User Guide for NICSLU
Contents
1. o AX 1 1 aZ 13513 2 2 9 9 8 8 3 3 4 4 11 11 5 5 10 1 12 12 6 6 4 1 4 1 2 3 2 4 0 5 7 8 10 13 Figure 1 Example to illustrate the CSR format Ai 3 Ap 0 3 NICSLU uses the compressed sparse row CSR format to store a sparse matrix as illustrated in Fig 1 illustrates CSR uses five parameters to describe a sparse matrix as listed in the below e n unsigned integer matrix dimension i e the matrix is nxn NICSLU only supports square matrices e nnz unsigned integer the number of nonzeros in the matrix e Ai unsigned integer array of length nnz storing the column indices of all nonzeros e Ax floating point array of length nnz storing the values of all nonzeros e Ap unsigned integer array of length n 1 storing the location of the first nonzeros of each row in Ai and Ax The first and last elements must be Ap 0 0 and Ap n nnz Values of the ith row of the matrix are stored in Ax Apli Ax Aplil 1 Ax Ap it 1 1 and the corresponding column indices of the nonzeros are stored in AifAp i Ai Aplil 11 Ai Ap i 1 1 number of nonzeros of the ith row is Ap it 1 Apli The matrix is zero based stored which means the row and column indices are in the range from 0 to n 1 The transposed format of CSR is compressed sparse column CSC which is stored in column major 4 Using NICSLU in a2. C C Program 4 1 Data Types Used in NICSLU NICSLU uses several self defined data types as listed in Table 1 in which the first column lists the data types used in NICSLU and the second column lists the corresponding data types in standard C The detailed definitions of the data types can be found in nics_config h Table 1 Data types used in NICSLU data type C type meaning int__t int 32 bit integer uint_t unsigned int 32 bit unsigned integer real_t double double precision floating point bool__t unsigned char boolean value TRUE or FALSE size_t size_t size_t 32 bit or 64 bit unsigned long integer byte_t unsigned char byte 8 bit According to the hardware platform or the compiling configurations you used 4 2 The SNicsLU Structure The SNicsLU structure contains all the configurations matrix data LU factors and statistical information for LU factorization in NICSLU This object appears in most of NICSLU functions as the first parameter Details of SNicsLU are given in nicslu h Only a few member parameters of SNicsLU can be read or written by users which are listed below 4 2 1 Readable Members All the members in floating point array stat are readable and the meanings of each indexed member is as follows e real_t stat 0 analysis time runtime in seconds of NicsLU_Analyze function e real_t stat 1 factorization time runtime in seconds of NicsLU_Factorize or NicsLU_Factorize MT function accordi
3. We have developed a novel parallel algorithm which obtains effective acceleration on shared memory multi core processors 5 7 There are also some other similar software packages such as SuperLU 8 10 PAR DISO 11 etc NICSLU is different from these software packages NICSLU is well suited for extremely sparse matrices such as the matrices in circuit simulation problems Currently NICSLU can be executed on Intel x86 or AMD64 x86 64 hardware plat forms both Windows and Linux are supported If you are using NICSLU in your research please cite the following three papers 1 Xiaoming Chen Wei Wu Yu Wang Hao Yu Huazhong Yang An EScheduler based Data Dependence Analysis and Task Scheduling for Parallel Circuit Simulation Circuits and Systems II Express Briefs IEEE Transactions on vol 58 no 10 pp 702 706 oct 2011 2 Xiaoming Chen Yu Wang Huazhong Yang NICSLU An Adaptive Sparse Ma trix Solver for Parallel Circuit Simulation Computer Aided Design of Integrated Circuits and Systems IEEE Transactions on vol 32 no 2 pp 261 274 feb 2013 3 Xiaoming Chen Yu Wang Huazhong Yang An Adaptive LU Factorization Algo rithm for Parallel Circuit Simulation Design Automation Conference ASP DAC 2012 17th Asia and South Pacific pp 359 364 Jan 30 2012 Feb 2 2012 3 Matrix Format 11 0 0 7 7 1313 0 22 0 0 99 0 8 8 3 3 0 0 0 4 4 0 0 0 1111 0 5 5 0 1 0 0 1212 0 66 ona oo 8 o
