# SRC/sgssvx.c File Reference

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

`#include "slu_sdefs.h"`

## 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, 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,
|  | 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.

     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.