real juggernauts with very large code streams taking up considerable space A good example is the Curve Fitting section about 1 5k in size in total but also some others fall in the same category as well TAYLOR takes about 1k and IERF takes about 650 bytes by itself to mention just two These are ideal candidates for bank switching The secondary FAT has absorbed the majority of the new functions with just a few changes made to the main FAT in the HL MATH section to include the most important functions in a more prominent location Also a new section TRANSFORM has been added to FCAT to facilitate the navigation around this catalog now containing 97 functions Defying those reports sta
172. nown values need to be input in counterclockwise order around the triangle as follows Knowns Suggested input order SSS S1 S2 S3 ASA A3 S2 Al SAS S1 A3 S2 SAA S1 A3 A1 SSA S1 S2 Al Example 1 Solve for all the unknown sides unknown angles and the area of the following triangle Solution 5 3 A 3 1 B 83 c Then press a to solve for A1 64 9903 b to solve for A2 32 0097 C to solve for S3 5 8048 and D to solve for AREA 8 1538 Once you have input the three defining knowns of a triangle you can change one or two values at a time to see how the other lengths are S3 affected c Angel M Martin Page 42 of 167 December 2014 SandMath_44 Manual Revision 3x3 Example 2 Given the following triangle with three known sides calculate all the angles and the area of the triangle 11 15 Solution This is aS S S problem so the input order is S1 S2 S3 With 4 24 as S1 Al 20 44 A2 46 25 A3 113 31 and AREA 17 07 Here are the keystrokes 4 24 A 8 77 B 11 15 C a b c D Example 3 Given the triangle below left side with two known angles and one known side calculate the unknowns Solution This is an ASA problem thus the input order is A3 S2 A1 With 15 as S2 press 65 c 15 B 75 a Then press A to see S1 22 54 C to see S3 21 15 b to see A2 40 and D to see the area 153 22 435 7
173. ns or roots You need to input the three coefficients into the stack registers Z Y X using a ENTER b ENTER c The roots are obtained using the well known formula X1 2 b 2a sqrt b 2a 2 c a Depending on the sign of the discriminant i e the argument of the square root the result will be real or complex roots If the discriminant is positive then the roots are real and their values x1 and x2 will be left in Y and X registers upon execution Register Z will contain a non zero value which can be used in program mode to determine the case Example Calculate the roots of the equation x 2 2x 3 0 1 ENTER 2 ENTER 3 CHS QROOT gt xl 1 x2 3 In RUN mode the SandMath will show both values in the display separated by the ampersand sign Moreover should the values be integers then the representation will omit the superfluous decimal places c Angel M Martin Page 27 of 167 December 2014 SandMath_ 44 Manual Revision 3x3 If the discriminant is negative then the roots z and z are complex and conjugated symmetrical over the X axis with Real and Imaginary parts defined by Re Z b 2a Z Re z i Im z Im Z sqrtLabs b 2a 2 c a Z2 Re z i Im z Upon execution reg Z will be zero used in Programs Im z will be left in Y and Re z will be left in X In RUN mode the display will show the first root in a composite format showing one of the roots Example Calculate the
174. ns are equivalent to Spherical and vice versa 3 to spherical coordinates and then back to gt r 3 741657386 gt phi 0 640522313 gt theta 1 107148718 gt Original point in stack You can also use function VMOD in the secondary FAT to check the modulus result Its value should be slightly more accurate as it uses direct math routines not based on TOPOL and HMS complement the arithmetic handling of numbers in HMS format adding to the native HMS and HMS pair They multiply or divide the HH MMSSSS value in Y by an scalar in X As it s expected the result is also put in HMS format as well Example calculate the triple of 2 hours 45 minues and 25 seconds 2 4525 ENTER 3 XEQ HMS gt 8 161499999 That is 8 hours 16 minutes and 15 seconds almost exactly This function is useful in surveying calculations as a shortcut of the standard approach involving conversion to decimal format prior to the operation Note that to multiply or divide two numbers given in HMS format you need to convert them both to rdecimal form using HR perfrom the operation and convert the result back to HMS format to end c ngel M Martin Page 25 of 167 December 2014 SandMath_44 Manual Revision 3x3 Entering the base conversion section The following functions are available in the SandMath Function Description Author BS gt D Base to Decimal George Eldridge D gt BS Decimal to bas
175. ns with rapidly converging power series if the series is cut off after some term the total error arising from the cutoff is close to the first term after the cutoff That is the first term after the cutoff dominates all later terms The same is true if the expansion is in terms of Chebyshev polynomials If a Chebyshev expansion is cut off after T_N the error will take a form close to a multiple of T_ N 1 The Chebyshev polynomials have the property that they are level they oscillate between 1 and 1 in the interval 1 1 T_ N 1 has N 2 level extrema This means that the error between f x and its Chebyshev expansion out to T_N is close to a level function with N 2 extrema so it is close to the optimal Nth degree polynomial After this introduction we re better equipped to use the functions and programs included in the SandMath related to approximation The first three are related to the Chebyshev approximation If f is a function defined over 1 1 and if n is a positive integer the Chebyshev coefficients c0 cl PE PEER cn may be computed by the formula cj 2 n 1 X cos 180 j k 1 2 n 1 cos 180 k 1 2 n 1 k 0 1 n if j 0 cj 1 n 1 X cos 180 j k 1 2 n 1 cos 180 k 1 2 n 1 Ik 0 1 n if j 0 If f is defined over a b we make the change of variable u 2x a b b a expects a b stored in R11 and R12 the function name in ALPHA and the desired number of coeff
176. nsists of just MEAN and SDEV Function Description Author CORR Correlation Coefficient of an X Y sample JM Baillard a COV Covariance of an X Y sample JM Baillard x LR Linear Regression of an X Y sample JM Baillard LRY Y value for an X point JM Baillard Linear regression is a statistical method for finding a straight line that best fits a set of two or more data pairs thus providing a relationship between two variables Using the well known method of least squares will calculate the slope A and Y intercept B of the linear equation Y Ax B The results are placed in Y and X registers respectively When executed in RUN mode the display will show the straight line equation similar to the STLINE function described before Example find the y intercept and slope of the linear approximation for the data set given below x p q o o Assuming all data pairs values have been entered using Y value ENTER X value we type XEQ LR gt 0 038650000 and X lt gt Y gt 4 856000000 producing the following output in FIX 2 Beuex 486 FAD O1 3 As to the remaining functions calculates the sample covariance returns the correlation coefficient and the linear estimate zero intercept For the same sample still in the calculator s memory we obtain the values Covariance 38 65 CORR 0 987954828 LRY 4 894184454 c Angel M Martin Page 54 of 167 December 2014 SandMa
177. nts the inputted values are expected to be all integers and return the numerator and denominator of the result fraction in registers Y and X respectively In RUN mode the execution continues to show the fraction result in ALPHA according to the currently set number of decimals see below The fraction arithmetic functions can be used in chained calculations there s no need to re enter the intermediate results and the Stack enabled makes unnecessary to press ENTER Notice that fractions are entered using the Numerator first To re calculate the fraction after changing the decimal settings just press the divide key followed by D gt F to re generate the fraction values For example calculate 2 7 over 4 13 then add 9 17 to the result 2 ENTER 7 ENTER 4 ENTER 13 F 9 ENTER 17 F gt 347 238 in FIX 6 mode c Angel M Martin Page 32 of 167 December 2014 SandMath_44 Manual Revision 3x3 Needless to say the fractional representation display will not be produced in PRGM mode but it ll have a silent execution instead Note that the fraction math functions operate on integer numbers in the stack returning also the numerator and denominator as integers To get the decimal number just execute to divide them In fact that s exactly what the functions do in RUN mode upon completion the fraction is converted to a decimal number then D gt F presents the final output That s why the display s
178. nx x 41 n Fz 1 m n 3 2 m n 3 2 x7 4 LOMS1 and LOMS2 calculates s1 m n x and s2 m n x Here are the specifics DATA REGISTERS ROO thru R09 temp Flags Used FO1 Output z m fy Example 2 SQRT 3 SQRT PI F LOMS1 gt si sqrt 2 sqrt 3 x 3 003060384 2 SORT 3 SORT PI F LOMS2 gt s2 sart 2 sqrt 3 x 9 048798662 alternatively Ma1 H L for s1 and bil H SHIFT L for s2 c ngel M Martin Page 139 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Lerch Transcendent Function LERCH In mathematics the Lerch zeta function sometimes called the Hurwitz Lerch zeta function is a special function that generalizes the Hurwitz zeta function and the polylogarithm It is named after Mathias Lerch The Lerch zeta function L and a related function the Lerch Transcendent are given by exp 2min A L A a s 5 O z s a X _ Special cases The Lerch Transcendent generates other special functions as particular cases as it s shown in the table below s a L 0 a s 1 s a Xn Z 2 z z n 1 2 C s 1 s 1 Li 2 zP z s 1 nls P 1 s 1 The figures below depict the representation for x given the other two constant Re Diz 1 0 Re Diz 0 1 2 4 7 0 2 4 4 2 0 2 4 z a The SandMath implementation LERCH is for the Lerch Transcendent function It is a
179. obability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and or space if these events occur with a known average rate and independently of the time since the last event A discrete stochastic variable X is said to have a Poisson distribution with parameter A gt 0 if for k 0 1 2 the probability mass function of X is given by Me fik A Pr X k k Poisson distribution 3 Its inputs are k and in stack registers X and Y PSD s result is the probability 025 corresponding to the inputs O20 0 15 0 70 095 JO ee Q 1 2 3 a e 6 F i tO 7 12 Example 1 Calculate the probability mass function for a Poisson distribution with parameters l 4 k 5 Probability mass function 4 ENTER 5 XFS PSD Returns 0 156293452 Example 2 do the same for I 10 and k 10 10 ENTER F PSD or Pl LastF Returns 0 125110036 The horizontal axis is the index K the number of occurrences The function is only defined at integer values of K The connecting lines are only guides for the eye c Angel M Martin Page 58 of 167 December 2014 SandMath_44 Manual Revision 3x3 And what about prime factorization PFCT Function PFCT will do a very fast and simple prime factorization of the number in X using PRIME To look for the successive divisors until 1 is found PFCT uses the ALPHA registers to present
