SuperLU  5.0
Functions
sgssvx.c File Reference

Solves the system of linear equations A*X=B or A'*X=B. More...

#include "slu_sdefs.h"
Include dependency graph for sgssvx.c:

Functions

void sgssvx (superlu_options_t *options, SuperMatrix *A, int *perm_c, int *perm_r, int *etree, char *equed, float *R, float *C, SuperMatrix *L, SuperMatrix *U, void *work, int lwork, SuperMatrix *B, SuperMatrix *X, float *recip_pivot_growth, float *rcond, float *ferr, float *berr, GlobalLU_t *Glu, mem_usage_t *mem_usage, SuperLUStat_t *stat, int *info)
 

Detailed Description

– SuperLU routine (version 3.0) –
Univ. of California Berkeley, Xerox Palo Alto Research Center,
and Lawrence Berkeley National Lab.
October 15, 2003

Function Documentation

void sgssvx ( superlu_options_t options,
SuperMatrix A,
int *  perm_c,
int *  perm_r,
int *  etree,
char *  equed,
float *  R,
float *  C,
SuperMatrix L,
SuperMatrix U,
void *  work,
int  lwork,
SuperMatrix B,
SuperMatrix X,
float *  recip_pivot_growth,
float *  rcond,
float *  ferr,
float *  berr,
GlobalLU_t Glu,
mem_usage_t mem_usage,
SuperLUStat_t stat,
int *  info 
)

Purpose

SGSSVX solves the system of linear equations A*X=B or A'*X=B, using
the LU factorization from sgstrf(). Error bounds on the solution and
a condition estimate are also provided. It performs the following steps:
  1. If A is stored column-wise (A->Stype = SLU_NC):
     1.1. If options->Equil = YES, scaling factors are computed to
          equilibrate the system:
          options->Trans = NOTRANS:
              diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
          options->Trans = TRANS:
              (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
          options->Trans = CONJ:
              (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
          Whether or not the system will be equilibrated depends on the
          scaling of the matrix A, but if equilibration is used, A is
          overwritten by diag(R)*A*diag(C) and B by diag(R)*B
          (if options->Trans=NOTRANS) or diag(C)*B (if options->Trans
          = TRANS or CONJ).
     1.2. Permute columns of A, forming A*Pc, where Pc is a permutation
          matrix that usually preserves sparsity.
          For more details of this step, see sp_preorder.c.
     1.3. If options->Fact != FACTORED, the LU decomposition is used to
          factor the matrix A (after equilibration if options->Equil = YES)
          as Pr*A*Pc = L*U, with Pr determined by partial pivoting.
     1.4. Compute the reciprocal pivot growth factor.
     1.5. If some U(i,i) = 0, so that U is exactly singular, then the
          routine returns with info = i. Otherwise, the factored form of 
          A is used to estimate the condition number of the matrix A. If
          the reciprocal of the condition number is less than machine
          precision, info = A->ncol+1 is returned as a warning, but the
          routine still goes on to solve for X and computes error bounds
          as described below.
     1.6. The system of equations is solved for X using the factored form
          of A.
     1.7. If options->IterRefine != NOREFINE, iterative refinement is
          applied to improve the computed solution matrix and calculate
          error bounds and backward error estimates for it.
     1.8. If equilibration was used, the matrix X is premultiplied by
          diag(C) (if options->Trans = NOTRANS) or diag(R)
          (if options->Trans = TRANS or CONJ) so that it solves the
          original system before equilibration.
  2. If A is stored row-wise (A->Stype = SLU_NR), apply the above algorithm
     to the transpose of A:
     2.1. If options->Equil = YES, scaling factors are computed to
          equilibrate the system:
          options->Trans = NOTRANS:
              diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
          options->Trans = TRANS:
              (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
          options->Trans = CONJ:
              (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
          Whether or not the system will be equilibrated depends on the
          scaling of the matrix A, but if equilibration is used, A' is
          overwritten by diag(R)*A'*diag(C) and B by diag(R)*B 
          (if trans='N') or diag(C)*B (if trans = 'T' or 'C').
     2.2. Permute columns of transpose(A) (rows of A), 
          forming transpose(A)*Pc, where Pc is a permutation matrix that 
          usually preserves sparsity.
          For more details of this step, see sp_preorder.c.
     2.3. If options->Fact != FACTORED, the LU decomposition is used to
          factor the transpose(A) (after equilibration if 
          options->Fact = YES) as Pr*transpose(A)*Pc = L*U with the
          permutation Pr determined by partial pivoting.
     2.4. Compute the reciprocal pivot growth factor.
     2.5. If some U(i,i) = 0, so that U is exactly singular, then the
          routine returns with info = i. Otherwise, the factored form 
          of transpose(A) is used to estimate the condition number of the
          matrix A. If the reciprocal of the condition number
          is less than machine precision, info = A->nrow+1 is returned as
          a warning, but the routine still goes on to solve for X and
          computes error bounds as described below.
     2.6. The system of equations is solved for X using the factored form
          of transpose(A).
     2.7. If options->IterRefine != NOREFINE, iterative refinement is
          applied to improve the computed solution matrix and calculate
          error bounds and backward error estimates for it.
     2.8. If equilibration was used, the matrix X is premultiplied by
          diag(C) (if options->Trans = NOTRANS) or diag(R) 
          (if options->Trans = TRANS or CONJ) so that it solves the
          original system before equilibration.
  See supermatrix.h for the definition of 'SuperMatrix' structure.

