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<title>SuperLU: SRC/cgsisx.c File Reference</title>
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Main Page
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Data Structures
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Files
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SRC/cgsisx.c File ReferenceComputes an approximate solutions of linear equations A*X=B or A'*X=B. More...
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#include "slu_cdefs.h"
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Functions
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void cgsisx (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, mem_usage_t *mem_usage, SuperLUStat_t *stat, int *info)
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Detailed Description
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-- SuperLU routine (version 4.1) --
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Lawrence Berkeley National Laboratory.
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November, 2010
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Function Documentation
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void cgsisx
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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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float *
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R,
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float *
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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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float *
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recip_pivot_growth,
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float *
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rcond,
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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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CGSISX computes an approximate solutions of linear equations
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A*X=B or A'*X=B, using the ILU factorization from cgsitrf().
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An estimation of the condition number is provided.
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The routine 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 or options->RowPerm = LargeDiag, scaling
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factors are computed to 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 fills a small number on the diagonal entry, that is
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U(i,i) = ||A(:,i)||_oo * options->ILU_FillTol ** (1 - i / n),
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and info will be increased by 1. The factored form of A is used
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to estimate the condition number of the preconditioner. If the
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reciprocal of the condition number is less than machine precision,
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info = A->ncol+1 is returned as a warning, but the routine still
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goes on to solve for X.
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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. options->IterRefine is not used
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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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1.9. options for ILU only
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1) If options->RowPerm = LargeDiag, MC64 is used to scale and
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permute the matrix to an I-matrix, that is Pr*Dr*A*Dc has
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entries of modulus 1 on the diagonal and off-diagonal entries
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of modulus at most 1. If MC64 fails, dgsequ() is used to
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equilibrate the system.
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( Default: LargeDiag )
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2) options->ILU_DropTol = tau is the threshold for dropping.
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For L, it is used directly (for the whole row in a supernode);
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For U, ||A(:,i)||_oo * tau is used as the threshold
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for the i-th column.
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If a secondary dropping rule is required, tau will
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also be used to compute the second threshold.
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( Default: 1e-4 )
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3) options->ILU_FillFactor = gamma, used as the initial guess
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of memory growth.
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If a secondary dropping rule is required, it will also
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be used as an upper bound of the memory.
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( Default: 10 )
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4) options->ILU_DropRule specifies the dropping rule.
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Option Meaning
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====== ===========
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DROP_BASIC: Basic dropping rule, supernodal based ILUTP(tau).
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DROP_PROWS: Supernodal based ILUTP(p,tau), p = gamma*nnz(A)/n.
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DROP_COLUMN: Variant of ILUTP(p,tau), for j-th column,
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p = gamma * nnz(A(:,j)).
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DROP_AREA: Variation of ILUTP, for j-th column, use
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nnz(F(:,1:j)) / nnz(A(:,1:j)) to control memory.
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DROP_DYNAMIC: Modify the threshold tau during factorizaion:
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If nnz(L(:,1:j)) / nnz(A(:,1:j)) > gamma
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tau_L(j) := MIN(tau_0, tau_L(j-1) * 2);
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Otherwise
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tau_L(j) := MAX(tau_0, tau_L(j-1) / 2);
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tau_U(j) uses the similar rule.
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NOTE: the thresholds used by L and U are separate.
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DROP_INTERP: Compute the second dropping threshold by
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interpolation instead of sorting (default).
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In this case, the actual fill ratio is not
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guaranteed smaller than gamma.
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DROP_PROWS, DROP_COLUMN and DROP_AREA are mutually exclusive.
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( Default: DROP_BASIC | DROP_AREA )
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5) options->ILU_Norm is the criterion of measuring the magnitude
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of a row in a supernode of L. ( Default is INF_NORM )
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options->ILU_Norm RowSize(x[1:n])
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================= ===============
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ONE_NORM ||x||_1 / n
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TWO_NORM ||x||_2 / sqrt(n)
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INF_NORM max{|x[i]|}
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6) options->ILU_MILU specifies the type of MILU's variation.
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= SILU: do not perform Modified ILU;
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= SMILU_1 (not recommended):
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U(i,i) := U(i,i) + sum(dropped entries);
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= SMILU_2:
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U(i,i) := U(i,i) + SGN(U(i,i)) * sum(dropped entries);
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= SMILU_3:
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U(i,i) := U(i,i) + SGN(U(i,i)) * sum(|dropped entries|);
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NOTE: Even SMILU_1 does not preserve the column sum because of
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late dropping.
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( Default: SILU )
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7) options->ILU_FillTol is used as the perturbation when
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encountering zero pivots. If some U(i,i) = 0, so that U is
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exactly singular, then
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U(i,i) := ||A(:,i)|| * options->ILU_FillTol ** (1 - i / n).