180. of the first kind Angel Martin cA JBS Bessel J n x of the first kind Angel Martin cA KBS Bessel K n x of the second kind Angel Martin SIBS Spherical Bessel i n x Angel Martin Al SJBS Spherical Bessel j n x Angel Martin cA SYBS Spherical Bessel y n x Angel Martin cA YBS Bessel Y n x of the second kind Angel Martin JNX1 Bessel J for large arguments integer orders only Keith Jarret The SandMath Module includes a set of functions written with the harmonic analysis in mind specifically to facilitate the calculation of the Bessel functions in their more general sense for any real number for order and argument Bessel functions of the First kind I n x and J n x The formulae used are as follows M armas 3 m 0 m Or T x i J tx Wie ako x iz 2 mID m a 1 z Where denotes the Gamma function These expressions are valid for any real number as order although there are issues for negative integers due to the singularities in the poles of the gamma function as there s always a term for which m n 1 equals zero or negative integers all of them being problematic To avoid this we use the following expression for negative integer orders my iy Jale 1 J z Whilst L x 1 x for every real number order This definition is also valid for negative values for X as there s no singularity for any x value The SandMath implementation uses a recurrence formula instead of the
181. of the manual covers many other functions included in the Sub functions group with entries located in the secondary hidden FAT go ahead and review the accessibility information from the introduction for a quick refresher if needed Let s use the Carlson and Hankel launchers as grouping criteria The first sub function launcher is the Carlson group It s loosely centered on the Carlson s integrals plus related functions The launcher prompt is activated by pressing O at the main XFL prompt and offers the following 14 choices in two line ups controlled by the SHIFT key Note the different leadings on each screen Keeping the choices constant regardless The table below shows in the first column the letter used for each of the functions within this group CR Function Description Author E ELIPF Elliptic Integral Angel Martin F CRF Carlson Integral 1 Kind JM Baillard G CRG Carlson Integral 2 Kind JM Baillard J CRJ Carlson Integral 3 Kind JM Baillard C CSX Fresnel Integrals C x amp S x JM Baillard W WEBAN Weber and Anger functions __ JM Baillard L ALF Associated Legendre function 1 kind Pnm x JM Baillard Y AIRY Airy Functions Ai x amp Bi x JM Baillard 1 LEIL Incomplete Legendre Integral of l kind F _ Angel Martin 2 LEI2 Incomplete Legendre Integral of 2 Kind E Angel Martin 3 LEI3 Incomplete legendre Integral of 3 kind II Angel Mart
182. ording to the expressions S x n x 6 1F 3 4 3 2 1 4 n x 16 and C x x F 1 4 1 2 5 4 n x 16 The figure below shows both functions plotted for 0 lt x lt 5 na a a ya as Gu BS y 7 e i SS a REGISTERS ROO thru R04 FLAGS none Output Examples 1 5 EF CSX gt 4 EF CSX gt C 4 Or Mal 0 C S 1 5 0 697504960 C 1 5 0 445261176 X lt gt yY 0 420515754 0 498426033 X lt gt Y S 4 c Angel M Martin Page 133 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Weber and Anger functions H In mathematics the Anger function introduced by C T Anger 1855 is a function defined as 1 T J z cos v zsin 0 d WT Jo The Weber function introduced by H F Weber 1879 is a closely related function defined by 1 f E z J sin v z sin f d If v is an integer then Anger functions Jv are the same as Bessel functions Jv and Weber functions can be expressed as finite linear combinations of Struve functions Hn and Ln With n and x in the stack WEBAN will return both J n x and E n x in the Y and X stack registers respectively The figures below show four of these functions for 4 orders 0 0 5 1 and 1 5 Anger on the left plots and Weber on the right Check J 0 0 1 and E 0 0 1 3f g Note that WEBAN will return both values to the stack REGISTERS ROO th
183. ortcuts for 1 10 Note that No STK functionality is implemented even if you can force the prompt at the IND step Typically you ll get a DATA ERROR message Rather not try it c Angel M Martin Page 15 of 167 December 2014 SandMath_44 Manual Revision 3x3 Function index at a glance And without further ado here s the list of functions included in the module First the main functions Name Description Name Description 0 0 1 2AX 1 Powers of 2 1 1 GME Reciprocal Gamma Cont Fre 2 DI N Harmonic Numbers 2 FL Function Launcher 3 DGT Sum of mantissa digits 3 F Launcher by Name 4 QNAX Geometric Sums 4 F Launcher by index 5 AINT Alpha Integer Part 5 BETA Beta Function 6 ATAN2 Dual argument ATAN 6 CHBAP Chebyshev s Approximation 7 BS gt D Base to Decimal 7 da Cosine Integral 8 CBRT Cubic Root 8 DHST Discrete Hartley Transform 9 CEIL Ceil function 9 El Exponential Integral 10 CHSYX CHSY by X 10 ENX Generalized Exponential Integrals 11 CROOT Cubic Equation Roots 11 ERF Error Function 12 CVIETA Driver for CROOT 12 FFOUR Fourier Series 13 D gt BS Decimal to Base 13 FINTG Numerical Integration 14 D gt H Dec to Hex 14 FLOOP Auxiliary function 15 E3 E 1 00X 15 FROOT Solution of f x 0 16 FLOOR Floor Function 16 GAMMA Gamma Function Lanczos 17 SOLVER Geometric and TVM Solvers 17 HCI Hyperbolic Cosine Integral 18 GEU Euler s Constant 18 HGF Generali
184. oth programs use pretty much the same code A Execution time tends to infinity as x tends to 27 This routine produces DATA ERROR if x 0 but f 0 0 Note also that CLAUS will change the angular mode to DEG thus you need to make sure it s set back in the appropriate mode before calling LOBACH T 0 2 i L l 4 so they both have the same amplitude and frequency i Home_assignment Being curious about their similar shapes calculate the differences between the Lobachevsky function and an equivalent Sine say f x G sin 2x where G Catalan s constant c Angel M Martin Page 109 of 167 December 2014 SandMath 44 Manual Revision_3x3 Function Approximations Function Description Author ZFL CHB CHB2 Chebyshev Polynomials Tn and Un JM Baillard ZFL CHBCF Chebyshev Coefficients JM Baillard CHBAP Chebyshev s Approximation JM Baillard XFL CdT Auxiliary for CHBAP JM Baillard TAYLOR Taylor Polynomial of order 10 and Approximation Martin Baillard FFOUR Fourier coefficients for x Angel Martin DHST Discrete Hartley Symmetrical Transform JM Baillard In mathematics approximation theory is concerned with how functions can best be approximated with simpler functions and with quantitatively characterizing the errors introduced thereby Note that what is meant by best and simpler will depend on the application A closely related topic is the approximat
185. ough a precision that doesn t require using the functional equation to adjust it for small values of the argument The obvious advantage is that without the required program loop the execution time is shorter and constant for any input This becomes of extreme importance when Gamma is used as a subroutine of more complex cases like the Bessel J and I functions where the cumulative additional time is very noticeable Ir z c ngel M Martin Page 79 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 Appendix 6 Accuracy comparison of different Gamma implementations The tables below provide a clear comparison between three methods used to calculate the Gamma function 1 Lanczos formula with k 6 2 Continuous fractions and 3 Windschitl Stirling Each of them implemented using both standard 10 digit and enhanced 13 digit precision routines The results clearly show that the best implementation is Lanczos and that the 13 digit routines provide a second order of magnitude improvement to the accuracy or in other words that it cannot compensate for the deficiencies of the used method We re lucky in that the more accurate method is faster that the second best albeit not as fast as Stirling s Obviously the extrapolation from integer case to the general case for the argument is assumed to follow the same trend albeit not shown in the summary tables Standard 10 digit Implementation Reference g Lanczos
186. ow a plot for both the hyperfactorial and its logarithm itself a convenient scale change very useful to avoid numeric range problems Note that they re extended to all real arguments and not only the natural numbers also called the K function Hin Int Hin 4 100 3 j 80 4 2 P 60 f F f 2 ES ue ont A 4 Byg ie 20 zE F 2 nal Mn Po TE n 2 4 6 8 10 See below a couple of simple FOCAL program to calculate the hyperfactorial which runs beyond the numeric range dramatically soon and its logarithm written by JM Baillard Understandably slow and limited as these programs are you can visit his web site for a comprehensive treatment using dedicated MCODE functions for the many different possible cases 0i LBL HFCT 07 01 LBL LOGHF 07 02 1 08 DSE Y 02 0 08 03 LBL 01 09 GTO 01 03 LBL 01 09 DSE Y 04 RCLY 10 END 04 RCLY 10 GTO 01 05 ENTER 05 ENTER 11 END 06 Y X 06 LOG Example calculate the Hyper factorial of 23 23 UF LOGHF 10 X lt gt Y Y X gt c Angel M Martin Page 71 of 167 December 2014 SandMath_44 Manual Revision 3x3 Logarithm Multi Factorial LOGMF The product of all odd integers up to some odd positive integer n is often called the double factorial of n even though it only involves about half the factors of the ordinary factorial and its value is therefore closer to the square root of the factorial It is denoted by n For an od
187. previous appendix now using this version of the functions Be aware that the execution time will be long but that s an acid test for the operation being a nested example of both For a second example refer to appendix in page 107 to calculate the Fourier coefficients for an explicit function f x Now this is what closes the circle Example Calculate the roots of Digamma and Exponential integral functions Nothing can be easier than writing this trivial program 0i LBL PSI2 02 PSI 03 RIN 04 LBL EI2 05 EI 06 END Enter the values 1 ENTER 5 then execute FROOT typing the program name in ALPHA at the prompt XEQ ALPHA FROOT ALPHA PSI2 ALPHA or XEQ FROOT ALPHA E12 ALPHA The corresponding solutions in FIX 9 are as follows X 1 461632145 for PSI and X 0 372507411 for Ei c Angel M Martin Page 158 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Example Calculate the root of the Kepler equation for E 0 2 and m 0 8 The equation is x E sin x m Programmed as follows assuming E is in RO1 and m in ROO data registers 01 LBL KPLR input data 0 2 STO 01 0 8 STO 00 02 RAD input interval 0 1 in YX 03 SIN XEQ FROOT KPLR gt 0 964333888 04 RCL O1 x 06 07 RCL substracts ROO 08 END Example Write a program to calculate Bessel J using the formula 1 iT J 2 cos nT x sinT dr To 12 LBL
188. programmed in main memory under its own global label The program prompts for the function name the first index to calculate and the number of desired coefficients The program also calculates the approximate value of the function at a given argument applying the Summation of the terms using the obtained coefficients ee C AMX wy fan FES 5 a0 gt a oad i gt by sin f To use it simply enter the value of x and press E XEQ E with user mode on this assumes that no function is assigned to that key The approximation will be more correct when a sufficient number of terms is included The goodness is also dependent on the argument itself c Angel M Martin Page 117 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Example calculate the first six coefficients for f x x2 assuming a period T 2r centered in x0 0 As it s known XA2 4 3 x42 LE 4 cos nx n 2 4r sin nx n n 0 1 Using an accuracy of 6 decimal places the program returns the following results x a0 13 1595 b0 0 al 4 b1 12 566 a2 1 b2 6 2830 a3 0 4444 b3 4 1888 a4 0 250 b4 3 1415 a5 0 160 b5 2 513 I 4 4 Pressing E will calculate an estimation of the function for the argument in X using the fourier temrs calculated previously In this case X 5 XEQ E gt f x 23 254423 X 1 XEQ E gt f x 0 154639 which obviously misses the point Typically the