Arguments

options (input) superlu_options_t*
        The structure defines the input parameters to control
        how the LU decomposition will be performed and how the
        system will be solved.
A       (input/output) SuperMatrix*
        Matrix A in A*X=B, of dimension (A->nrow, A->ncol). The number
        of the linear equations is A->nrow. Currently, the type of A can be:
        Stype = SLU_NC or SLU_NR, Dtype = SLU_D, Mtype = SLU_GE.
        In the future, more general A may be handled.
        On entry, If options->Fact = FACTORED and equed is not 'N', 
        then A must have been equilibrated by the scaling factors in
        R and/or C.  
        On exit, A is not modified if options->Equil = NO, or if 
        options->Equil = YES but equed = 'N' on exit.
        Otherwise, if options->Equil = YES and equed is not 'N',
        A is scaled as follows:
        If A->Stype = SLU_NC:
          equed = 'R':  A := diag(R) * A
          equed = 'C':  A := A * diag(C)
          equed = 'B':  A := diag(R) * A * diag(C).
        If A->Stype = SLU_NR:
          equed = 'R':  transpose(A) := diag(R) * transpose(A)
          equed = 'C':  transpose(A) := transpose(A) * diag(C)
          equed = 'B':  transpose(A) := diag(R) * transpose(A) * diag(C).
perm_c  (input/output) int*
        If A->Stype = SLU_NC, Column permutation vector of size A->ncol,
        which defines the permutation matrix Pc; perm_c[i] = j means
        column i of A is in position j in A*Pc.
        On exit, perm_c may be overwritten by the product of the input
        perm_c and a permutation that postorders the elimination tree
        of Pc'*A'*A*Pc; perm_c is not changed if the elimination tree
        is already in postorder.
        If A->Stype = SLU_NR, column permutation vector of size A->nrow,
        which describes permutation of columns of transpose(A) 
        (rows of A) as described above.
perm_r  (input/output) int*
        If A->Stype = SLU_NC, row permutation vector of size A->nrow, 
        which defines the permutation matrix Pr, and is determined
        by partial pivoting.  perm_r[i] = j means row i of A is in 
        position j in Pr*A.
        If A->Stype = SLU_NR, permutation vector of size A->ncol, which
        determines permutation of rows of transpose(A)
        (columns of A) as described above.
        If options->Fact = SamePattern_SameRowPerm, the pivoting routine
        will try to use the input perm_r, unless a certain threshold
        criterion is violated. In that case, perm_r is overwritten by a
        new permutation determined by partial pivoting or diagonal
        threshold pivoting.
        Otherwise, perm_r is output argument.
etree   (input/output) int*,  dimension (A->ncol)
        Elimination tree of Pc'*A'*A*Pc.
        If options->Fact != FACTORED and options->Fact != DOFACT,
        etree is an input argument, otherwise it is an output argument.
        Note: etree is a vector of parent pointers for a forest whose
        vertices are the integers 0 to A->ncol-1; etree[root]==A->ncol.
equed   (input/output) char*
        Specifies the form of equilibration that was done.
        = 'N': No equilibration.
        = 'R': Row equilibration, i.e., A was premultiplied by diag(R).
        = 'C': Column equilibration, i.e., A was postmultiplied by diag(C).
        = 'B': Both row and column equilibration, i.e., A was replaced 
               by diag(R)*A*diag(C).
        If options->Fact = FACTORED, equed is an input argument,
        otherwise it is an output argument.
R       (input/output) float*, dimension (A->nrow)
        The row scale factors for A or transpose(A).