4. 4 1988 T A Davis and E P Natarajan Algorithm 907 KLU A Direct Sparse Solver for Circuit Simulation Problems ACM Trans Math Softw 37 3 36 1 36 17 September 2010 I S Duff and J Koster The Design and Use of Algorithms for Permuting Large Entries to the Diagonal of Sparse Matrices SIAM J Matrix Anal Appl 20 4 889 901 July 1999 I S Duff and J Koster On Algorithms For Permuting Large Entries to the Diagonal of a Sparse Matrix SIAM J Matrix Anal Appl 22 4 973 996 July 2000 X Chen Y Wang and H Yang NICSLU An Adaptive Sparse Matrix Solver for Par allel Circuit Simulation Computer Aided Design of Integrated Circuits and Systems IEEE Transactions on 32 2 261 274 feb 2013 X Chen W Wu Y Wang H Yu and H Yang An EScheduler Based Data De pendence Analysis and Task Scheduling for Parallel Circuit Simulation Circuits and Systems II Express Briefs IEEE Transactions on 58 10 702 706 oct 2011 X Chen Y Wang and H Yang An adaptive LU factorization algorithm for parallel circuit simulation In Design Automation Conference ASP DAC 2012 17th Asia and South Pacific pages 359 364 30 2012 feb 2 2012 J W Demmel S C Eisenstat J R Gilbert X S Li and J W H Liu A Supernodal Approach to Sparse Partial Pivoting SIAM J Matrix Anal Appl 20 3 720 755 May 1999 J W Demmel J R Gilbert and X S Li An Asynchronous Parallel Supernodal Al gorithm for Sparse Gau
5. LU Flops SNicsLU nicslu real_t flops This function calculates the number of FLOPs that are required to factorize the matrix Argument flops returns the number of FLOPs if flops is not NULL The number of FLOPs can be also obtained by nicslu gt stat 5 4 4 21 NicsLU_ThreadLoad int_ t NicsLU_ThreadLoad SNicsLU nicslu uint__t thread real__t xthread_flops This function calculates the number of FLOPs of each thread such that one can evaluate load balance of the parallel algorithm Parameter thread specifies the number of threads and the pointer thread flops should be NULL On output this function will allocate memory for thread_flops which is a floating point array with the length of the thread number The number of FLOPs of thread No i is stored in thread_flops i The thread number specified here may not equal to the actual thread number used in parallel factorization Example real__t thread_flops thread_flops NULL factorizing the matrix here NicsLU_ThreadLoad amp nicslu 8 amp thread_load to obtain flops of thread i visit thread_load i 4 4 22 NicsLU_Transpose int_t NicsLU_Transpose uint__t n uint_t nnz real_t ax uint_t ai uint_t ap 13 This function transposes a matrix stored in CSR CSC format On input you should specify n nnz ax ai ap to be the original matrix on output ax ai ap will be overwritten by the transposed matrix 4 4 23 NicsLU_DumpA int_t Ni
6. MatrixValues 2 20048 12 Nie slLUsRefine lt lt 2 0 4 ote a ey ee ee ae a 12 NicsLU ResidUali ir fob ce we a Rae eg gods a de 12 NicsLU_ Throughput cuac aoe ee BAG ke RS Reon 13 NicSLUZELODS i ge 4459 Ai Soe o ee alg Ai fan be pan eae 13 NicsLU ThreadLoad wn as ek ae gon Rk SB EA a a 13 NicsLUzTranspOses nice td le Ma RE ah he Pc a 13 Ni GsSLUDumpA s ss ged ts ae eatin IS de Se ES 14 NicSLUDUMPpEO ici a ee eg el Se eel a 14 NicsLU_ConditionNumber 0008 4 15 15 1 License NICSLU Copyright 2011 2013 Tsinghua University All Rights Reserved This library is free software you can redistribute it and or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation either version 2 1 of the License or at your option any later version This library is distributed in the hope that it will be useful but WITHOUT ANY WARRANTY without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE See the GNU Lesser General Public License for more details You should have received a copy of the GNU Lesser General Public License along with this library if not write to the Free Software Foundation Inc 59 Temple Place Suite 330 Boston MA 02111 1307 USA 2 Introduction NICSLU is a high performance and robust software package for solving large scale sparse linear systems of equations Ax b on shared memory machines It