189. quently as solutions to differential equations There is no general formal definition but the list of mathematical functions contains functions which are commonly accepted as special Some elementary functions are also considered as special functions The implementations described in this manual do nothing but scratching the surface or more appropriately gingerly touching it of the Special Functions field where one can easily spend several life times and still be just half way through Implementing multiple special functions in a 41 ROM is clearly challenged by the available space in ROM the internal accuracy and the speed of the CPU It is therefore understandable that more commonality and re usable components identified will make it more self contained and powerful overcoming some of the inherent design limitations The Generalized Hyper geometric function is one of those rare instances that works in our favor as many of the special functions can be expressed as minor variations of the general case Indeed there are no less than 20 functions implemented as short FOCAL programs really direct applications of the general case saving tons of space and contributing to the general consistency and common approach in the SandMath We have Jean Marc Baillard to thank for writing the original HGF the Generalized Hyper geometric function real cornerstone of the next two sections The SandMath has an enhanced MCODE implementation that opt
190. quiring quite a number of coefficients to be calculated for good accuracy result Moreover that calculation involves a lot of registers to store the values since there isn t any iterative approach based on recursion erf x The expression below is definitely too inaccurate only three or four digits are correct to deserve a dedicated MCODE function Tr 1277 43697 348077 Hisi EP eel r ina SS bv 2 12 480 40320 5806080 182476800 A paper from 1968 by A Strecok lists the first 200 coefficients of a power series that represents the inverse error function While using this approach it became clear that at least 30 of them are needed for a 10 digit accuracy for 0 lt x lt 0 85 This only gets worse as x approaches 1 getting into a clear example of the law of diminishing results A better method for the vicinity of 1 is probably to use an asymptotic expansion such as l erf z x gt gt log log log _ 74 1 2 2 og lo a se A combination of both approaches would seem to be the best compromise depending on the argument Typing the 30 coefficients is not fun however thus the best is no doubt to use a data file in X Memory to keep them safe c Angel M Martin Page 105 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 Appendix 9b Inverse Error Function CUDA Library The author reportedly went
191. r a second overlay The Last Function functionality The latest releases of the SandMath and SandMatrix modules include support for the LASTF functionality This is a handy choice for repeat executions of the same function i e to execute again the last executed function without having to type its name or navigate the different launchers to access it The implementation is not universal it only covers functions invoked using the dedicated launchers but not those called using the mainframe XEQ function It does however support two scenarios a functions in the main FATs as well as b those sub functions from the auxiliary FATs Because the latter group cannot be assigned to a key in the user keyboard the LASTF solution is especially useful in this case The following table summarizes the launchers that have this feature Module Launchers LASTF Method SandMath 3x3 XFL HYP FRC RCL Captures sub fnc id revision M Captures sub fnc NAME Captures sub fnc ie revision N FCAT XEQ Captures sub fnc id Note that the Alphabetical launcher F will switch to ALPHA mode automatically Spelling the function name is terminated pressing ALPHA which will either execute the function in RUN mode or enter it using two program steps in PRGM mode by means of the XF function plus the corresponding index using the so called non merged approach This conversion happens entirely automatically The LAST
192. r s Gamma Function H Hankel s Launcher I Bessel I n x J Bessel J n x Hyperbolics Launcher K Bessel K n x L Bessel Y n x M Lambert s W SST Incomplete Gamma N Root Finder 0O Carlson Launcher R Exponential integral S Numeric integral X Polygamma PsiN V Cosine Integral W Spherical Y n x Z Sine Integral Spherical J n x Incomplete Beta 0 Fractions Launcher R S View Mantissa L Activates the Last Function USER Time Value of Money launcher TVM ALPHA Sub function Alpha launcher F PRGM Sub function Index Launcher F 0N Turns the calculator OFF 28 Cancels out to the OS or retruns from 2 A green H on the overlay prefixing the function name represents the Hyperbolic functions This also includes the Hyperbolic Sine and Cosine integrals in addition to the three standard ones Using the SHIFT key will toggle between the direct and inverse functions Pressing lt will take you back to the main F prompt Note that the RCL Math functions are also linked to the main launcher to invoke them use the el launcher sort of Hyper RCL thus need to press M81 HYP to get the RCL _ _ prompt Typically the secondary launchers have the possible choices in their prompt we ll see them later on The STAT menu differs from the others in that it consists of two line ups toggled with the SHIFT key providing acc
193. r the program name in the Alpha register followed by 0 2 ENTER 0 XEQ TAYLOR gt 5 4 3 2 1 RUNNING and for the second case the program name is still in ALPHA 0 2 ENTER 1 XEQ TAYLOR gt 5 4 3 2 1 RUNNING The display shows the progress in the calculations with the first phase obtaining the 5 pairs of value functions followed by the approximation of the coefficients When it s complete in shorter time that expected due to the MCODE speed advantage the execution stops with the first four coefficients in the stack and all ten of them stored in registers RO1 to R10 In these particular cases the results are summarized in the table below together with the exact values and the accuracy of the estimations which deteriorates as the order of the derivative increases RG T1 Approx T1 Exact T1 delta TO Approx TO Exact TO delta R01 2 18281828 2 718281828 0 000000000 0 999999999 1 0 000000001 R02 1 359140927 1 359140914 0 000000010 0 499999994 0 5 0 000000012 R03 0 453046958 0 453046971 0 000000029 0 166666691 0 166666667 0 000000144 R04 0 113261624 0 113261743 0 000001051 0 04166676 0 041666667 0 000002232 R05 0 022652373 0 022652349 0 000001059 0 008333231 0 008333333 0 000012240 RO6 0 003775805 0 003775391 0 000109658 0 001388497 0 001388889 0 000282240 RO7 0 00053928 0 000539342 0 000114955 0 000198548 0 000198413 0 000680399
194. rally this approach requires a good implementation of both Gamma and Psi which is the case with the SandMath Clearly the challenging region is going to be the negative axis where Gamma has all the singularities and thus the calculation will have some difficult times to obtain the result for values near the origin even returning negative arguments Example calculate the non integer argument that yields T x 2 Type 2 XEQ IGMMA gt 0 442877396 To check it simply XEQ GAMMA gt 2 000000001 c Angel M Martin Page 85 of 167 December 2014 SandMath_44 Manual Revision 44 _3x3 The programs below show the two versions of the implementation very similar in the approach but with a different initial estimation which makes a difference as shown in the table from previous page Note that in the SandMath case the calculation of the L x factor is done in MCODE which increases accuracy and saves bytes in the main bank O01 LBL GOMMA 01 LBL IGMMA 02 STOOL argument to ROI 0 STO 01 argument to ROI 05 2 border fine 03 CF 01 0 X amp Y p 2 05 X gt Y i5 X gt 27 05 Xe gt y 06 GTO 02 yes gO over 06 Mc 07 N Lnjx 07 SF 01l 08 X 07 wgs x 17 08 LN 09 JE yes replace with 1 g9 FS 0l 10 S5STO store in ROO 10 GTO 02 11 GTO 00 go to loop 11 716 12 LBLO2 S 12 XoY 13 03653 magic factor 13 7 14 add to argument 14 LASTX 15 Pl 15 SORT 16 STX 2p 16 2 21 17 SORT sgri
195. ram in any way t j To clear the menu at any time press lt This shows you the contents of the X register but does not end aA 8 the program You can perform calculations then recall the main menu by pressing 0 However you do not E a need to clear the program s display or recall the menu i before performing calculations Remarks This program uses local Alpha labels as explained in the owner s manual fox the HP 41 assigned to keys A E and their shifted counterparts except Jic and J These local assignments are overridden by any User key assignments you might have made to these same keys thereby defeating this program Therefore be sure to clear any existing User key assigments of these keys before using this program and avoid redefining these keys in the future within possible The financial varlable keys will only store a value if you enter it from the keyboard If for example you recall a value from a register then press a variable key the program wil calculate that variable instead of storing the recalled value To store a value that was placed in the X register by some other means than actually keying it in press STO before pressing the variable key c ngel M Martin Page 48 of 167 December 2014 SandMath_44 Manual Revision 3x3 Examples Example 1 A borrower can afford a 650 00 monthly payment on a 30 year 14 25 mortgage How much can he borrow The first payment is
196. rect addressing SHIFT 3 should utilize the top 2 rows for index entry shortcut The first condition is easy to implement in RUN mode as it s just a matter of selecting the appropriate prompting bits in the function MCODE name but it gets very tricky when used under program mode This has been elegantly resolved using a method first used by Doug Wilder by means of using the program line following the instruction as the index argument Somewhat similar to the way the HEPAX implemented it although here there s some advantages in that the length of the index argument doesn t need to be fixed dropping leading zeroes and even omitting it altogether if it s zero assuming the following line isn t a numeric one which could be misinterpreted The indirect addressing is actually quite simple as it simply consists of an offset added to the register number in the index All the function code must do is remove it from the entry data provided by the OS and the task is done The offset value is hex 80 or 128 decimal We ll revisit this when discussing the RCL launcher And the third objective is provided for free by the OS as well no need for extra code at all just using the appropriate prompting bits in the function s name c Angel M Martin Page 36 of 167 December 2014 SandMath_44 Manual Revision 3x3 Stack arguments are more involved than the indirect addressing No attempt has been made to use the mainframe
197. reserved in LASTx credit is due to JM Baillard once more PY n n Y db n x L 0 xX Example 3 ENTER 0 7 XF DBY gt DB 0 7 3 6 406833597 Alternatively A81 0 SHIFT Y using the main launcher instead c ngel M Martin Page 151 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Dawson Integral The Dawson function or Dawson integral named for John M Dawson is either Pe D x ee e dt Or D x e f e dt 0 DAW computes F x by a series expansion F x e x x3 3 x5 5 2 x7 7 3 The figures below show both functions in graphical form A T E att EX A e S H fo Here as well no data registers are used Stack Output L ee ex Examples 1 94 EF DAW gt F 1 94 0 3140571659 10 EFL gt F 10 0 05025384716 15 EFL gt F 15 0 03340790676 For x gt 15 there will be an OUT OF RANGE condition For large arguments the execution is rather slow taking a couple of seconds even with TURBO mode on V41 so be patient c ngel M Martin Page 152 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Hyper geometric Functions HGF HGF and RHGF are the ordinary and the Regularized Hyler geometric functions The Gaussian or ordinary hypergeometric function 2F4 a b c z is a special function represented by the hypergeometric series that includes many other special func
