#include "slu_cdefs.h"
Functions | |
| 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) |
-- SuperLU routine (version 4.1) -- Lawrence Berkeley National Laboratory. November, 2010
| 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 | |||
| ) |
Purpose =======
CGSISX computes an approximate solutions of linear equations A*X=B or A'*X=B, using the ILU factorization from cgsitrf(). 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, 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 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 ) 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, 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_C, 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 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, cgsequ() is used to equilibrate the system,
and A is scaled as above, there is no permutation involved.
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_C, 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_C, 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_C, 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.
If options->RowPerm = LargeDiag, MC64 is used to scale and permute
the matrix A to an I-matrix. Then, in addition to the scaling
above, B is further permuted by P*B if options->Trans = NOTRANS,
where P is obtained from MC64.
X (output) SuperMatrix* X has types: Stype = SLU_DN, Dtype = SLU_C, 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.
mem_usage (output) mem_usage_t* Record the memory usage statistics, consisting of following fields:
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.
1.5.5