7. User Guide for NICSLU Xiaoming Chen chenxm05 mails tsinghua edu cn Nano Integrated Circuit and System NICS Laboratory Tsinghua National Laboratory for Information Science and Technology Department of Electronic Engineering Tsinghua University April 16 2013 Contents 1 License 3 2 Introduction 3 3 Matrix Format 4 4 Using NICSLU in a C C Program 5 4 1 Data Types Used in NICSLU o e o 5 4 2 The SNicsLU Structure 6 4 2 1 Readable Members e 6 4 2 2 Writable Members e 7 4 3 Function Return Values o 8 did NICGSLU ROQUES dei A ae eR ey Re ee ES 9 AAt NicsLUIMitialize a a Be ae A BAe eh id da 9 44 2 NicsLU Destroy a cs ic 00000 ae we a a 9 44 3 NicsLU CreateMatriz uta y n wa A Bae d 9 AAA NicsLU_DestroyMatrix sacs u aa toati Boge Ba a AE p ee 10 4 4 5 NicsLU CreateThreads coin aa a 10 4 4 6 NicsLU_DestroyThreadS a 10 4 4 7 NicsLU_CreateScheduler a 10 4 48 NicsLUDestroyScheduler a 10 AAO Nic sLUCANGLyZe porsi Sok dd A e 11 44 10 NicsLUFactobiZze esos a a dado cd 11 MAL NicsLUcReFactord Ze rri A a AY 11 44 12 NicsLU Factorize MT py ci p A A A a A de la 11 4 4 13 NicsLU ReFactorizeMO sac a a oe a 11 4 4 14 4 4 15 4 4 16 4 4 17 4 4 18 4 4 19 4 4 20 4 4 21 4 4 22 4 4 23 4 4 24 4 4 25 5 History NicshLUSo0LVG Vcc A ee ee 11 NiCSLU SOLVeFASE sop soi a Bg Mote St oda Sea AM A ees 12 NicsLU Reset
8. ation 4 4 7 NicsLU_CreateScheduler int__t NicsLU_CreateScheduler SNicsLU nicslu This function creates the task scheduler for parallel LU factorization If you want to run parallel factorization or parallel re factorization it should be called after NicsLU_Analyze If this function returns 1 it indicates that the matrix is not suitable for parallel factorization i e the parallel performance may be even worse then the sequential performance for the specified matrix It returns 0 if the ma trix is suitable for parallel factorization So we suggest you choose the proper factorization function according to the return value of this function The sug gestion is only for factorization but not re factorization NicsLU_ReFactorize_MT can always achieve speedups than NicsLU_ReFactorize The suggestion can be also obtained by nicslu gt stat 13 4 4 8 NicsLU_DestroyScheduler int_t NicsLU DestroyScheduler SNicsLU nicslu 10 This function destroys the scheduler and frees memory used by the scheduler It is contained in NicsLU_Destroy so it can be skipped when you finish your computation 4 4 9 NicsLU_Analyze int_t NicsLU Analyze SNicsLU nicslu This function analyzes the matrix including row column ordering and MC64 scal ing It must be called after NicsLU_CreateMatrix and before any factorization or re factorization 4 4 10 NicsLU_Factorize int__t NicsLU_Factorize SNicsLU nicslu This function performs the numer
9. caling 2 indicates sum scaling and other values indicate no scaling Based on our experiments the scaling methods may have effect in frequency domain simulation but they generally have no effect in time domain transient simulation uint_t cfgil3 default value is 10 It is a scheduling threshold for parallel LU factorization It should be larger than or equal to the number of threads uint_t cfgil4 default value is 2 It is used to pre allocate memory for parallel LU factorization If it is larger NICSLU will use more memory but during parallel LU factorization less memory re allocation will happen e uint_t cfgil7 default value is the number of created threads This number indicates the actual number of threads used for parallel computation You can set it before NicsLU_Factorize_MT or NicsLU_ReFactorize_MT It cannot exceed the number of created threads e real_t cfgf 0 default value is 0 001 It is the partial pivoting tolerance If the diagonal entry has a magnitude greater than or equal to cfgf 0 times the largest magnitude of entries in the pivot row then the diagonal entry is selected as the pivot otherwise an off diagonal pivot will be chosen e real_t cfgf 1 default value is 3 0 It is also used to pre allocate memory for parallel LU factorization If it is larger NICSLU will use more memory but during parallel LU factorization less memory re allocation will happen If not necessary these configuration
