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<meta content="text/html;charset=UTF-8" http-equiv="Content-Type">
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<title>SuperLU: SRC/zgssvx.c File Reference</title>
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<link href="doxygen.css" rel="stylesheet" type="text/css">
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Main Page
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Data Structures
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Files
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SRC/zgssvx.c File ReferenceSolves the system of linear equations A*X=B or A'*X=B. More...
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#include "slu_zdefs.h"
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Functions
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void zgssvx (superlu_options_t *options, SuperMatrix *A, int *perm_c, int *perm_r, int *etree, char *equed, double *R, double *C, SuperMatrix *L, SuperMatrix *U, void *work, int lwork, SuperMatrix *B, SuperMatrix *X, double *recip_pivot_growth, double *rcond, double *ferr, double *berr, mem_usage_t *mem_usage, SuperLUStat_t *stat, int *info)
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Detailed Description
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-- SuperLU routine (version 3.0) --
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Univ. of California Berkeley, Xerox Palo Alto Research Center,
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and Lawrence Berkeley National Lab.
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October 15, 2003
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Function Documentation
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void zgssvx
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(
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superlu_options_t *
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options,
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SuperMatrix *
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A,
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int *
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perm_c,
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int *
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perm_r,
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int *
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etree,
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char *
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equed,
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double *
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R,
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double *
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C,
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SuperMatrix *
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L,
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SuperMatrix *
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U,
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void *
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work,
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int
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lwork,
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SuperMatrix *
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B,
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SuperMatrix *
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X,
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double *
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recip_pivot_growth,
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double *
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rcond,
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double *
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ferr,
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double *
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berr,
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mem_usage_t *
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mem_usage,
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SuperLUStat_t *
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stat,
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int *
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info
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)
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Purpose
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=======
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ZGSSVX solves the system of linear equations A*X=B or A'*X=B, using
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the LU factorization from zgstrf(). Error bounds on the solution and
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a condition estimate are also provided. It performs the following steps:
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1. If A is stored column-wise (A->Stype = SLU_NC):
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1.1. If options->Equil = YES, scaling factors are computed to
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equilibrate the system:
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options->Trans = NOTRANS:
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diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
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options->Trans = TRANS:
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(diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
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options->Trans = CONJ:
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(diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
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Whether or not the system will be equilibrated depends on the
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scaling of the matrix A, but if equilibration is used, A is
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overwritten by diag(R)*A*diag(C) and B by diag(R)*B
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(if options->Trans=NOTRANS) or diag(C)*B (if options->Trans
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= TRANS or CONJ).
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1.2. Permute columns of A, forming A*Pc, where Pc is a permutation
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matrix that usually preserves sparsity.
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For more details of this step, see sp_preorder.c.
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1.3. If options->Fact != FACTORED, the LU decomposition is used to
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factor the matrix A (after equilibration if options->Equil = YES)
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as Pr*A*Pc = L*U, with Pr determined by partial pivoting.
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1.4. Compute the reciprocal pivot growth factor.
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1.5. If some U(i,i) = 0, so that U is exactly singular, then the
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routine returns with info = i. Otherwise, the factored form of
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A is used to estimate the condition number of the matrix A. If
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the reciprocal of the condition number is less than machine
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precision, info = A->ncol+1 is returned as a warning, but the
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routine still goes on to solve for X and computes error bounds
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as described below.
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1.6. The system of equations is solved for X using the factored form
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of A.
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1.7. If options->IterRefine != NOREFINE, iterative refinement is
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applied to improve the computed solution matrix and calculate
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error bounds and backward error estimates for it.
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1.8. If equilibration was used, the matrix X is premultiplied by
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diag(C) (if options->Trans = NOTRANS) or diag(R)
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(if options->Trans = TRANS or CONJ) so that it solves the
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original system before equilibration.
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2. If A is stored row-wise (A->Stype = SLU_NR), apply the above algorithm
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to the transpose of A:
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2.1. If options->Equil = YES, scaling factors are computed to
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equilibrate the system:
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options->Trans = NOTRANS:
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diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
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options->Trans = TRANS:
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(diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
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options->Trans = CONJ:
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(diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
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Whether or not the system will be equilibrated depends on the
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scaling of the matrix A, but if equilibration is used, A' is
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overwritten by diag(R)*A'*diag(C) and B by diag(R)*B
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(if trans='N') or diag(C)*B (if trans = 'T' or 'C').