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( Default: 1e-2 )
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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 or options->RowPerm = LargeDiag, scaling
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factors are computed to 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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7d535a |
routine fills a small number on the diagonal entry, that is
|
|
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7d535a |
U(i,i) = ||A(:,i)||_oo * options->ILU_FillTol ** (1 - i / n).
|
|
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7d535a |
And info will be increased by 1. The factored form of A is used
|
|
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7d535a |
to estimate the condition number of the preconditioner. If the
|
|
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7d535a |
reciprocal of the condition number is less than machine precision,
|
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7d535a |
info = A->ncol+1 is returned as a warning, but the routine still
|
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7d535a |
goes on to solve for X.
|
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7d535a |
2.6. The system of equations is solved for X using the factored form
|
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7d535a |
of transpose(A).
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7d535a |
2.7. If options->IterRefine is not used.
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7d535a |
2.8. If equilibration was used, the matrix X is premultiplied by
|
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7d535a |
diag(C) (if options->Trans = NOTRANS) or diag(R)
|
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7d535a |
(if options->Trans = TRANS or CONJ) so that it solves the
|
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7d535a |
original system before equilibration.
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7d535a |
See supermatrix.h for the definition of 'SuperMatrix' structure.
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7d535a |
Arguments
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7d535a |
=========
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7d535a |
options (input) superlu_options_t*
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7d535a |
The structure defines the input parameters to control
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7d535a |
how the LU decomposition will be performed and how the
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7d535a |
system will be solved.
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7d535a |
A (input/output) SuperMatrix*
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7d535a |
Matrix A in A*X=B, of dimension (A->nrow, A->ncol). The number
|
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7d535a |
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_C, Mtype = SLU_GE.
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7d535a |
In the future, more general A may be handled.
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7d535a |
On entry, If options->Fact = FACTORED and equed is not 'N',
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7d535a |
then A must have been equilibrated by the scaling factors in
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7d535a |
R and/or C.
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7d535a |
On exit, A is not modified
|
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7d535a |
if options->Equil = NO, or
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7d535a |
if options->Equil = YES but equed = 'N' on exit, or
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7d535a |
if options->RowPerm = NO.
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7d535a |
Otherwise, if options->Equil = YES and equed is not 'N',
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7d535a |
A is scaled as follows:
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7d535a |
If A->Stype = SLU_NC:
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7d535a |
equed = 'R': A := diag(R) * A
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7d535a |
equed = 'C': A := A * diag(C)
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7d535a |
equed = 'B': A := diag(R) * A * diag(C).
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7d535a |
If A->Stype = SLU_NR:
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7d535a |
equed = 'R': transpose(A) := diag(R) * transpose(A)
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7d535a |
equed = 'C': transpose(A) := transpose(A) * diag(C)
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7d535a |
equed = 'B': transpose(A) := diag(R) * transpose(A) * diag(C).
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7d535a |
If options->RowPerm = LargeDiag, MC64 is used to scale and permute
|
|
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7d535a |
the matrix to an I-matrix, that is A is modified as follows:
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7d535a |
P*Dr*A*Dc has entries of modulus 1 on the diagonal and
|
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7d535a |
off-diagonal entries of modulus at most 1. P is a permutation
|
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obtained from MC64.
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7d535a |
If MC64 fails, cgsequ() is used to equilibrate the system,
|
|
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7d535a |
and A is scaled as above, there is no permutation involved.
|
|
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7d535a |
perm_c (input/output) int*
|
|
kusano |
7d535a |
If A->Stype = SLU_NC, Column permutation vector of size A->ncol,
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|
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7d535a |
which defines the permutation matrix Pc; perm_c[i] = j means
|
|
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7d535a |
column i of A is in position j in A*Pc.
|
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7d535a |
On exit, perm_c may be overwritten by the product of the input
|
|
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7d535a |
perm_c and a permutation that postorders the elimination tree
|
|
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7d535a |
of Pc'*A'*A*Pc; perm_c is not changed if the elimination tree
|
|
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7d535a |
is already in postorder.
|
|
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7d535a |
If A->Stype = SLU_NR, column permutation vector of size A->nrow,
|
|
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7d535a |
which describes permutation of columns of transpose(A)
|
|
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7d535a |
(rows of A) as described above.
|
|
kusano |
7d535a |
perm_r (input/output) int*
|
|
kusano |
7d535a |
If A->Stype = SLU_NC, row permutation vector of size A->nrow,
|
|
kusano |
7d535a |
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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7d535a |
position j in Pr*A.
|
|
kusano |
7d535a |
If A->Stype = SLU_NR, permutation vector of size A->ncol, which
|
|
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7d535a |
determines permutation of rows of transpose(A)
|
|
kusano |
7d535a |
(columns of A) as described above.