10. csLU_DumpA SNicsLU nicslu real_t ax uint_t ai uint_t ap This function stores matrix A into CSR format after factorization The exported matrix is different from the original matrix since row column ordering and MC64 scaling may be performed after analysis and factorization Pointers ax ai ap must be NULL otherwise a memory exception or memory leak will occur This function will allocate memory for these pointers Example real__t ax uint__t ai ap ax NULL ai NULL ap NULL factorizing the matrix here NicsLU_DumpA nicslu tax amp ai ap 4 4 24 NicsLU_DumpLU int_t NicsLU_DumpLU SNicsLU nicslu real_t 1x uint_t li size_t l1p real_t ux uint_t ui size_t up This function stores the factorized LU factors into CSR format Pointers 1x li lp ux ui up must be NULL otherwise a memory exception or memory leak will occur This function will allocate memory for these pointers The exported CSR arrays contain the diagonals of L and U The number of nonzeros of L and U can be obtained from nicslu gt 1_nnz and nicslu gt w_nnz Example real__t lx ux uint__t li ui size_t lp up lx ux NULL li ui NULL lp up NULL factorizing the matrix here NicsLU_DumpLU nicslu amp lx amp li amp lp amp ux amp ui amp up 14 4 4 25 NicsLU_ConditionNumber int_t NicsLU_ConditionNumber SNicsLU nicslu real_t cond This function
11. estimates the condition number of the matrix using the 1 norm If MC64 scaling is used the condition number is reported based on the scaled matrix otherwise it s calculated based on the original matrix Argument cond returns the condition number if cond is not NULL The condition number can be also obtained by nicslu gt stat 6 5 History 2013 Apr 16 version 2 0 NICSLU is distributed under the GNU LGPL license 2012 Dec 16 version 1 2 the framework of NICSLU has a few changes current framework is the final one and will not be changed in future versions add function NicsLU_MemoryUsage add function NicsLU_DumpA add function NicsLU_SolveFast NicsLU_ResetMatrixData is changed to NicsLU_ResetMatrixValues NicsLU_ResidualError is changed to NicsLU_Residual the arguments are also changed memory usage optimization demo programs are changed bug fixes 2011 Oct 19 version 1 1 add function NicsLU_Throughput add function NicsLU_ThreadLoad NicsLU_CreateScheduler doesn t need to be called after NicsLU_CreateThreads anymore correct an error in NicsLU_ResidualError some small improvements some small bug fixes 15 2011 Jul 20 version 1 0 the first version is released References 1 2 10 11 J R Gilbert and T Peierls Sparse partial pivoting in time proportional to arithmetic operations SIAM J Sci Statist Comput 9 5 862 87
12. ical LU factorization i e A LU with partial pivoting It must be called after NicsLU_Analyze 4 4 11 NicsLU_ReFactorize int_t NicsLU_ReFactorize SNicsLU nicslu real_t ax If you want to factorize another matrix with different entry values but with the same nonzero structure this function can be used This function is without partial pivoting so it uses the same pivoting order as the last NicsLU_Factorize or NicsLU_Factorize_MT called It must be called after NicsLU_Factorize or NicsLU_Factorize_MT is called at least once This function executes faster than NicsLU_Factorize however it may cause numerical stability problem Array ax specifies the new matrix values in CSR CSC format 4 4 12 NicsLU_Factorize_MT int_t NicsLU_Factorize MT SNicsLU nicslu It is the parallel version of NicsLU_Factorize NicsLU_CreateScheduler and NicsLU_CreateThreads should be called before this function 4 4 13 NicsLU_ReFactorize_MT int_t NicsLU_ReFactorize MT SNicsLU nicslu real_t ax It is the parallel version of NicsLU_ReFactorize NicsLU_CreateScheduler and NicsLU_CreateThreads should be called before this function 4 4 14 NicsLU_Solve int__t NicsLU_Solve SNicsLU nicslu real_t rhs This function performs right hand solving i e Ly b and Ux y to obtain the solution of Ax b It can be called after any factorization or re factorization functions Array rhs is used for both input and output On input it should store the