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2.2. Permute columns of transpose(A) (rows of A),
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forming transpose(A)*Pc, where Pc is a permutation matrix that
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usually preserves sparsity.
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For more details of this step, see sp_preorder.c.
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2.3. If options->Fact != FACTORED, the LU decomposition is used to
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factor the transpose(A) (after equilibration if
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options->Fact = YES) as Pr*transpose(A)*Pc = L*U with the
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permutation Pr determined by partial pivoting.
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2.4. Compute the reciprocal pivot growth factor.
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2.5. If some U(i,i) = 0, so that U is exactly singular, then the
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routine returns with info = i. Otherwise, the factored form
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of transpose(A) is used to estimate the condition number of the
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matrix A. If the reciprocal of the condition number
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is less than machine precision, info = A->nrow+1 is returned as
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a warning, but the routine still goes on to solve for X and
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computes error bounds as described below.
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2.6. The system of equations is solved for X using the factored form
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of transpose(A).
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2.7. If options->IterRefine != NOREFINE, iterative refinement is
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applied to improve the computed solution matrix and calculate
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error bounds and backward error estimates for it.
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2.8. If equilibration was used, the matrix X is premultiplied by
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diag(C) (if options->Trans = NOTRANS) or diag(R)
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(if options->Trans = TRANS or CONJ) so that it solves the
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original system before equilibration.
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See supermatrix.h for the definition of 'SuperMatrix' structure.
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Arguments
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=========
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options (input) superlu_options_t*
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The structure defines the input parameters to control
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how the LU decomposition will be performed and how the
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system will be solved.
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A (input/output) SuperMatrix*
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Matrix A in A*X=B, of dimension (A->nrow, A->ncol). The number
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of the linear equations is A->nrow. Currently, the type of A can be:
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Stype = SLU_NC or SLU_NR, Dtype = SLU_D, Mtype = SLU_GE.
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In the future, more general A may be handled.
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On entry, If options->Fact = FACTORED and equed is not 'N',
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then A must have been equilibrated by the scaling factors in
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R and/or C.
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On exit, A is not modified if options->Equil = NO, or if
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options->Equil = YES but equed = 'N' on exit.
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Otherwise, if options->Equil = YES and equed is not 'N',
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A is scaled as follows:
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If A->Stype = SLU_NC:
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equed = 'R': A := diag(R) * A
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equed = 'C': A := A * diag(C)
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equed = 'B': A := diag(R) * A * diag(C).
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If A->Stype = SLU_NR:
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equed = 'R': transpose(A) := diag(R) * transpose(A)
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equed = 'C': transpose(A) := transpose(A) * diag(C)
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equed = 'B': transpose(A) := diag(R) * transpose(A) * diag(C).
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perm_c (input/output) int*
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If A->Stype = SLU_NC, Column permutation vector of size A->ncol,
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which defines the permutation matrix Pc; perm_c[i] = j means
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column i of A is in position j in A*Pc.
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On exit, perm_c may be overwritten by the product of the input
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perm_c and a permutation that postorders the elimination tree
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of Pc'*A'*A*Pc; perm_c is not changed if the elimination tree
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is already in postorder.
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If A->Stype = SLU_NR, column permutation vector of size A->nrow,
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which describes permutation of columns of transpose(A)
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(rows of A) as described above.
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perm_r (input/output) int*
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If A->Stype = SLU_NC, row permutation vector of size A->nrow,
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which defines the permutation matrix Pr, and is determined
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7d535a |
by partial pivoting. perm_r[i] = j means row i of A is in
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kusano |
7d535a |
position j in Pr*A.
|
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kusano |
7d535a |
If A->Stype = SLU_NR, permutation vector of size A->ncol, which
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kusano |
7d535a |
determines permutation of rows of transpose(A)
|
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kusano |
7d535a |
(columns of A) as described above.
|
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kusano |
7d535a |
If options->Fact = SamePattern_SameRowPerm, the pivoting routine
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kusano |
7d535a |
will try to use the input perm_r, unless a certain threshold
|
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kusano |
7d535a |
criterion is violated. In that case, perm_r is overwritten by a
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kusano |
7d535a |
new permutation determined by partial pivoting or diagonal
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kusano |
7d535a |
threshold pivoting.
