HYPRE_Int hypre_AMESolve(void *esolver)
{
hypre_AMEData *ame_data = esolver;
HYPRE_Int nit;
lobpcg_BLASLAPACKFunctions blap_fn;
lobpcg_Tolerance lobpcg_tol;
HYPRE_Real *residuals;
#ifdef HYPRE_USING_ESSL
blap_fn.dsygv = dsygv;
blap_fn.dpotrf = dpotrf;
#else
blap_fn.dsygv = hypre_F90_NAME_LAPACK(dsygv,DSYGV);
blap_fn.dpotrf = hypre_F90_NAME_LAPACK(dpotrf,DPOTRF);
#endif
lobpcg_tol.relative = ame_data -> tol;
lobpcg_tol.absolute = ame_data -> tol;
residuals = hypre_TAlloc(HYPRE_Real, ame_data -> block_size);
lobpcg_solve((mv_MultiVectorPtr) ame_data -> eigenvectors,
esolver, hypre_AMEMultiOperatorA,
esolver, hypre_AMEMultiOperatorM,
esolver, hypre_AMEMultiOperatorB,
NULL, blap_fn, lobpcg_tol, ame_data -> maxit,
ame_data -> print_level, &nit,
ame_data -> eigenvalues,
NULL, ame_data -> block_size,
residuals,
NULL, ame_data -> block_size);
hypre_TFree(residuals);
return hypre_error_flag;
}
static HYPRE_Int dpotrf_interface (char *uplo, HYPRE_Int *n, double *a, HYPRE_Int *
lda, HYPRE_Int *info)
{
#ifdef HYPRE_USING_ESSL
dpotrf(uplo, *n, a, *lda, info);
#else
hypre_F90_NAME_LAPACK( dpotrf, DPOTRF )(uplo, n, a, lda, info);
#endif
return 0;
}
static HYPRE_Int dsygv_interface (HYPRE_Int *itype, char *jobz, char *uplo, HYPRE_Int *
n, double *a, HYPRE_Int *lda, double *b, HYPRE_Int *ldb,
double *w, double *work, HYPRE_Int *lwork, HYPRE_Int *info)
{
#ifdef HYPRE_USING_ESSL
dsygv(*itype, a, *lda, b, *ldb, w, a, *lda, *n, work, *lwork );
#else
hypre_F90_NAME_LAPACK( dsygv, DSYGV )( itype, jobz, uplo, n,
a, lda, b, ldb,
w, work, lwork, info );
#endif
return 0;
}
void
dgssvx(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 )
{
/*
* Purpose
* =======
*
* DGSSVX solves the system of linear equations A*X=B or A'*X=B, using
* the LU factorization from dgstrf(). Error bounds on the solution and
* a condition estimate are also provided. It performs the following steps:
*
* 1. If A is stored column-wise (A->Stype = SLU_NC):
*
* 1.1. If options->Equil = YES, scaling factors are computed to
* equilibrate the system:
* options->Trans = NOTRANS:
* diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
* options->Trans = TRANS:
* (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
* options->Trans = CONJ:
* (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
* Whether or not the system will be equilibrated depends on the
* scaling of the matrix A, but if equilibration is used, A is
* overwritten by diag(R)*A*diag(C) and B by diag(R)*B
* (if options->Trans=NOTRANS) or diag(C)*B (if options->Trans
* = TRANS or CONJ).
*
* 1.2. Permute columns of A, forming A*Pc, where Pc is a permutation
* matrix that usually preserves sparsity.
* For more details of this step, see sp_preorder.c.
*
* 1.3. If options->Fact != FACTORED, the LU decomposition is used to
* factor the matrix A (after equilibration if options->Equil = YES)
* as Pr*A*Pc = L*U, with Pr determined by partial pivoting.
*
* 1.4. Compute the reciprocal pivot growth factor.
*
* 1.5. If some U(i,i) = 0, so that U is exactly singular, then the
* routine returns with info = i. Otherwise, the factored form of
* A is used to estimate the condition number of the matrix A. If
* the reciprocal of the condition number is less than machine
* precision, info = A->ncol+1 is returned as a warning, but the
* routine still goes on to solve for X and computes error bounds
* as described below.
*
* 1.6. The system of equations is solved for X using the factored form
* of A.
*
* 1.7. If options->IterRefine != NOREFINE, iterative refinement is
* applied to improve the computed solution matrix and calculate
* error bounds and backward error estimates for it.
*
* 1.8. If equilibration was used, the matrix X is premultiplied by
* diag(C) (if options->Trans = NOTRANS) or diag(R)
* (if options->Trans = TRANS or CONJ) so that it solves the
* original system before equilibration.
*
* 2. If A is stored row-wise (A->Stype = SLU_NR), apply the above algorithm
* to the transpose of A:
*
* 2.1. If options->Equil = YES, scaling factors are computed to
* equilibrate the system:
* options->Trans = NOTRANS:
* diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
* options->Trans = TRANS:
* (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
* options->Trans = CONJ:
* (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
* Whether or not the system will be equilibrated depends on the
* scaling of the matrix A, but if equilibration is used, A' is
* overwritten by diag(R)*A'*diag(C) and B by diag(R)*B
* (if trans='N') or diag(C)*B (if trans = 'T' or 'C').
*
* 2.2. Permute columns of transpose(A) (rows of A),
* forming transpose(A)*Pc, where Pc is a permutation matrix that
* usually preserves sparsity.
* For more details of this step, see sp_preorder.c.
*
* 2.3. If options->Fact != FACTORED, the LU decomposition is used to
* factor the transpose(A) (after equilibration if
* options->Fact = YES) as Pr*transpose(A)*Pc = L*U with the
* permutation Pr determined by partial pivoting.