        If equed = 'R' or 'B', A (if A->Stype = SLU_NC) or transpose(A)
            (if A->Stype = SLU_NR) is multiplied on the left by diag(R).
        If equed = 'N' or 'C', R is not accessed.
        If options->Fact = FACTORED, R is an input argument,
            otherwise, R is output.
        If options->zFact = FACTORED and equed = 'R' or 'B', each element
            of R must be positive.
C       (input/output) float*, dimension (A->ncol)
        The column scale factors for A or transpose(A).
        If equed = 'C' or 'B', A (if A->Stype = SLU_NC) or transpose(A)
            (if A->Stype = SLU_NR) is multiplied on the right by diag(C).
        If equed = 'N' or 'R', C is not accessed.
        If options->Fact = FACTORED, C is an input argument,
            otherwise, C is output.
        If options->Fact = FACTORED and equed = 'C' or 'B', each element
            of C must be positive.
L       (output) SuperMatrix*
        The factor L from the factorization
            Pr*A*Pc=L*U              (if A->Stype SLU_= NC) or
            Pr*transpose(A)*Pc=L*U   (if A->Stype = SLU_NR).
        Uses compressed row subscripts storage for supernodes, i.e.,
        L has types: Stype = SLU_SC, Dtype = SLU_S, Mtype = SLU_TRLU.
U       (output) SuperMatrix*
        The factor U from the factorization
            Pr*A*Pc=L*U              (if A->Stype = SLU_NC) or
            Pr*transpose(A)*Pc=L*U   (if A->Stype = SLU_NR).
        Uses column-wise storage scheme, i.e., U has types:
        Stype = SLU_NC, Dtype = SLU_S, Mtype = SLU_TRU.
work    (workspace/output) void*, size (lwork) (in bytes)
        User supplied workspace, should be large enough
        to hold data structures for factors L and U.
        On exit, if fact is not 'F', L and U point to this array.
lwork   (input) int
        Specifies the size of work array in bytes.
        = 0:  allocate space internally by system malloc;
        > 0:  use user-supplied work array of length lwork in bytes,
              returns error if space runs out.
        = -1: the routine guesses the amount of space needed without
              performing the factorization, and returns it in
              mem_usage->total_needed; no other side effects.
        See argument 'mem_usage' for memory usage statistics.
B       (input/output) SuperMatrix*
        B has types: Stype = SLU_DN, Dtype = SLU_S, Mtype = SLU_GE.
        On entry, the right hand side matrix.
        If B->ncol = 0, only LU decomposition is performed, the triangular
                        solve is skipped.
        On exit,
           if equed = 'N', B is not modified; otherwise
           if A->Stype = SLU_NC:
              if options->Trans = NOTRANS and equed = 'R' or 'B',
                 B is overwritten by diag(R)*B;
              if options->Trans = TRANS or CONJ and equed = 'C' of 'B',
                 B is overwritten by diag(C)*B;
           if A->Stype = SLU_NR:
              if options->Trans = NOTRANS and equed = 'C' or 'B',
                 B is overwritten by diag(C)*B;
              if options->Trans = TRANS or CONJ and equed = 'R' of 'B',
                 B is overwritten by diag(R)*B.
X       (output) SuperMatrix*
        X has types: Stype = SLU_DN, Dtype = SLU_S, Mtype = SLU_GE. 
        If info = 0 or info = A->ncol+1, X contains the solution matrix
        to the original system of equations. Note that A and B are modified
        on exit if equed is not 'N', and the solution to the equilibrated
        system is inv(diag(C))*X if options->Trans = NOTRANS and
        equed = 'C' or 'B', or inv(diag(R))*X if options->Trans = 'T' or 'C'
        and equed = 'R' or 'B'.