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kusano |
7d535a |
Otherwise, perm_r is output argument.
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kusano |
7d535a |
etree (input/output) int*, dimension (A->ncol)
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kusano |
7d535a |
Elimination tree of Pc'*A'*A*Pc.
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kusano |
7d535a |
If options->Fact != FACTORED and options->Fact != DOFACT,
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kusano |
7d535a |
etree is an input argument, otherwise it is an output argument.
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kusano |
7d535a |
Note: etree is a vector of parent pointers for a forest whose
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kusano |
7d535a |
vertices are the integers 0 to A->ncol-1; etree[root]==A->ncol.
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kusano |
7d535a |
equed (input/output) char*
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kusano |
7d535a |
Specifies the form of equilibration that was done.
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kusano |
7d535a |
= 'N': No equilibration.
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kusano |
7d535a |
= 'R': Row equilibration, i.e., A was premultiplied by diag(R).
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kusano |
7d535a |
= 'C': Column equilibration, i.e., A was postmultiplied by diag(C).
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kusano |
7d535a |
= 'B': Both row and column equilibration, i.e., A was replaced
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kusano |
7d535a |
by diag(R)*A*diag(C).
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kusano |
7d535a |
If options->Fact = FACTORED, equed is an input argument,
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7d535a |
otherwise it is an output argument.
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kusano |
7d535a |
R (input/output) double*, dimension (A->nrow)
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kusano |
7d535a |
The row scale factors for A or transpose(A).
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kusano |
7d535a |
If equed = 'R' or 'B', A (if A->Stype = SLU_NC) or transpose(A)
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kusano |
7d535a |
(if A->Stype = SLU_NR) is multiplied on the left by diag(R).
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kusano |
7d535a |
If equed = 'N' or 'C', R is not accessed.
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kusano |
7d535a |
If options->Fact = FACTORED, R is an input argument,
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7d535a |
otherwise, R is output.
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kusano |
7d535a |
If options->zFact = FACTORED and equed = 'R' or 'B', each element
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7d535a |
of R must be positive.
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kusano |
7d535a |
C (input/output) double*, dimension (A->ncol)
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kusano |
7d535a |
The column scale factors for A or transpose(A).
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kusano |
7d535a |
If equed = 'C' or 'B', A (if A->Stype = SLU_NC) or transpose(A)
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kusano |
7d535a |
(if A->Stype = SLU_NR) is multiplied on the right by diag(C).
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kusano |
7d535a |
If equed = 'N' or 'R', C is not accessed.
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kusano |
7d535a |
If options->Fact = FACTORED, C is an input argument,
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kusano |
7d535a |
otherwise, C is output.
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kusano |
7d535a |
If options->Fact = FACTORED and equed = 'C' or 'B', each element
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|
kusano |
7d535a |
of C must be positive.
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kusano |
7d535a |
L (output) SuperMatrix*
|
|
kusano |
7d535a |
The factor L from the factorization
|
|
kusano |
7d535a |
Pr*A*Pc=L*U (if A->Stype SLU_= NC) or
|
|
kusano |
7d535a |
Pr*transpose(A)*Pc=L*U (if A->Stype = SLU_NR).
|
|
kusano |
7d535a |
Uses compressed row subscripts storage for supernodes, i.e.,
|
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kusano |
7d535a |
L has types: Stype = SLU_SC, Dtype = SLU_Z, Mtype = SLU_TRLU.
|
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kusano |
7d535a |
U (output) SuperMatrix*
|
|
kusano |
7d535a |
The factor U from the factorization
|
|
kusano |
7d535a |
Pr*A*Pc=L*U (if A->Stype = SLU_NC) or
|
|
kusano |
7d535a |
Pr*transpose(A)*Pc=L*U (if A->Stype = SLU_NR).