198. riate function information either index codes or full names in registers within a dedicated buffer with id 9 The buffer is maintained automatically by the modules created if not present when the calculator is switched ON and its contents are preserved while it is turned off during deep Sleep No user interaction is required c Angel M Martin Page 14 of 167 December 2014 SandMath_44 Manual Revision 3x3 If no last function information yet exists the error message NO LASTF is shown If the buffer 9 is not present the error message is NO BUF instead Appendix 1 Launcher Maps The figures below provide a better overview illustrating the hierarchy between launchers and their interconnectivity For the most part it is always possible to return to the main launcher pressing the back arrow key improving so the navigation features rather useful when you re not certain of a particular function s location The first one is the Main SandMath Launcher The first mapping doesn t show all the direct execute function keys Use the SandMath overlay as a reference for them names written in BLUE aside the functions Launchers invoked trom the Main XFL Note that XFL requires pressing ALPHA a second time in order to type the sub function name And here s the Enhanced RCL MATH group RCL Math Launchers Here all the prompts expect a numeric entry The two top rows keys can be used as sh
199. riodic signals into the sum of a possibly infinite set of simple oscillating functions namely sines and cosines or complex exponentials The study of Fourier series is a branch of Fourier analysis The partial sums for f are trigonometric polynomials One expects that the functions XN f approximate the function f and that the approximation improves as N tends to infinity The infinite sum hq af 5 la cos nr bn sin nr n l is called the Fourier series of f The Fourier series does not always converge and even when it does converge for a specific value x0 of x the sum of the series at x0 may differ from the value f x0 of the function It is one of the main questions in harmonic analysis to decide when Fourier series converge and when the sum is equal to the original function FFOUR Calculates the Fourier coefficients for a periodic function F x defined as f fi j 3 In Pic cos dx i r fix L dx i ih i l f r si Z5 i Pe ix sin dx ea ES with the following characteristics ff RERA TT centered in x x0 with period 2L on an interval x0 x0 2L with a given precision for calculations significant decimal places FFOUR is a rather large FOCAL program despite having a MCODE FAT entry It calculates all integrals internally not making use of general purpose numeric integrators like INTEG IG etc so it s totally self contained The function must be
200. rogram will proceed to its next line If no root resuits the next program line will be skipped This is the do if true rule of HP 41 programming Knowing this you can write your program to handle the case of FROOT not finding a root such as by choosing new initial estimates or changing a func tion parameter c Angel M Martin Page 163 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 FROOT uses one of the six pending subroutine returns that the calculator has leaving five returns for a program that calls FROOT Note that FROOT cannot be used recursively calling itself If it does the program stops and displays RECURSION You can however use FROOT with FINTG the integration program Numerical Integration The FINTG program finds the definite integral I of a function f x ffx within the interval bounded by a and b This is expressed mathematically and graphically as ji I b f x dx Executing the FINTG program employs an advanced numerical technique to find the definite integral of a function You supply the equation for the function in a program and the interval of integration and FINTG does the rest a b Method The algorithm for FINTG uses a Romberg method for accumulating the value of an integral The algorithm evaluates f x at many values of x between the limits of integration It takes the program from several seconds to several minutes to do this and produce a result
201. rrow the section from the HP 41 Advantage s Pac user s manual a superb vintage document that avoids re inventing the wheel Bear in mind that whereas the FOCAL version relies on the local keys within the program the SandMath implementation uses the TMV launcher options for each value input this is the main difference between both implementations c Angel M Martin Page 46 of 167 December 2014 SandMath_44 Manual Revision 3x3 The TVM program solves different problems involving time money and interest the compound interest functions The following variables can be inputs or results e N the number of compounding periods or payments For a 30 year loan with monthly payments N 12 x 30 360 e I the periodic interest rate as a percent For other than annual compounding this represents the annual percentage rate divided by the number of compounding periods per year For instance 9 annually compounded monthly equal 9 12 0 75 e PV the present value This can also be an initial cash flow or a discounted value of a series of future cash flows Always occurs at the beginning of the first period e PMT The periodic payment e FV The future value This can also be a final cash flow or a compounded value of a series of cash flows Always occurs at the end of the Nth period You can specify the timing of the payments to be either at the end of the compounding period End mode or at the beginni
202. rtin Page 35 of 167 December 2014 SandMath_44 Manual Revision 3x3 dH RLL MATH The SandMath Module includes a set of functions written to extend the native RCL functionality mainly in the direct math operations missing when compared to the STO equivalents but also increasing its versatility and ease of use There are five new RCL Math functions plus a launcher to access them in a convenient and useful way I ae al Ic on a a ae O D B J TATA RCL ___ RCL Math Launcher ngel Martin Xe RCL __ RCL Plus Angel Martin G e RCL RCL Minus ngel Martin R RcL __ RCL Multiply Angel Martin R RCL __ RCL Division ngel Martin o BIRCI _RCL4__ RCL Power Angel Martin IRC AIRCL__ ARCLinteger Part of number in Register nn Angel Martin f 2 4 1 Individual Recall Math functions The five RCL Math new functions cover the range of four arithmetic operations like STO does plus a new one added for completion sake The functions would recall the number in the register specified by the prompt performing the appropriate math using the value in register X as first argument and the recalled number as the second argument Design criteria for these were 1 should be prompting functions 2 should support indi
203. ru R06 FLAGS none Y n WQ Example 2 SQRT PI EF WEBAN gt E sqrt 2 x 0 315594385 X lt gt Y gt J sqrt 2 x 0 366086559 Alternatively A81 O W using the main launcher instead c ngel M Martin Page 134 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 3 5 2 Hankel Struve and similar functions The second sub function launcher is the Hankel group It s loosely centered on the Hankel functions plus related sort The launcher prompt is activated by pressing H at the main XFL prompt and offers the following 14 choices in two line ups controlled by the SHIFT key Note the different leadings on each screen keeping the choices constant regardless The table below shows in the first column the letter used for each of the functions within this group HK Function Description Author 1 HK1 Hankel1 Function ngel Martin 2 HK2 Hankel2 Function ngel Martin W WLO Lambert WO Angel Martin H HNX Struve H Function JM Baillard L LOMS1 Lommel s1 function JM Baillard R LERCH Lerch Transcendental function JM Baillard T TMNR Toronto function JM Baillard K KLV Kelvin Functions 1st kind JM Baillard 1 SHK1 Spherical Hankel1 ngel Martin 2 SHK2 Spherical Hankel2 ngel Martin W WL1 Lambert W1 ngel Martin H LNX Struve Ln Function JM Baillard L LOMS2 Lommel s2 function JM Baillard R RCWF Regular Coulomb Wave Function JM Baillard T
204. s The result is also a dB value Use a negative sign in X for subtractions Examples 3 dB 5 dB 7 124426028 5 dB 3 dB 0 670765667 c ngel M Martin Page 148 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Arithmetic Geometric Mean In mathematics the arithmetic geometric mean AGM of two positive real numbers x and y is defined as follows First compute the arithmetic mean of x and y and call it al Next compute the geometric mean of x and y and call it g1 this is the square root of the product xy 1 ay a t y gi Zy TY Then iterate this operation with al taking the place of x and g1 taking the place of y In this way two sequences an and gn are defined ji n 1 5 an T gn n T y inin These two sequences converge to the same number which is the arithmetic geometric mean of x and y it is denoted by M x y or sometimes by agm x y Y a Zz Y Z X bO agm a0 b0 bt W Note that DATA ERROR will be triggered when one of the arguments is negative but not if both are Example 1 To find the arithmetic geometric mean of a0 24 and gO 6 simply input 24 ENTER 6 F AGM gt 13 45817148 Example 2 Gauss Constant The reciprocal of the arithmetic geometric mean of 1 and the square root of 2 is called Gauss s constant after Carl Friedrich Gauss Calculate it using AGM 2 SQRT 1 F AGM gt 1 198140235 1 X
205. s own function table this would only allow to index 128 functions where are the others and how can they be accessed The answer is called the Multi Function groups Multi Functions XF and F provide access to an entire group of sub functions grouped by their affinity or similar nature The sub functions can be invoked either by its index within the group using F or by its direct name using XF This is implemented in such a way that they are also programmable and can be entered into a program line using a technique called non merged functions You may already be familiar with this technique originally developed by the HEPAX programmers In the HEPAX there were two of those groups one for the XF M functions and another for the HEPAX A extensions The PowerCL Module also contains its own and now the SandMath joins them this time applied to the mathematical extensions particularly for the Special Functions group A sub function catalog is also available listing the functions included within the group Direct execution or programming if in PRGM mode is possible just by stopping the catalog at a certain entry and pressing the XEQ key The catalog behaves very much live the native ones in the machine you can stop them using R S SST BST them press ENTER to move to the next sub section cancel or resume the listing at any time As additional bonus the sub function launcher XF will also search the main mo
206. s are based on the PPC routines JC and CJ ported to an all MCODE implementation to make effective use of the available ROM space in the secondary banks The formulas used are as follows see PPC ROM manual for details JDN int int D int 367 x int x 0 75 int x 0 75 int int x 100 1 721 115 where X Y M 2 85 12 Let N JDN 1 721 119 C int N 0 2 36 524 25 if Gregorian N N C int C or if Julian N N 2 Y M int N 0 2 365 25 N N int 365 25 Y int N 0 5 30 6 D int N 30 6 M 0 5 A few polynomial functions follow next You should refer to the SandMatrix module for a much comprehensive coverage on this subject is a simple but useful routine to get the polynomial degree from the control word in X in the form bbb eee It is used by INPUT and obviously we have degree eee bbb As an additional bonus PDEG also leaves in LastX the address of the next free register eee 1 and are full fledged MCODE functions used to evaluate polynomials and to calculate the first derivative of a polynomial which coefficients are stored in data registers It requires the control word bbb eee in Y and the evaluation point x in X Example evaluate and calculate the derivative of P x 5x 3 4 x 2 3 in x 2 First we input the coefficients in registers ROO to R03 using INPUT 0 003 XFS INPUT followed by 5 ENTER 4 CHS E