13. is the right hand vector input eps is the precision when the residual is smaller than eps the refinement ends maxiter is used to control the refinement iterations If maxiter is nonzero the refinement will end when the number of iterations reaches maxiter otherwise the number of iterations has no limit but it will also end when the residual reaches a minimum value 4 4 18 NicsLU_Residual int_t NicsLU_Residual uint_t n real_t ax uint_t ai uint_t ap real_t x real_t b real_t error int_t norm uint_t mode This function calculates the residual error of Aa b 12 ax ai and ap are the CSR CSC storage of matrix A Array x is the solution vector and b is the right hand vector both are inputs norm indicates the norm of the residual 1 indicates the 1 norm 2 indicates the 2 norm and other values indicate the infinite norm mode indicates the CSR CSC mode zero indicates CSR and non zero indicates CSC On output error returns the residual error error cannot be a NULL pointer 4 4 19 NicsLU_Throughput int_t NicsLU_Throughput SNicsLU nicslu real_t thr This function estimates the memory throughput in bytes that is required to factorize the matrix Parameter thr returns the throughput if thr is not NULL It is an estimation of the throughput the actual memory throughput may not be equal to the estimated value The throughput can be also obtained by nicslu gt stat 12 4 4 20 NicsLU Flops int_t Nics
14. is written by C and can be easily used in C C programs NICSLU solves Ax b by Gaussian elimination method LU factorization It factor izes matrix A into product of a lower triangular matrix L and an upper triangular matrix U i e A LU numerical factorization step and then the solution of Ax b is obtained by solving two triangular equations Ly b and Ux y right hand solving step Matrix A doesn t need to be symmetric or definite but it must be square and full rank otherwise NICSLU cannot solve it Generally speaking a simple description of sparse Gaussian elimination is as follows Matrix A is factorized to EMn 1Rn 1 MR PD AD Q where n is the dimension of A D and De are two diagonal matrices to scale A to enhance numerical stability P and Q are row and column permutation matrices which are used to maintain sparsity i e reduce fill ins Ry is the column permutation matrix generated by partial pivoting that occurs at step k during numerical factorization My is an upper triangular matrix whose kth row contains the multipliers So Ax b can be solved by x Aq d Dr P LMy 1Rn 1 Mi RiQ7 Dz b D OR My sR ML Pb NICSLU is based on the sparse left looking algorithm proposed by Gilbert and Peierl s 1 and KLU algorithm proposed by Davis 2 We use a more efficient static pivoting algorithm HSL_MC64 algorithm 3 4 which is combined with partial pivoting to achieve higher numerical stability
15. ld be set to a non zero value If you need to change the configurations of NICSLU they should be set after calling this function 4 4 4 NicsLU_DestroyMatrix int_t NicsLU DestroyMatrix SNicsLU nicslu This function destroys the matrix and frees memory used by the matrix It is contained in NicsLU Destroy so it can be skipped when you finish your computation In this function all the configurations are set to default values 4 4 5 NicsLU_CreateThreads int__t NicsLU_CreateThreads SNicsLU nicslu uint__t thread bool__t check This function creates the threads for parallel computation The second argument thread specifies the number of threads including the main thread The last argument check specifies whether to check the number of threads or not If it is TRUE then this function will check your specified thread number and if the thread number is larger than the number of cores of your computer the thread number will be set to core number We strongly recommend check TRUE If you only want to run single threaded LU factorization i e sequential factorization this function is not required you should directly call the sequential version of factorization and re factorization functions 4 4 6 NicsLU_DestroyThreads int_t NicsLU DestroyThreads SNicsLU nicslu This function destroys the threads and frees memory used by the threads It is con tained in NicsLU Destroy so it can be skipped when you finish your comput
