dgsisx.c File Reference

Computes an approximate solutions of linear equations A*X=B or A'*X=B. More...

`#include "slu_ddefs.h"`
Include dependency graph for dgsisx.c:

## Functions

void | dgsisx (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_t lwork, SuperMatrix *B, SuperMatrix *X, double *recip_pivot_growth, double *rcond, GlobalLU_t *Glu, mem_usage_t *mem_usage, SuperLUStat_t *stat, int_t *info)

## Detailed Description

Copyright (c) 2003, The Regents of the University of California, through Lawrence Berkeley National Laboratory (subject to receipt of any required approvals from U.S. Dept. of Energy)

All rights reserved.

The source code is distributed under BSD license, see the file License.txt at the top-level directory.

    -- SuperLU routine (version 7.0.0) --
    Lawrence Berkeley National Laboratory.
    November, 2010
    August 2024

## Function Documentation

## dgsisx()

void dgsisx | ( | 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_t | lwork,
|  | SuperMatrix * | B,
|  | SuperMatrix * | X,
|  | double * | recip_pivot_growth,
|  | double * | rcond,
|  | GlobalLU_t * | Glu,
|  | mem_usage_t * | mem_usage,
|  | SuperLUStat_t * | stat,
|  | int_t * | info
| ) |  |

    Purpose
    =======

    DGSISX computes an approximate solutions of linear equations
    A*X=B or A'*X=B, using the ILU factorization from dgsitrf().
    An estimation of the condition number is provided.
    The routine performs the following steps:

      1. If A is stored column-wise (A->Stype = SLU_NC):

         1.1. If options->Equil = YES or options->RowPerm = LargeDiag_MC64, 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 fills a small number on the diagonal entry, that is
            U(i,i) = ||A(:,i)||_oo * options->ILU_FillTol ** (1 - i / n),
              and info will be increased by 1. The factored form of A is used
              to estimate the condition number of the preconditioner. 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.

         1.6. The system of equations is solved for X using the factored form
              of A.

         1.7. options->IterRefine is not used

         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.

         1.9. options for ILU only
              1) If options->RowPerm = LargeDiag_MC64, MC64 is used to scale and
            permute the matrix to an I-matrix, that is Pr*Dr*A*Dc has
            entries of modulus 1 on the diagonal and off-diagonal entries
            of modulus at most 1. If MC64 fails, dgsequ() is used to
            equilibrate the system.
                 ( Default: LargeDiag_MC64 )
              2) options->ILU_DropTol = tau is the threshold for dropping.
            For L, it is used directly (for the whole row in a supernode);
            For U, ||A(:,i)||_oo * tau is used as the threshold
                 for the    i-th column.
            If a secondary dropping rule is required, tau will
                 also be used to compute the second threshold.
                 ( Default: 1e-4 )
              3) options->ILU_FillFactor = gamma, used as the initial guess
            of memory growth.
            If a secondary dropping rule is required, it will also
                 be used as an upper bound of the memory.
                 ( Default: 10 )
              4) options->ILU_DropRule specifies the dropping rule.
            Option        Meaning
            ======        ===========
            DROP_BASIC:   Basic dropping rule, supernodal based ILUTP(tau).
            DROP_PROWS:   Supernodal based ILUTP(p,tau), p = gamma*nnz(A)/n.
            DROP_COLUMN:  Variant of ILUTP(p,tau), for j-th column,
                          p = gamma * nnz(A(:,j)).
            DROP_AREA:    Variation of ILUTP, for j-th column, use
                          nnz(F(:,1:j)) / nnz(A(:,1:j)) to control memory.
            DROP_DYNAMIC: Modify the threshold tau during factorizaion:
                          If nnz(L(:,1:j)) / nnz(A(:,1:j)) > gamma
                              tau_L(j) := MIN(tau_0, tau_L(j-1) * 2);
                          Otherwise
                              tau_L(j) := MAX(tau_0, tau_L(j-1) / 2);
                          tau_U(j) uses the similar rule.
                          NOTE: the thresholds used by L and U are separate.
            DROP_INTERP:  Compute the second dropping threshold by
                          interpolation instead of sorting (default).
                          In this case, the actual fill ratio is not
                          guaranteed smaller than gamma.
            DROP_PROWS, DROP_COLUMN and DROP_AREA are mutually exclusive.
            ( Default: DROP_BASIC | DROP_AREA )
              5) options->ILU_Norm is the criterion of measuring the magnitude
            of a row in a supernode of L. ( Default is INF_NORM )
            options->ILU_Norm       RowSize(x[1:n])
            =================       ===============
            ONE_NORM                ||x||_1 / n
            TWO_NORM                ||x||_2 / sqrt(n)
            INF_NORM                max{|x[i]|}
              6) options->ILU_MILU specifies the type of MILU's variation.
            = SILU: do not perform Modified ILU;
            = SMILU_1 (not recommended):
                U(i,i) := U(i,i) + sum(dropped entries);
            = SMILU_2:
                U(i,i) := U(i,i) + SGN(U(i,i)) * sum(dropped entries);
            = SMILU_3:
                U(i,i) := U(i,i) + SGN(U(i,i)) * sum(|dropped entries|);
            NOTE: Even SMILU_1 does not preserve the column sum because of
            late dropping.
                 ( Default: SILU )
              7) options->ILU_FillTol is used as the perturbation when
            encountering zero pivots. If some U(i,i) = 0, so that U is
            exactly singular, then
               U(i,i) := ||A(:,i)|| * options->ILU_FillTol ** (1 - i / n).
                 ( Default: 1e-2 )

      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 or options->RowPerm = LargeDiag_MC64, 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 fills a small number on the diagonal entry, that is
             U(i,i) = ||A(:,i)||_oo * options->ILU_FillTol ** (1 - i / n).
              And info will be increased by 1. The factored form of A is used
              to estimate the condition number of the preconditioner. 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.

         2.6. The system of equations is solved for X using the factored form
              of transpose(A).

         2.7. If options->IterRefine is not used.

         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, or
            if options->RowPerm = NO.

            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).

            If options->RowPerm = LargeDiag_MC64, MC64 is used to scale and permute
               the matrix to an I-matrix, that is A is modified as follows:
               P*Dr*A*Dc has entries of modulus 1 on the diagonal and
               off-diagonal entries of modulus at most 1. P is a permutation
               obtained from MC64.
               If MC64 fails, dgsequ() is used to equilibrate the system,
               and A is scaled as above, but no permutation is involved.
               On exit, A is restored to the orginal row numbering, so
               Dr*A*Dc is returned.

    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 MC64 first then followed 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) double*, 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->Fact = FACTORED and equed = 'R' or 'B', each element
                of R must be positive.

    C          (input/output) double*, 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_D, 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_D, 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_D, 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_D, 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) double*
            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) double*
            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.

    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\U 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: number of zero pivots. They are replaced by small
                  entries due to options->ILU_FillTol.
            = 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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