207. s auxiliary code repository to make it all fit in an 8k footprint factor and to allow reuse with other modules This is totally transparent to the end user just make sure it is installed in your system and that the revisions match See the relevant Library 4 documentation if interested Function Launchers and Mass key assignments As any good theme module worth its name the SandMath has its own mass Key assignment routine MKEYS Use it to assign the most common functions within the ROM to their dedicated keys for a convenient mapping to explore the functions Besides that a distinct feature of this module is the function launchers used to access diverse functions grouped by categories These include the Hyperbolic the Fractions the RCL Math and the Special Function groups This saves memory registers for key assignments whilst maintaining the standard keyboard available also in USER mode for other purposes This is the eighth incarnation of the SandMath project which in turn has had a fair number of revisions and iterations on its own One distinct addition has been a secondary Function address Table FAT to provide access to many more functions exceeding the limit imposed by the operating system 64 functions per page Some other refinements consisted in a rationalization of the backbone architecture as well as a more modular approach to each of pages of the module Gone are the 8k and 12k distinctions of the past
208. s be placed in the Z register Conversely if the calculator is set in DEG mode then registers Z Y and X have the three real roots Example1 Calculate the three solutions of the equation x x x 1 0 1 ENTER ENTER ENTER CROOT gt Z 1 000 Y 1 000 X 1 E 10 Shown as rounded number for the real part c Angel M Martin Page 28 of 167 December 2014 SandMath_44 Manual Revision 3x3 Example 2 Calculate the roots of the equation Ax 2x 3x 3x 2 2 ENTER 3 ENTER ENTER 2 CROOT gt Z 0 500 Y 1 000 X 2 000 r Tiai ae y i _ i xj 2x 3x 3x 42 x t 4 4 4 4 94 44 44 4 4 4 4 4 4 4 4 aH f1 05 0 05 1 15 2 25 is Doe Th From the final prompt you know all roots are real since the two last roots are real and the first one must always be real in a cubic equation The value in Z blinks briefly in the display before the final prompt above is presented use RCL Z or RDN RDN to retrieve it No user registers are used outputs the contents of the X and Y registers to the display interpreted by the value in Z to determine whether there are tow real roots or the Real amp Imaginary parts of the complex roots It will be automatically invoked by QROOT all cases and by CROOT real roots when they are executed in RUN mode Note that CROOT will not display the first real root which will be located in Z CVIET
209. s it 14 K3 a2 3 27 S 77 QROUT F 15 ST X 3 2 q243 27 78 STO 01 F 16 RCL O1 a1 T 79 x lt 17 RCL 02 a2 3 80 STO 02 18 a1 a2 3 81 RTN f9 l 2 a2 3 27 a1 a2 3 B all real roots 20 21 84 LASTX 2 85 CHS F 23 STO 03 a0 2 a2 3 27 a1 a2 6 86 SQRT T 24 Xm a0 2 a2 3 27 a1 a2 6 2 F 87 ST X 3 F 25 RCL 01 a1 T 88 X 0 F 26 RCL 02 a2 3 89 1 X 27 XA2 a2 2 9 90 RCL 03 a0 2 a2 3 27 a1 a2 6 23 53 91 ST X 3 a0 2 a243 27 a1 a2 3 T 29 a2 2 3 92 CHS f 30 l a1 a2 2 3 T 93 gt F 31 STO 01 a1 a2 2 3 94 ACOS f 32 j3 95 roe 1 3 a1 a2 2 3 96 34 XA 97 F 35 1 27 al a2 2 3 3 a0 2 a243 27 al a2 e 98 F 36 X lt 0 99 37 GTO 01_____ yes all real roots __ __ ____ 100 STO 05 1 003 38 SQRT complexroots 101 RCL O1 a1 a2 2 3 F 39 ENTER 102 53 40 JENTERA ReX OOO 103 a1 3 a2 2 9 F 41 RCLO03 a0 2 a2 3 27 a1 a2 6 104 CHS a2 2 9 a1 3 a l 105 SQRT 43 CBRI a O 106 ST X 3 F 44 STO 01 cbrt x R3 2 107 2 SQR a2 2 9 a1 3 F 45 XoY 108 YBO 4E 46 CHS 109 RCL 03 47 RCL 03 a0 2 a2 3 27 a1 a2 6 110 COS F 4g 111 49 CBRI as ae 50 STO 03 cbrt x R3 2 413 RCL 02 a2 3 r 51 r i14 F 52 RCL 02 a2 3 F 115 Ps3 l F 116 O 54 X1 a7 Bf i 55 ARCLX 3 118 56 119 ARCL X 3 57 STO 00 real root 120 58 RCL 01 cbrt x R3 2 121 STO IND 05 59 R
210. s it refers to the required programming resources and techniques Function Description Author cA 1 GMF Reciprocal Gamma Continuous fractions JM Baillard Al BETA Euler s Beta function Angel Martin Al GAMMA Euler s Gamma function Lanczos Angel Martin EA ICBT Incomplete Beta function JM Baillard Al ICGM Incomplete Gamma function JM Baillard IGMMA Inverse Gamma function Angel Martin DAS LNGM Logarithm Gamma function ngel Martin Al PSI Digamma Psi function Angel Martin IEA PSIN Polygamma function JM Baillard In mathematics the Gamma function represented by the capital Greek letter F is an extension of the factorial function with its argument shifted down by 1 to real and complex numbers If n is a positive integer then n n 1 Showing the connection to the factorial function For a complex number z with positive real part the Gamma function is defined by C z J tte dt Things become much more interesting in the negative semi plane as can be seen in the plot on the right for real arguments The Gamma function has become standard in pocket calculators either as extended factorials or as proper gamma definition It s already available in the HP 11C and of course on the 15C and that has continued to today s models Implementing it isn t the issue but achieving a reasonable accuracy is the challenge A popular method uses the Stirling approximation to compute Gamma This is relatively
211. s show up in important structures and have direct relevance to some applications e g the equation of a pendulum They also have useful analogies to the functions of trigonometry as indicated by the matching notation sn for sin They were introduced by Carl Gustav Jakob Jacobi 1829 Definition as inverses of elliptic integrals There is a simpler but completely equivalent definition giving the elliptic functions as inverses of the incomplete elliptic integral of the first kind Let i d u ae o 1l msin 6 Then the elliptic functions sn u m cn u m and dn u m are given by sn u m sin cn u m cos and dn u 4 1 msin Here the angle is called the amplitude On occasion dn u A u is called the delta amplitude In the above the value m is a free parameter usually taken to be real 0 lt m lt 1 and so the elliptic functions can be thought of as being given by two variables the amplitude and the parameter m The elliptic functions can be given in a variety of notations which can make the subject unnecessarily confusing Elliptic functions are functions of two variables The first variable might be given in terms of the amplitude or more commonly in terms of v given below The second variable might be given in terms of the parameter n or as the elliptic modulus k where K m or in terms of the modular angle a where m sin a Formulae and Methodology The SandMath implementatio
212. s that transfers into the FOCAL code calculating EI so strictly speaking it s a sort of hybrid natured function e DAYS is taken from the HP Securities Pac It calculates the number of days between two dates The input format is MM DDYYY with the later date in Y and the earlier in X The result is returned to the X reg Example Calculate the number of days elapsed between July 21 1959 and May 21 2014 5 212014 ENTER 7 211959 XF DAYS gt 20 014 00000 and are reciprocal date functions to convert a given date into the Julian day number and back to the calendar date Use flag 00 to select either Julian or gregorian calendars in the conversions The date format is also MM DDYYYY regardless of the time module settings if there s one c Angel M Martin Page 146 of 167 December 2014 SandMath 44 Manual Revision_3x3 Example the date May 21 2014 corresponds to 2 456 799 Gregorian calendar or 2 456 812 Julian calendar day numbers You can also use JDAY to calculate the elapsed number of days between two dates simply converting both to their Julian day numbers and subtracting them If you do that you ll notice a Small discrepancy 18 days between this approach and the resuts from DAYS which leads me to believe that DAYS has some different convention but unfortunately it appears to be a stealth function as there is no documentation for it in the Securities Pac at all These two function
213. stored in the corresponding register and left in the X register as well Using TVM in Programs Notice that TVM is designed to be used interactively but it can also be entered in a program utilizing the merged functions scheme whereby the specific option is specified as an index or argument in the next program step following TMV This will be taken as the function argument of the argument in Rnn always assuming it is a calculation action and not data input regardless of the current status of UF 22 Simply use STO for storing the values in a program The valid values for this argument line are logically 0 to 10 corresponding to the same indexes used in the register allocation and local keys within the menu Had TVM been a sub function and therefore already using 2 steps in a program XF plus index you d appreciate the fact that td take three program lines and 5 bytes to access to any of the financial sub routines This compounded scheme is nothing short of amazing if you ask me c Angel M Martin Page 50 of 167 December 2014 SandMath_44 Manual Revision 3x3 3 Upper Page Functions in detail It s time now to move on to the second page within the SandMath holding the Special Functions and the Statistical and Probability groups Let s see first the Statistical section easier to handle and of much less extension and later on we ll move into high level math taking advantage of the extended lau
214. t 1 x Ta Sat du n l n or the integral form The Riemann zeta function satisfies the functional equation C s 27 sin ZE T t s 1 8 valid for all complex numbers s excluding 0 and 1 which relates its values at points s and 1 s The plots below of the real Zeta function show the negative side with some trivial zeros as well as the pole at x 1 g x il I i Li l TOE ww E N J l 1f i Il l The direct implementation in the SandMath module uses the alternative definitions shown below in a feeble attempt to get a faster convergence which in theory it does although not very noticeably given the long execution times involved The summations are called the Dirichlet Lambda and Eta functions respectively 1 See ms ig a P ick 1 ra x Var al n s _ gl s y n l H 5 n Go ahead and try ZETA with FIX 9 set in the calculator you ll see the successive iterations being shown for each additional term until the final result doesn t change Be aware than MCODE or not t7 take a very long time for small arguments approaching infinite as x approaches zero For values lower than 1 we make use of the following ctx relationship a sort of reflection formula if you wish The interesting fact about this is how it has been De implemented if x lt 1 then the MCODE function branches to a FOCAL program that as part of