|
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kusano |
7d535a |
Uses column-wise storage scheme, i.e., U has types:
|
|
kusano |
7d535a |
Stype = SLU_NC, Dtype = SLU_Z, Mtype = SLU_TRU.
|
|
kusano |
7d535a |
work (workspace/output) void*, size (lwork) (in bytes)
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|
kusano |
7d535a |
User supplied workspace, should be large enough
|
|
kusano |
7d535a |
to hold data structures for factors L and U.
|
|
kusano |
7d535a |
On exit, if fact is not 'F', L and U point to this array.
|
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kusano |
7d535a |
lwork (input) int
|
|
kusano |
7d535a |
Specifies the size of work array in bytes.
|
|
kusano |
7d535a |
= 0: allocate space internally by system malloc;
|
|
kusano |
7d535a |
> 0: use user-supplied work array of length lwork in bytes,
|
|
kusano |
7d535a |
returns error if space runs out.
|
|
kusano |
7d535a |
= -1: the routine guesses the amount of space needed without
|
|
kusano |
7d535a |
performing the factorization, and returns it in
|
|
kusano |
7d535a |
mem_usage->total_needed; no other side effects.
|
|
kusano |
7d535a |
See argument 'mem_usage' for memory usage statistics.
|
|
kusano |
7d535a |
B (input/output) SuperMatrix*
|
|
kusano |
7d535a |
B has types: Stype = SLU_DN, Dtype = SLU_Z, Mtype = SLU_GE.
|
|
kusano |
7d535a |
On entry, the right hand side matrix.
|
|
kusano |
7d535a |
If B->ncol = 0, only LU decomposition is performed, the triangular
|
|
kusano |
7d535a |
solve is skipped.
|
|
kusano |
7d535a |
On exit,
|
|
kusano |
7d535a |
if equed = 'N', B is not modified; otherwise
|
|
kusano |
7d535a |
if A->Stype = SLU_NC:
|
|
kusano |
7d535a |
if options->Trans = NOTRANS and equed = 'R' or 'B',
|
|
kusano |
7d535a |
B is overwritten by diag(R)*B;
|
|
kusano |
7d535a |
if options->Trans = TRANS or CONJ and equed = 'C' of 'B',
|
|
kusano |
7d535a |
B is overwritten by diag(C)*B;
|
|
kusano |
7d535a |
if A->Stype = SLU_NR:
|
|
kusano |
7d535a |
if options->Trans = NOTRANS and equed = 'C' or 'B',
|
|
kusano |
7d535a |
B is overwritten by diag(C)*B;
|
|
kusano |
7d535a |
if options->Trans = TRANS or CONJ and equed = 'R' of 'B',
|
|
kusano |
7d535a |
B is overwritten by diag(R)*B.
|
|
kusano |
7d535a |
X (output) SuperMatrix*
|
|
kusano |
7d535a |
X has types: Stype = SLU_DN, Dtype = SLU_Z, Mtype = SLU_GE.
|
|
kusano |
7d535a |
If info = 0 or info = A->ncol+1, X contains the solution matrix
|
|
kusano |
7d535a |
to the original system of equations. Note that A and B are modified
|
|
kusano |
7d535a |
on exit if equed is not 'N', and the solution to the equilibrated
|
|
kusano |
7d535a |
system is inv(diag(C))*X if options->Trans = NOTRANS and
|
|
kusano |
7d535a |
equed = 'C' or 'B', or inv(diag(R))*X if options->Trans = 'T' or 'C'
|
|
kusano |
7d535a |
and equed = 'R' or 'B'.
|
|
kusano |
7d535a |
recip_pivot_growth (output) double*
|
|
kusano |
7d535a |
The reciprocal pivot growth factor max_j( norm(A_j)/norm(U_j) ).
|
|
kusano |
7d535a |
The infinity norm is used. If recip_pivot_growth is much less
|
|
kusano |
7d535a |
than 1, the stability of the LU factorization could be poor.