|
|
kusano |
7d535a |
If options->Fact = SamePattern_SameRowPerm, the pivoting routine
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|
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7d535a |
will try to use the input perm_r, unless a certain threshold
|
|
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7d535a |
criterion is violated. In that case, perm_r is overwritten by a
|
|
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7d535a |
new permutation determined by partial pivoting or diagonal
|
|
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7d535a |
threshold pivoting.
|
|
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7d535a |
Otherwise, perm_r is output argument.
|
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7d535a |
etree (input/output) int*, dimension (A->ncol)
|
|
kusano |
7d535a |
Elimination tree of Pc'*A'*A*Pc.
|
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7d535a |
If options->Fact != FACTORED and options->Fact != DOFACT,
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7d535a |
etree is an input argument, otherwise it is an output argument.
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|
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7d535a |
Note: etree is a vector of parent pointers for a forest whose
|
|
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7d535a |
vertices are the integers 0 to A->ncol-1; etree[root]==A->ncol.
|
|
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7d535a |
equed (input/output) char*
|
|
kusano |
7d535a |
Specifies the form of equilibration that was done.
|
|
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7d535a |
= 'N': No equilibration.
|
|
kusano |
7d535a |
= 'R': Row equilibration, i.e., A was premultiplied by diag(R).
|
|
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7d535a |
= 'C': Column equilibration, i.e., A was postmultiplied by diag(C).
|
|
kusano |
7d535a |
= 'B': Both row and column equilibration, i.e., A was replaced
|
|
kusano |
7d535a |
by diag(R)*A*diag(C).
|
|
kusano |
7d535a |
If options->Fact = FACTORED, equed is an input argument,
|
|
kusano |
7d535a |
otherwise it is an output argument.
|
|
kusano |
7d535a |
R (input/output) float*, dimension (A->nrow)
|
|
kusano |
7d535a |
The row scale factors for A or transpose(A).
|
|
kusano |
7d535a |
If equed = 'R' or 'B', A (if A->Stype = SLU_NC) or transpose(A)
|
|
kusano |
7d535a |
(if A->Stype = SLU_NR) is multiplied on the left by diag(R).
|
|
kusano |
7d535a |
If equed = 'N' or 'C', R is not accessed.
|
|
kusano |
7d535a |
If options->Fact = FACTORED, R is an input argument,
|
|
kusano |
7d535a |
otherwise, R is output.
|
|
kusano |
7d535a |
If options->zFact = FACTORED and equed = 'R' or 'B', each element
|
|
kusano |
7d535a |
of R must be positive.
|
|
kusano |
7d535a |
C (input/output) float*, dimension (A->ncol)
|
|
kusano |
7d535a |
The column scale factors for A or transpose(A).
|
|
kusano |
7d535a |
If equed = 'C' or 'B', A (if A->Stype = SLU_NC) or transpose(A)
|
|
kusano |
7d535a |
(if A->Stype = SLU_NR) is multiplied on the right by diag(C).
|
|
kusano |
7d535a |
If equed = 'N' or 'R', C is not accessed.
|
|
kusano |
7d535a |
If options->Fact = FACTORED, C is an input argument,
|
|
kusano |
7d535a |
otherwise, C is output.
|
|
kusano |
7d535a |
If options->Fact = FACTORED and equed = 'C' or 'B', each element
|
|
kusano |
7d535a |
of C must be positive.
|
|
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.,
|
|
kusano |
7d535a |
L has types: Stype = SLU_SC, Dtype = SLU_C, Mtype = SLU_TRLU.
|
|
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).
|
|
kusano |
7d535a |
Uses column-wise storage scheme, i.e., U has types:
|
|
kusano |
7d535a |
Stype = SLU_NC, Dtype = SLU_C, Mtype = SLU_TRU.
|
|
kusano |
7d535a |
work (workspace/output) void*, size (lwork) (in bytes)
|
|
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.
|
|
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_C, 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 |
If options->RowPerm = LargeDiag, MC64 is used to scale and permute
|
|
kusano |
7d535a |
the matrix A to an I-matrix. Then, in addition to the scaling
|
|
kusano |
7d535a |
above, B is further permuted by P*B if options->Trans = NOTRANS,
|
|
kusano |
7d535a |
where P is obtained from MC64.
|
|
kusano |
7d535a |
X (output) SuperMatrix*
|
|
kusano |
7d535a |
X has types: Stype = SLU_DN, Dtype = SLU_C, 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) float*
|
|
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) float*
|
|
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 |
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: number of zero pivots. They are replaced by small
|
|
kusano |
7d535a |
entries due to options->ILU_FillTol.
|
|
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 10:23:47 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 |
|