215. t 1 However the SandMath uses the implementation developed by JM Baillard using a series expansion for x lt 1 5 and continuous fractions for x gt 1 5 as shown below MP G3 a Ge 3 Pl E pin nz 2 ki 1 n k and E z e FSF z4 1 27 1 Z Examples Calculate ENX for x 1 4 and n 0 2 100 0 ENTER 1 4 XEQ ENX gt E0 1 4 0 176140689 2 ENTER 1 4 XEQ ENX gt E2 1 4 0 0838899263 100 ENTER 1 4 XEQ ENX gt E100 1 4 0 0024558006 Examples Calculate ENX for x 2 and n 3 and for x n 100 3 ENTER 2 XEQ ENX gt E3 2 0 03013337978 100 ENTER XEQ ENX gt E100 100 1 864676429 E 46 Note that we can use ENX to reverse calculate UICGM the upper incomplete gamma which obviously should satisfy the equation shown in the ICGM section LICGM s x UICGM s x I s 01 LBL UICGM 10 1 02 X lt gt Y 11 03 CHS 12 CHS 04 1 13 Y X 05 14 06 X lt gt Y 15 END 07 ENX 08 RCL 00 A short and simple program does it just type 09 RCLO1 n ENTER x XEQ UICGM c Angel M Martin Page 103 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 In mathematics the error function also called the Gauss error function is a special function non elementary of sigmoid shape which occurs in probability statistics and partial differential equations Its definition and the expression based on the Hyper geometric function
216. t you consider accurate in the funct ion s values If you set n smaller the calculator will compute the integral more quickly but it will also presume that the function is accurate to no more than the number of digits shown in the display format FIX and ENG determine an uncertainty in the function that is proportional to the function s magnitude while SCI determines an uncertainty that is independent of the function s magnitude At the same time that the FINTG program returns the resulting integral to the X register the display it returns the Uncertainty of that approximation to the Y register To view this uncertainty value press X lt gt Y No algorithm for numerical integration can compute the exact difference between its approximation and the actual integral But this algorithm estimates an upper bound on this difference which is returned as the uncertainly of the approximation If the uncertainty of an approximation is greater than what you choose to tolerate you can decrease it by specifying more digits in the display format and rerunning FINTG Programming Information You can incorporate FINTG as part of a larger program you create Be sure that your program provides upper and lower limits in the X and Y registers just before it executes FINTG Remember also that INTEG will look in the Alpha register for the name of the program that calculates your function INTEG uses one of the six pending subrout ine returns that the calculator
217. th_44 Manual Revision 3x3 Ratios Sorting and Register Maxima and in the secondary FAT are miniature functions to calculate the percent of a number relative to another one its reference and the delta percentual between the numbers in Y reference and X new value The formulas are T y x 100x y D 100 x y x Example the relative percent of 4 over 25 is 16 You type 25 ENTER 4 XEQ T Example the delta percentual of a change from 85 to 75 is 11 765 and are fundamental functions also inexplicably absent in the original function set They are short and sweet and certainly not complex to calculate The algorithms for these functions are based on the PPC routines GC and LM conveniently modified to get the most out of MCODE environment If a and b are not both zero the greatest common divisor of a and b can be computed by using least common multiple lcm of a and b ee cd a b _ i 2 lem a b Examples GCD 13 17 1 primes GCD 12 18 6 GCD 15 33 3 Examples LCM 13 17 221 LCM 12 18 36 LCM 15 33 165 sorts the contents of the registers specified in the control number in X defined as bbb eee where bbb is the begin register number and eee is the end register number If the control number is positive the sorting is done in ascending order if negative it is done in descending order This function was written by HaJo David and published in PPCCJ V
218. the results grouping the repetitions of a given factor in its corresponding exponent For example for x 126 the three prime factors are 2 3 and 7 with 3 repeated two times For large numbers or when many prime factors exist the display will scroll left to a maximum length of 24 characters This is sufficient for the majority of cases and only runs into limiting situations in very few instances if at all remember that exceeding 24 characters will shift off the display the left characters first that is the original number which doesn t result into any data loss Obviously prime numbers don t have any other factors than themselves For instance for x 17777 PFCT will return E which indeed is hardly debatable Note that only the last two prime factors found will be stored in Y and Z and that the original number will remain in X after the execution terminates A more capable prime factorization program is available in the ALGEBRA module using the matrix functions of the Advantage and Advanced Matrix ROMs to save the solutions in a results matrix See the appendices for a listing of the program used in the SandMath and the more comprehensive one LBL PRME Shown on the left there s an even simpler version that doesn t 1 7 consolidate the multiple factors which will aggravate the length Fa limitation of the ALPHA registers of 24 chrs max The core of the action F A is performed by PRIME therefore t
219. the point for the approximation Repeat this last step as needed by entering the value then R S The accuracy of the approximation depends on the number of coefficients used Choosing a larger n value would give a better precision but will also increase the calculation time The Chebyshev polynomials are useful to approximate f x if f is very complicated like the planetary positions but it s also interesting to use these programs to evaluate the derivative f x where the results are often more accurate than those given by other numerical differentiation methods c Angel M Martin Page 112 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Chebyshev Polynomials j Integral part of the approximation is the calculation of the Chebyshev polynomials which is done internally in the CdT function The SandMath also includes separate functions to calculate Tn x and Un x the first and second kind respectively The Chebyshev polynomials of the first and second kinds are defined by the recurrence relations Ta x 1 Up x 1 hlz 2 U x 2r Tayi lt 207 T a 2 Un i r 22U x Up_i z There are also explicit expressions based on different approaches to defining them trigonometric Square roots and even an expression using the Hypergeometric Function cos narccos z r 1 1 T 2 cosh narccosh x gt 1 1 cosh n arccosh 2 2 lt 1 T vat 1 a oF
220. tines and other usability enhancements 2 5 1 The three geometric solvers CIRCLE different geometric properties of a circle segment SARR Slope Angle Rise and Run also connects to the Triangle solver TRIA Knowing three elements it resolves the other 3 unknown and the area Notice that SOLVER can be launched directly from the mail MA1 launcher using the key as shortcut Upon selecting the desired solver an information message is shortly shown on the display while the key is held depressed followed by NULL if kept pressed to cancel the action If not then the initial execution of the chosen solver starts always by presenting the default menu of choices which can always be recalled by pressing the menu option on the E key It s important to mention that these three are FOCAL programs albeit quite unusual and also stealth to the FAT triggering the different choices for these solvers as local labels therefore the top row keys Should not have any key assignments for this approach to work Note also that the USER mode will be activated automatically by the function Rather than attempt to explain these functions let s refer to the original AECROM user s manual for a first hand and inimitable description of their functionality 1 lt SARR SLOPE ANGLE RISE AND RUN SOLVER The SARR solver computes slopes angles rise and run Your HP 41 must be in USER mode and if you have anything assigned to the
221. ting that it could not be done this new revision includes the all time favorite Solve and Integrate functionality first released by HP in the Advantage Module and now available here as FROOT and FINTG The twist has been the modification of the original code to run in a bank switched configuration located in bank 3 of the upper page The challenge was irresistible and the end result really is a beauty to behold Revision 3x3 also includes the Geometry Solvers from the AECROM The three solvers TRIA CIRC and SARR are consolidated into a single function SOLVER so only one FAT entry was needed No surprisingly it is a launcher by itself The icing on the cake is a full implementation of the Last Function functionality Similar to LastX but applied to the last function executed it allows repeated execution of the same function using a convenient shortcut that bypasses all the launcher paths Very useful for sub functions which cannot be assigned to any key in USER mode The LastFunction is recorded either by name or index using EF and SF Substantial enhancements were made to the main launchers and the sub function handling such as the automated display of the sub function name during a single step SST execution of a program Sub function names are also briefly shown during the execution in RUN mode or when entering in a program using XF providing visual feedback to the user Last but not least revision M also managed
222. tion w 1 z is sometimes called the trigamma function The polygamma function satisfies the follwing Recurrence relation nim am 1 m we z 4 1 wh z and the following Reflection formula d Un z D Un a cot rz The SandMath implements the FOCAL program written by JM Baillard The asymptotic expansion of the Psi function is derived n times and the recurrence relation is used for values lower then 8 to achieve a good accuracy in the result Note also that it uses ALPHA and the stack but no data registers The figure below shows the graphis for the first few values on m color coded as follows Blacik n 0 Red n 1 Yellow n 2 Green n 3 W X Examples Calculate Digamma 1 6 Trigamma 1 6 Tetragamma 1 6 Pentagamma 1 6 0 ENTER 1 6 XEQ PSIN gt Digam 1 6 0 269717877 Psi 1 6 1 ENTER 1 6 XEQ PSIN gt Trigam 1 6 10 44375936 2 ENTER 1 6 XEQ PSIN gt Tetragam 1 6 22 49158811 3 ENTER 1 6 XEQ PSIN gt Pentagam 1 6 283 4070827 c Angel M Martin Page 84 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 3 3 6 Inverse Gamma function Not to be confused with the reciprocal the inverse gamma function is a bit of an elusive one in terms of literature and references perhaps due to a relatively small applicability From a theoretical point of view however it represents an interesting challenge which in
223. tions as specific or limiting case It is defined for z lt 1 by the power series A a n 8 n 2 g TL TL 2Fi a b c z ce ml ao Ch OT provided that c does not equal 0 1 2 Here q n is the Pochhammer symbol which is defined by fi ifn 0 Vn qig 1 n 1 ifn gt 0 Many of the common mathematical functions can be expressed in terms of the hypergeometric function or as limiting cases of it Some typical examples are In 1l z z9F 1 1 2 z 1 z Fla 1 1 z 2 arcsin z zaf 5 a z The relation gt F a b c x 1 x a 2F a c b c x 1 x is used if x lt 0 The Regularized Hypergeometric function has a similar expression for each summing term just divided by Gamma of the corresponding Pochhamer symbol plus the index n REGISTERS RO1 thru RO3 They are to be initialized before executing HGF or RGHF ROO is not used RO1 a R02 b R03 c Output OX X lt 1 F b c x HGF Examples e 1 2 STOO01 2 3 STO02 3 7 STO 03 0 4 XFS HGF gt 1 435242953 a3 2Fe z gt 0 309850661 RHGF Examples e 2 STOOI 3 STO02 7 STOO3 0 4 2F RHGF gt 5353330 290 3 XFL gt 2128 650875 c Angel M Martin Page 153 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Regularized Generalized Hypergeometric Function In mathematics a generalized hypergeometric series is a power series in which the ratio of successive