16. ng to your last calling e real_t stat 2 re factorization time runtime in seconds of NicsLU_ReFactorize or NicsLU ReFactorize MT function according to your last calling e real_t stat 3 right hand solving time runtime in seconds of NicsLU_Solve or NicsLU_SolveFast function according to your last calling e real_t stat 4 initialization time of the scheduler runtime in seconds of NicsLU_CreateSchedule function e real_t stat 5 total number of floating point operations FLOPs to factorize the matrix which is generated by NicsLU_Flops function e real_t stat 6 condition number of the matrix which is estimated by NicsLU ConditionNumber function If MC64 scaling is used the condition number is estimated after MC64 scaling to the matrix e real_t stat 7 estimated speedup if all the cores of the CPU are used which is calculated by NicsLU_CreateScheduler function e real_t stat 8 estimated upper bound of speedup attainable by NICSLU re gardless of the number of cores which is calculated by NicsLU_CreateScheduler function e real_t stat 9 number of cores on the computer If super threading is supported and enabled stat 9 is twice of the number of physical cores e real_t stat 10 estimated number of FLOPs to factorize the matrix which is calculated by NicsLU_CreateScheduler function real_t stat 11 estimated NNZ L U which is calculated by NicsLU_CreateScheduler function real_
17. reated yet e NICSLU_SCHEDULER_NOT_INITIALIZED value 11 The scheduler is not created yet e NICSLU_SINGLE_THREAD value 12 When creating only 1 thread this error occurs since the main thread does not require to be explicitly created e NICSLU_THREADS_INIT_FAIL value 13 The specified threads cannot be created e NICSLU_MATRIX_NOT_ANALYZED value 14 The matrix is not analyzed yet e NICSLU_MATRIX_NOT_FACTORIZED value 15 The matrix is not factorized yet e NICSLU_NUMERIC_OVERFLOW value 16 Numerical overflow has occurred during fac torization 4 4 NICSLU Routines 4 4 1 NicsLU_Initialize int__t NicsLU_Initialize SNicsLU nicslu This function initializes the SNicsLU structure and sets the default configurations It must be called first before any other NICSLU function called 4 4 2 NicsLU Destroy int_t NicsLU Destroy SNicsLU nicslu This function destroys the SNicsLU structure and frees all the memory allocated by NICSLU It must be called at last otherwise memory leak will occur 4 4 3 NicsLU_CreateMatrix int__t NicsLU_CreateMatrix SNicsLU nicslu uint_t n uint_t nnz real_t ax uint_t ai uint_t ap This function initializes the matrix which will be used by NICSLU The matrix is described by the CSR CSC format i e n nnz ax ai ap which is described in Section 3 If your matrix is stored in CSC format you can also directly use it and after calling this function nicslu gt cfgi 0 shou
18. right hand vector b on output it is overwritten by the solution vector x 11 4 4 15 NicsLU_SolveFast int_t NicsLU_SolveFast SNicsLU nicslu real_t rhs It is a faster version of NicsLU_Solve When there are many zeros in the right hand vector b this function can be faster than NicsLU_Solve 4 4 16 NicsLU_ResetMatrixValues int__t NicsLU_ResetMatrixValues SNicsLU nicslu real_t ax Since NicsLU_ReFactorize and NicsLU_ReFactorize_MT are performed without par tial pivoting they may cause numerical stability problem If you want to factorize a new matrix with the same nonzero pattern and with partial pivoting to avoid the potential