recip_pivot_growth (output) float*
        The reciprocal pivot growth factor max_j( norm(A_j)/norm(U_j) ).
        The infinity norm is used. If recip_pivot_growth is much less
        than 1, the stability of the LU factorization could be poor.
rcond   (output) float*
        The estimate of the reciprocal condition number of the matrix A
        after equilibration (if done). If rcond is less than the machine
        precision (in particular, if rcond = 0), the matrix is singular
        to working precision. This condition is indicated by a return
        code of info > 0.
FERR    (output) float*, dimension (B->ncol)   
        The estimated forward error bound for each solution vector   
        X(j) (the j-th column of the solution matrix X).   
        If XTRUE is the true solution corresponding to X(j), FERR(j) 
        is an estimated upper bound for the magnitude of the largest 
        element in (X(j) - XTRUE) divided by the magnitude of the   
        largest element in X(j).  The estimate is as reliable as   
        the estimate for RCOND, and is almost always a slight   
        overestimate of the true error.
        If options->IterRefine = NOREFINE, ferr = 1.0.
BERR    (output) float*, dimension (B->ncol)
        The componentwise relative backward error of each solution   
        vector X(j) (i.e., the smallest relative change in   
        any element of A or B that makes X(j) an exact solution).
        If options->IterRefine = NOREFINE, berr = 1.0.
Glu      (input/output) GlobalLU_t *
         If options->Fact == SamePattern_SameRowPerm, it is an input;
             The matrix A will be factorized assuming that a 
             factorization of a matrix with the same sparsity pattern
             and similar numerical values was performed prior to this one.
             Therefore, this factorization will reuse both row and column
        scaling factors R and C, both row and column permutation
        vectors perm_r and perm_c, and the L & U data structures
        set up from the previous factorization.
         Otherwise, it is an output.
mem_usage (output) mem_usage_t*
        Record the memory usage statistics, consisting of following fields:

  • for_lu (float) The amount of space used in bytes for L data structures.
  • total_needed (float) The amount of space needed in bytes to perform factorization.
  • expansions (int) The number of memory expansions during the LU factorization.
stat   (output) SuperLUStat_t*
       Record the statistics on runtime and floating-point operation count.
       See slu_util.h for the definition of 'SuperLUStat_t'.
info    (output) int*
        = 0: successful exit   
        < 0: if info = -i, the i-th argument had an illegal value   
        > 0: if info = i, and i is   
             <= A->ncol: U(i,i) is exactly zero. The factorization has   
                   been completed, but the factor U is exactly   
                   singular, so the solution and error bounds   
                   could not be computed.   
             = A->ncol+1: U is nonsingular, but RCOND is less than machine
                   precision, meaning that the matrix is singular to
                   working precision. Nevertheless, the solution and
                   error bounds are computed because there are a number
                   of situations where the computed solution can be more
                   accurate than the value of RCOND would suggest.   
             > A->ncol+1: number of bytes allocated when memory allocation
                   failure occurred, plus A->ncol.

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