224. to include the 7ime Value of Money functionallity from the just released TVM ROM an all MCODE implementation of the classic functions that rivals with that in the HP 12C in speed and accuracy c Angel M Martin Page 7 of 167 December 2014 SandMath_44 Manual Revision 3x3 Rather than re invent the wheel the SandMath uses optimized versions of the best math software available for the 41 platform The Geometric Solvers and Curve Fitting programs from the AECROM are a good example as well as all the excellent programs developed by Jean Marc Baillard that have found its way here Very often I added a few enhancements to the code like using 13 digit OS routines or other MCODE tweaks but all credit should go to the original authors All in all I hope you d agree this new incarnation of the SandMath takes good advantage of the developments made and reaches an even balance between enhancements and usability with few compromises to speak of Note that the changes from previous revisions caused a re arrangement of the function entries in the upper page the High Level Math both in the main and auxiliary FATs Be advised that the individual function codes are different in case you have written some programs using the older ones Configuring the SandMath_3x3 Revision N Plugging the SandMath 3x3 module requires using the bank switching configuration options on the 41 CL as well as on Clonix NoVRAM or the MLDL 2k For t
225. torial JM Baillard XFCT Extended FACT _ _ JM Baillard POCH Pochhammer symbol Angel Martin FFCT Falling factorial Angel Martin Large numbers in a calculator like the HP 41 represent a challenge Not only the numeric range represents a problem but also the reduced accuracy limits the practical application of the field Nevertheless the few functions that follow contribute to add further examples of the ingenuity and what s possible using this venerable platform This was the last section added to the SandMath in revision E It also required compacting the few gaps available and transferring some code to the last available space in the Library 4 module Make sure you have matching revision of those two The functions in the table above operate only on integers i e no extension to real numbers using GAMMA Below one of such extensions the Hyperfactorial in a 3D visualization from WolframWorld Re H 2 Im A IH z The figure on the left shows a plot of the four functions on the real line Fibonacci in blue double factorial in red superfactorial in green hyperfactorial in purple Dont expect quantum leaps in number theory here it is after all one of the most difficult branches of math c ngel M Martin Page 68 of 167 December 2014 SandMath_44 Manual Revision 3x3 Pochhammer symbol Rising and falling empires FFCT In mathematics the Pochhammer symbol introd
226. trictions imposed on the parameters which are L is a non negative integer n is real r is non negative c Angel M Martin Page 155 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 Integrals of Bessel Functions j One of the usual approaches is to use the following recurrent relations for the calculation T K f J t dt S I k With Re n gt 0 More specifically for positive integer orders n 1 2 we have r 7 n J Jon t di Jolt dt 2 x Jari 7 0 kf and also i J Jon i t dt Jal 2 Jop X a tt k t There s however another approach based yes here as well on the Generalized Hypergeometric function HGF In fact the applicability of this method extends to the Integro Differential forms of the Bessel functions and so could be used to calculate second primitives or derivatives as well The expressions used in the SandMath for functions ITJ and ITI are as follows D L x K x T 1 1 2F 3 1 2 n 2 2 n 1 u 2 n 2 u 2 n 1 x7 4 D J x K x T n 1 o0F 3 m4 1 2 n 2 2 nt1 p 2 n 2 u 2 n 1 x 4 Where K 287 sqrt z and u 1 for the integral primitive in case you don t believe such a convenience take a look at this WolframAlpha s link http www wolframalpha com input 7i integrate 28bessell 28EN WY2CX 29 29 n 1 n l n4 3 x In dx ai ent 2 n snel n ra a 2 2 2 4 1 fn 1 3 x m00 ax ai
227. ttp cose math bas bg webMathematica webComputing BesselZeros jsp 1 LBL ITIBS k J Ae Ae be aw B 2 XoY 3 STOOO0 1 24048 3 8317 5 1356 6 3802 7 5883 8 7715 4 XoY 5 0 2 55201 7 0156 84172 97610 11 0647 12 3386 Hi i es l l FE 3 86537 10 1735 11 6198 13 0152 14 3725 15 7002 Aha 4 117915 133237 14 7960 16 2235 17 6160 18 9801 10 LBL IZER 5 14 9309 16 4706 17 9598 19 4094 20 8269 22 2178 11 HL MATH Orders 12 STOP 13 STO 00 Ai UREY 14 0 0 6 15 0 ji Aa 16 a da A O2H ai Gane TETE 18 aV a IOW 04 19 f 20 SOLVE ve 21 STOP r x 22 INCX 0AE A 23 GTO00 Nort 24 LBL IBS Ef 2 nw 1s to ts 5 fi 26 X lt Y y oh LP 27 JBS v 4 28 END Note that the program listing also includes code to calculate the Integral of JBS defined as incomplete function with the argument in the upper integration limit Granted it isn t the fastest one in town but such isn t an issue on a modern day emulator and the economy of code cannot be stronger EAG dt ERG dt 1 Jp x EAG dt 0 w Jo Which allegedly satisfies the equation 9 Jn t dt 2 Jnai X Jng3 X Jnas X 0000 c Angel M Martin Page 157 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 3 7 Solve and Integrate Reloaded 3 7 1 Functions description and examples Last but not least what an understatement in this case let s go with a bang welcome to the bank switched implementation of So
228. two values of a physical quantity often power or intensity One of these quantities is often a reference value and in this case the decibel can be used to express the absolute level of the physical quantity as in the case of sound pressure The number of decibels is ten times the logarithm to base 10 of the ratio of two power quantities 1 or of the ratio of the squares of two field amplitude quantities 2 One decibel is one tenth of one bel named in honor of Alexander Graham Bell 1 Power quantities When referring to measurements of power or intensity a ratio can be expressed in decibels by evaluating ten times the base 10 logarithm of the ratio of the measured quantity to the reference level Thus the ratio of a power value P1 to another power value PO is represented by LdB that ratio expressed in decibels 19 which is calculated using the formula below P Lag 10logig gt 0 2 Field quantities When referring to measurements of field amplitude it is usual to consider the ratio of the squares of A1 measured amplitude and AO reference amplitude This is because in most applications power is proportional to the square of amplitude and it is desirable for the two decibel formulations to give the same result in such typical cases Thus the following definition is used A A Lag 10 logy 3 20 logo The function calculates the result of adding or subtracting two values in X and Y expressed in decibel
229. uced by Leo August Pochhammer is the notation x where n is a non negative integer Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below Care needs to be taken to check which interpretation is being used in any particular article The symbol x is used for the rising factorial Sometimes called the Pochhammer function Pochhammer polynomial ascending factorial rising sequential product or upper factorial a a x 1 2 2 x n 1 The symbol X n is used to represent the falling factorial sometimes called the descending factorial 2 falling sequential product lower factorial 2 u x 1 a 2 2 n 1 These conventions are used in combinatory However in the theory of special functions in particular the hypergeometric function the Pochhammer symbol x n is used to represent the rising factorial Extreme caution is therefore needed in interpreting the meanings of both notations The figures below show the rising left and falling factorials for n 0 1 2 3 4 and 2 lt x lt 2 x n 4 n 3 n 2 j Function POCH calculates the rising factorial It expects n and x to be in the Y and X registers respectively i e the usual convention For large values of n the execution time may be very long you can hit any key to stop the execution at any time The falling factorial is related to it a k a Pochh
230. ue to convert depends on the destination base since besides the math numeric factors it s also a function of the Alpha characters available up to Z and the number of them in the display length lt 12 For b 16 the maximum is 9999 E9 or 0x91812D7D600 T gt BS is an enhanced version of the original function also included in Ken Emery s book MCODE for Beginners The author added the PRGM compatible prompting as well as some display trickery to eliminate the visual noise of the original implementation Also provision for the case x 0 was added trivially returning the character 0 for any base The prompt can be filled using the two top keys as shortcuts from 1 to 10 A J or the numeric keys 0 9 TSBS L BASE 3495 USER RAD O12 4 USER RAD Hie 4 Direct DEC lt gt HEX Conversion Because of its importance in computer science the dec to hexadecimal conversions have dedicated MCODE functions in the SandMath and H gt D Use them to convert the number in X to its Hex value in Alpha and vice versa Both functions are mutually reversed and H gt D does an stack lift as well The maximum number allowed is 0x2540BE3FF or 9 99999999 E9 decimal much smaller than with T gt BS so there s a price to pay for convenience These functions were written by William Graham and published in PPCJ V12N6 p19 enhancing in turn the initial versions first published by Derek Amos in PPCCJ V12N1 p3 c Angel M Martin Page 26
231. understand where all this comes from we write the polynomial p x a0 a1 x al10 x 10 so that it takes the same values as f for x 0 x 1 X 2 sesececenes and x 5 With this we get a 11x11 linear system to solve which requires finding the inverse of a Vandermonde matrix like the one below 10 0 0 rl 1 1 1 1 1 1 1 12 4 1024 1 2 4 8 eceeeeeeees 1024 1 5 25 125 cece 5410 1 5 25 125 5 10 J Once the coefficients are calculated we can evaluate the 10 degree Taylor Polynomial as a check to verify the accuracy of the approximation Note that this accuracy will decrease as the argument chosen to evaluate it gets further away from the center i e the value a used to generate them which intuitively can be explained by the need of more terms of the polynomial certainly more than the 10 available to us Let s see a couple of examples of utilization The first one using perhaps the best behaved non rational function We ll use TAYLOR twice to obtain the coefficients around a 0 and a 1 then evaluate the resulting polynomials TO and T1 for x 1 x 2 and x 3 in each case After programming the function as 01 LBL EXP 02 E X 03 RTN let s use h 0 2 as step size for the derivative approximations For the first case then you type c ngel M Martin Page 115 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 ALPHA EXP ALPHA to ente
232. uring the initialization and its address is placed in the RPN return stack just above the other address for the global LBL that calculates f x This will ensure that SIRTN will run after LBL f x is finished ok SO we ve got control back what to do with it Say now that the second auxiliary function SILOOP is the very first and only line in SIRTN that ll send the execution back to our MCODE way to go but this is a new function that has no recollection of the past or knows nothing about whatever was done before unless Unless of course we use the buffer as data structure to do the parameter passing Isn t this brilliant Yes of course that s the answer SILOOP will retrieve from buffer 14 the necessary information to resume picking up exactly where it was left off prior to calling LBL f x Mind you it ll also have to make sure things are as expected when it wakes up is the buffer there which function was run SOLVE or INTEG and react adequately if some of the information is not there This can happen if a user programs SILOOP unadvertingly of course although they could have made it non programmable I suspect they didn t care anyway The last touch of sophistication to speed up things was to also store the addresses of both LBL f x and SIRTN in the buffer itself thus there s no need to search for them in every iteration of the solution and we know there may be from several to many depending of the difficulty of the