|
|
kusano |
7d535a |
rcond (output) double*
|
|
kusano |
7d535a |
The estimate of the reciprocal condition number of the matrix A
|
|
kusano |
7d535a |
after equilibration (if done). If rcond is less than the machine
|
|
kusano |
7d535a |
precision (in particular, if rcond = 0), the matrix is singular
|
|
kusano |
7d535a |
to working precision. This condition is indicated by a return
|
|
kusano |
7d535a |
code of info > 0.
|
|
kusano |
7d535a |
FERR (output) double*, dimension (B->ncol)
|
|
kusano |
7d535a |
The estimated forward error bound for each solution vector
|
|
kusano |
7d535a |
X(j) (the j-th column of the solution matrix X).
|
|
kusano |
7d535a |
If XTRUE is the true solution corresponding to X(j), FERR(j)
|
|
kusano |
7d535a |
is an estimated upper bound for the magnitude of the largest
|
|
kusano |
7d535a |
element in (X(j) - XTRUE) divided by the magnitude of the
|
|
kusano |
7d535a |
largest element in X(j). The estimate is as reliable as
|
|
kusano |
7d535a |
the estimate for RCOND, and is almost always a slight
|
|
kusano |
7d535a |
overestimate of the true error.
|
|
kusano |
7d535a |
If options->IterRefine = NOREFINE, ferr = 1.0.
|
|
kusano |
7d535a |
BERR (output) double*, dimension (B->ncol)
|
|
kusano |
7d535a |
The componentwise relative backward error of each solution
|
|
kusano |
7d535a |
vector X(j) (i.e., the smallest relative change in
|
|
kusano |
7d535a |
any element of A or B that makes X(j) an exact solution).
|
|
kusano |
7d535a |
If options->IterRefine = NOREFINE, berr = 1.0.
|
|
kusano |
7d535a |
mem_usage (output) mem_usage_t*
|
|
kusano |
7d535a |
Record the memory usage statistics, consisting of following fields:
|
|
kusano |
7d535a |
for_lu (float)
|
|
kusano |
7d535a |
The amount of space used in bytes for L data structures.total_needed (float)
|
|
kusano |
7d535a |
The amount of space needed in bytes to perform factorization.expansions (int)
|
|
kusano |
7d535a |
The number of memory expansions during the LU factorization.
|
|
kusano |
7d535a |
|
|
kusano |
7d535a |
stat (output) SuperLUStat_t*
|
|
kusano |
7d535a |
Record the statistics on runtime and floating-point operation count.
|
|
kusano |
7d535a |
See slu_util.h for the definition of 'SuperLUStat_t'.
|
|
kusano |
7d535a |
info (output) int*
|
|
kusano |
7d535a |
= 0: successful exit
|
|
kusano |
7d535a |
< 0: if info = -i, the i-th argument had an illegal value
|
|
kusano |
7d535a |
> 0: if info = i, and i is
|
|
kusano |
7d535a |
<= A->ncol: U(i,i) is exactly zero. The factorization has
|
|
kusano |
7d535a |
been completed, but the factor U is exactly
|
|
kusano |
7d535a |
singular, so the solution and error bounds
|
|
kusano |
7d535a |
could not be computed.
|
|
kusano |
7d535a |
= A->ncol+1: U is nonsingular, but RCOND is less than machine
|
|
kusano |
7d535a |
precision, meaning that the matrix is singular to
|
|
kusano |
7d535a |
working precision. Nevertheless, the solution and
|
|
kusano |
7d535a |
error bounds are computed because there are a number
|
|
kusano |
7d535a |
of situations where the computed solution can be more
|
|
kusano |
7d535a |
accurate than the value of RCOND would suggest.
|
|
kusano |
7d535a |
> A->ncol+1: number of bytes allocated when memory allocation
|
|
kusano |
7d535a |
failure occurred, plus A->ncol.
|
|
kusano |
7d535a |
|
|
kusano |
7d535a |
|
|
kusano |
7d535a |
|
|
kusano |
7d535a |
|
|
kusano |
7d535a |
<address style="text-align: right;"><small>Generated on Mon Nov 22 10:23:48 2010 for SuperLU by </small></address>
|
|
kusano |
7d535a |
|
|
kusano |
7d535a |
 1.5.5
|
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kusano |
7d535a |
|
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kusano |
7d535a |
|