nu merical stability problem then this function should be used to reset the matrix data And then NicsLU_Factorize or NicsLU_Factorize MT can be used to factorize the new matrix with partial pivoting Array ax specifies the new matrix values in CSR CSC format 4 4 17 NicsLU_Refine int_t NicsLU Refine SNicsLU nicslu real_t x real_t b real_t eps uint_t maxiter When necessary this function can be used to refine the solution However it is not always successful The refinement is implemented as follows compute residual r Ax b while r gt eps solve Ad r update solution 1 x d update residual r Ar b end while The residual is based on the 1 norm Array x should be the solution vector on input on output it will be updated by the refinement Array b
19. s writable members are not recommended to be changed 4 3 Function Return Values Each NICSLU function returns an integer int__t to indicate whether the function is executed successfully or not The return values and their meanings are listed in the below You should check the return value of each function to avoid failures of NICSLU e NICS_OK value 0 The function is executed successfully e NICSLU_GENERAL_FAIL value 1 A simple failure has occurred e NICSLU_ARGUMENT_ERROR value 2 There are some errors with the function argu ments for example you specify NULL to a pointer that is not allowed to be NULL e NICSLU_MEMORY_OVERFLOW value 3 No enough memory e NICSLU_FILE_CANNOT_OPEN value 4 The specified file cannot be opened It is not used in NICSLU e NICSLU_MATRIX_STRUCTURAL_SINGULAR value 5 The matrix is structural singular i e the matrix is not structural full rank e NICSLU_MATRIX_NUMERIC_SINGULAR value 6 The matrix is numerical singular i e there is one row column that does not contain any nonzero elements e NICSLU_MATRIX_INVALID value 7 The matrix is invalid because there are some errors in the CSR CSC storage For example an index is out of range e NICSLU MATRIX ENTRY DUPLICATED value 8 The matrix has duplicated entries in the CSR CSC storage e NICSLU_THREADS_NOT_INITIALIZED value 9 The threads are not created yet e NICSLU_MATRIX_NOT_INITIALIZED value 10 The matrix is not c
20. ssian Elimination SIAM J Matrix Analysis and Applications 20 4 915 952 1999 X S Li and J W Demmel SuperLU_DIST A scalable distributed memory sparse direct solver for unsymmetric linear systems ACM Trans Math Softw 29 2 110 140 June 2003 O Schenk and K Gartner Solving unsymmetric sparse systems of linear equations with PARDISO Future Gener Comput Syst 20 3 475 487 April 2004 16
21. t stat 12 estimated memory throughput in bytes which is calculated by NicsLU_Throughput function real_t stat 13 a suggestion Non zero suggests using NicsLU_Factorize func tion and zero suggests using NicsLU_Factorize_MT function The suggestion is gen erated by NicsLU_CreateScheduler function real_t stat 14 number of off diagonal pivots real_t stat 15 refinement time runtime in seconds of NicsLU Refine real_t stat 16 number of iterations of NicsLU_Refine Besides the above members in stat array the following members are also readable size_t l_nnz u_nnz the two members indicate the number of nonzeros in L and U after factorization including the diagonals of L and U respectively size_t lu_nnz number of nonzeros in L U I after factorization which is equal to l_nnz unnz n 4 2 2 Writable Members All the writable members are in unsigned integer array cfgi and floating point array cfgf uint_t cfgil0 default value is 0 A flag to indicate the CSR or CSC mode Zero indicates CSR and non zero indicates CSC If your matrix is stored in CSC format NICSLU can also directly deal with it In this case NICSLU solves A x b uint_t cfgi 1 default value is 1 A flag to indicate whether using MC64 to scale the matrix before factorization MC64 scaling is strongly recommended uint t cfgi 2 default value is 0 A flag to indicate the scaling method when factorizing the matrix 1 indicates max s
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