233. ved to LastX The program below gives a FOCAL equivalent note the clever programming done by JM Baillard to only perform Y X once per term which reduces the execution times significantly 01 LBL LIN 09 LBLO1 17 RCL 02 02 STO 01 10 RCLO1 18 Y X 03 X lt gt Y 11 RCL 03 19 04 STO 02 12 20 05 1 13 STO 03 21 X Y 06 STO 03 14 ISG 00 22 GTO 01 07 CLX 15 CLX 23 END 08 STO 00 16 RCL 00 2 ENTER 0 7 XEQ LINX gt 0 889377624 3 ENTER 0 7 XEQ LINX gt 0 780063934 c Angel M Martin Page 107 of 167 December 2014 SandMath_ 44 Manual Revision_3x3 3 4 4 Clausen and Lobachevsky Functions CLAUS LOBACH Very closely related to each other by just a change of variable but implemented in the SandMath using different approaches in independent programs for a better coverage allowing comparison between both In mathematics the Clausen function was introduced by Thomas Clausen 1832 and is defined by the following integral sin n Cl 0 tog 2sin t 2 a Cl 6 0 Or n l The expression on the right is a more general definition valid for complex s with Re s gt 1 This definition may be extended to all of the complex plane through analytic continuation however it s not practical for programming as thousands of terms would be required to return accurate results if we used this formula x 2 Integrating by parts gives Cl2 x x Ln 2 sin x 2 2
234. via ascending series are given in the table below The complementary error function denoted erfc is defined as erfc 1 erf x Both functions are shown below for an overview O r Generalized Error Functions ERFN Some authors discuss the more general functions yhprl ni f Bale Ff FI V Tapt DP Notable cases are amp X is a straight line through the origin E 0 x x e sqrt p F X is the equation 1 e x sqrt p gray curve F X is the error function erf x red curve green curve E3 x blue curve E4 x and gold curve E5 x Examples Calculate the first four error functions for x 5 and x 0 9 comparing E 2 x to the results obtained by ERF erf1 erf2 erf3 erf4 i 5 0 221991303 0 520499878 1 641511206 6 687094868 0 520499878 0 000000000 0 9 0 334807217 0 796908213 2 589816366 10 839692051 0 796908213 0 000000000 Note that because ERFN is located in the auxiliary FAT you need to use F to execute it or alternatively XF 061 its corresponding sub function index c Angel M Martin Page 104 of 167 December 2014 SandMath 44 Manual Revision 44_3x3 Appendix 9a Inverse Error Function coefficients galore The inverse error function can be defined in terms of the Maclaurin series 20o 2k 1 erf z 2 TEE ar Where c0 1 and 7 127 _ gt Cm Ck 1 m O f 1 w A m 1 Qn 1 6 90 This really is a bear to handle re
235. vision is made for those cases where n 0 and r 0 returning zero and one as results respectively This avoids DATA ERROR situations in running programs and is consistent with the functions definitions for those singularities Note as well that there is no final rounding made to the result This was the subject of heated debates in the HP Museum forum with some good arguments for a final rounding to ensure that the result is an integer The SandMath implementation however does not perform such final conditioning as the algorithm used seems to always return an integer already Pls Report examples of non conformance if you run into them Example Calculate the number of sets from a sample of 335 objects taken in quantities of 167 Type 335 ENTER 167 XEQ NCR gt 3 0443587 99 Example How many different arrangements are possible of five pictures which can be hung on the wall three at a time Type 5 ENTER 3 XEQ NPR gt 60 00000000 The execution time for these functions may last several seconds depending on the magnitude of the inputs The display will show RUNNING during this time c ngel M Martin Page 53 of 167 December 2014 SandMath_44 Manual Revision 3x3 Linear Regression Let s not digress The following four functions deal with the Linear Regression the simplest type of the curve fitting approximations for a set of data points They complement the native set which basically co
236. when the execution starts and killing it upon termination So as far as the rest of the machine is concerned OS included it is as if buffer 14 had never existed But why all that hassle you d ask Couldn t they have used the normal approach to hold whatever data that needed to be stored in a standard type buffer like every other implementation does I believe the reason was to have an absolute location for the buffer registers with the starting location for buffer 14 always being Ox0CO 192 dec the access and retrival of the values stored there becomes a much easier affair just using their fixed register numbers This may have made using the 15C algorithms simpler and avoids altogether the relative addressing problem present when the buffers are placed in their regular space which incidentally I became very aware of while writing the 41Z complex stack buffer implementation However one of the implications of wedging a buffer below the key assignments area is that the code would first need to move them all as well as all other buffers already present up in memory to make room for the newcomer And conversely this will have to be undone upon termination of the function execution Now you can imagine the housekeeping chores required and the intricacies of the implementation in the code That s why the IO_SRVC event is constantly checking for the presence of buffer 14 proceeding to its removal if found at a non suitable t
237. y corrected with the indirect comparison functions of the CX model X gt NN but unfortunately not quite the same is a quick and simple way to check whether the value in X equals one As usual program execution skips one step if the answer is false establishes the comparison of the rounded values of both X and Y according to the current decimal digits set in the calculator Use it to reduce the computing time albeit at a loss of precision when the algorithms have slow convergence or show oscillating results for larger number of decimals and are two more test functions which criteria is the integer or fractional nature of the number in X Having them available comes very handy for decision branching in FOCAL programs The Fractions section of the module is the natural placement for them e EVEN and Test the divisibility by 2 of the number in X i e whether it is an even or an odd number For non integer values the fractional part will be ignored in the test e ZOUT has been used in FOCAL programs in the SandMath Its most interesting features are perhaps displaying integer values in either real or imaginary parts without any decimals as well as omitting them when equal to zero showing Z 0 if both are null c ngel M Martin Page 31 of 167 December 2014 SandMath_44 Manual Revision 3x3 ec FRAL TIONS 2 2 1 Fraction Arithmetic and Displaying A rudimentary set of fraction arithmetic
238. y too much available space in bank 3 of the upper page to leave it unused so this 500 bytes MCODE rendition of the game written by Roos Cooling see PPCJ V14 N2 p31 was begging to be included As to how to access it the discovery is part of the enjoyment Hint even if it s not geometric it certainly is a Solver of a SHIFT ed type Choose any combination from the available digits on the right which sum matches the target on the left repeating until there s no more digits left YOU WIN or there aren t possible combinations YOU LOSE Use R S to proceed back arrow to delete digits The game will ask you to try again and will keep the count of the scores Finall Disclaimer With just an EE background the author has had his dose of relatively special functions from college to today However not being a mathematician doesn t qualify him as a field expert by any stretch of the imagination Therefore the descriptions that follow are mainly related to the implementation details and not to the general character of the functions This is not a mathematical treatise but just a summary of the important aspects of the project highlighting their applicability to the HP 41 platform c Angel M Martin Page 10 of 167 December 2014 SandMath_44 Manual Revision 3x3 Getting Started Accessing the Functions There are about 230 functions in the SandMath Module With each of the main two pages containing it
239. y trick is what you d use to register it in a program as long as the RCL key is not assigned to another function Notice also that indirect addressing is indeed supported by this scheme just add hex 80 that is decimal 128 to the register number you want to use as indirect register As simple as that So RCL IND 25 will be entered as the following two program lines RCL followed by 153 This however effectively limits the usefulness of the prompt lengthener to the range R100 to R127 because from R128 and on the index is interpreted as an indirect register address instead However the function will allow pressing SHIF and EEX for a combination of IND and prompt lengthener which will work as expected provided that the 128 limit isn t reached enough to make your head spin a little bit Example Store 5 in register R101 and 55555 000 in register R5 This requires some indirect addressing as well say using register Y the sequence would be 101 ENTER 5 STO IND Y and then 55555 STO 5 Then execute RCL IND 101 press RAK SHIFT EEX 0 1 gt to obtain 55555 00 in X Note general purpose prompt lengtheners are a better alternative to the EEX implementation used here Their advantage of course is that they are applicable to all mainframe prompting functions not only to the enhanced RCL Thus for instance you could use it with STO as well removing the need for indirect addressing to store 5 in R101 The AM
240. zed Hypergeometric Funct 19 H gt D Hex to Dec 19 HSI Hyperbolic Sine Integral 20 HMS HMS Multiply by scalar 20 IBS Bessel In Function 21 HMS HMS Divide by scalar 21 ICBT Incomplete Beta Function 22 LOGYX LOG b of X 22 ICGM Incomplete Gamma Function 23 MKEYS Mass Key Assgn 23 IERF Inverse Error function 24 P gt R Complete P R 24 IGMMA Inverse Gamma 25 REM Quotient Reminder 25 JBS Bessel Jn Function 26 QROOT 2nd Degree Roots 26 KBS Bessel Kn Function 27 ROUT Ouput Roots 27 LINX Polylogarithm 28 R gt P Complete R P 28 LNGM Logarythm Gamma Function 29 R gt S Rectangular to Spheric 29 LOBACH Lobachevsky Function 30 S gt R Spheric to Rectangular 30 PSI Digamma Function 31 STLINE Straight Line from Stack 31 PSIN Polygamma 32 T gt BS__ Dec to Base 32 SI Sine Integral 33 VMANT View Mantissa 33 SJBS Spherical J Bessel 34 XA3 XA3 34 SYBS Spherical Y Bessel 35 X 1 IsX 1 35 TAYLOR Taylor Polynomial order 10 36 X YR Is X Y rounded 36 WLO Lambert W Function 37 X gt 0 is X gt 0 37 YBS Bessel Yn 38 X gt Y is X gt Y 38 ZETA Zeta Function Direct method 39 Y 1 X Xth Root of Y 39 ZETAX Zeta Function Borwein 40 YAAX Extended Y X 40 41 YXA Modified Y X 41 T Total Percentual 42 42 CORR Correlation Coefficient 43 D gt F Decimal to Frac 43 COV Sample Covariance 44 F Fraccion Addition 44 CURVE Curve Fitting AECROM 45 F Fraction Substract 45 EVEN is X Even 46 F Fraction Multiply 46 GCD Greatest Common Divisor 47 F

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