*
* 2.4. Compute the reciprocal pivot growth factor.
*
* 2.5. If some U(i,i) = 0, so that U is exactly singular, then the
* routine returns with info = i. Otherwise, the factored form
* of transpose(A) is used to estimate the condition number of the
* matrix A. If the reciprocal of the condition number
* is less than machine precision, info = A->nrow+1 is returned as
* a warning, but the routine still goes on to solve for X and
* computes error bounds as described below.
*
* 2.6. The system of equations is solved for X using the factored form
* of transpose(A).
*
* 2.7. If options->IterRefine != NOREFINE, iterative refinement is
* applied to improve the computed solution matrix and calculate
* error bounds and backward error estimates for it.
*
* 2.8. If equilibration was used, the matrix X is premultiplied by
* diag(C) (if options->Trans = NOTRANS) or diag(R)
* (if options->Trans = TRANS or CONJ) so that it solves the
* original system before equilibration.
*
* See supermatrix.h for the definition of 'SuperMatrix' structure.
*
* Arguments
* =========
*
* options (input) superlu_options_t*
* The structure defines the input parameters to control
* how the LU decomposition will be performed and how the
* system will be solved.
*
* A (input/output) SuperMatrix*
* Matrix A in A*X=B, of dimension (A->nrow, A->ncol). The number
* of the linear equations is A->nrow. Currently, the type of A can be:
* Stype = SLU_NC or SLU_NR, Dtype = SLU_D, Mtype = SLU_GE.
* In the future, more general A may be handled.
*
* On entry, If options->Fact = FACTORED and equed is not 'N',
* then A must have been equilibrated by the scaling factors in
* R and/or C.
* On exit, A is not modified if options->Equil = NO, or if
* options->Equil = YES but equed = 'N' on exit.
* Otherwise, if options->Equil = YES and equed is not 'N',
* A is scaled as follows:
* If A->Stype = SLU_NC:
* equed = 'R': A := diag(R) * A
* equed = 'C': A := A * diag(C)
* equed = 'B': A := diag(R) * A * diag(C).
* If A->Stype = SLU_NR:
* equed = 'R': transpose(A) := diag(R) * transpose(A)
* equed = 'C': transpose(A) := transpose(A) * diag(C)
* equed = 'B': transpose(A) := diag(R) * transpose(A) * diag(C).
*
* perm_c (input/output) int*
* If A->Stype = SLU_NC, Column permutation vector of size A->ncol,
* which defines the permutation matrix Pc; perm_c[i] = j means
* column i of A is in position j in A*Pc.
* On exit, perm_c may be overwritten by the product of the input
* perm_c and a permutation that postorders the elimination tree
* of Pc'*A'*A*Pc; perm_c is not changed if the elimination tree
* is already in postorder.
*
* If A->Stype = SLU_NR, column permutation vector of size A->nrow,
* which describes permutation of columns of transpose(A)
* (rows of A) as described above.
*
* perm_r (input/output) int*
* If A->Stype = SLU_NC, row permutation vector of size A->nrow,
* which defines the permutation matrix Pr, and is determined
* by partial pivoting. perm_r[i] = j means row i of A is in
* position j in Pr*A.
*
* If A->Stype = SLU_NR, permutation vector of size A->ncol, which
* determines permutation of rows of transpose(A)
* (columns of A) as described above.
*
* If options->Fact = SamePattern_SameRowPerm, the pivoting routine
* will try to use the input perm_r, unless a certain threshold
* criterion is violated. In that case, perm_r is overwritten by a
* new permutation determined by partial pivoting or diagonal
* threshold pivoting.
* Otherwise, perm_r is output argument.
*
* etree (input/output) int*, dimension (A->ncol)
* Elimination tree of Pc'*A'*A*Pc.
* If options->Fact != FACTORED and options->Fact != DOFACT,
* etree is an input argument, otherwise it is an output argument.
* Note: etree is a vector of parent pointers for a forest whose
* vertices are the integers 0 to A->ncol-1; etree[root]==A->ncol.
*
* equed (input/output) char*
* Specifies the form of equilibration that was done.
* = 'N': No equilibration.
* = 'R': Row equilibration, i.e., A was premultiplied by diag(R).
* = 'C': Column equilibration, i.e., A was postmultiplied by diag(C).
* = 'B': Both row and column equilibration, i.e., A was replaced
* by diag(R)*A*diag(C).
* If options->Fact = FACTORED, equed is an input argument,
* otherwise it is an output argument.
*
* R (input/output) 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->zFact = 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.
*
* FERR (output) double*, dimension (B->ncol)
* The estimated forward error bound for each solution vector
* X(j) (the j-th column of the solution matrix X).
* If XTRUE is the true solution corresponding to X(j), FERR(j)
* is an estimated upper bound for the magnitude of the largest
* element in (X(j) - XTRUE) divided by the magnitude of the
* largest element in X(j). The estimate is as reliable as
* the estimate for RCOND, and is almost always a slight
* overestimate of the true error.
* If options->IterRefine = NOREFINE, ferr = 1.0.
*
* BERR (output) double*, dimension (B->ncol)
* The componentwise relative backward error of each solution
* vector X(j) (i.e., the smallest relative change in
* any element of A or B that makes X(j) an exact solution).
* If options->IterRefine = NOREFINE, berr = 1.0.
*
* mem_usage (output) mem_usage_t*
* Record the memory usage statistics, consisting of following fields:
* - for_lu (float)
* The amount of space used in bytes for L\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 util.h for the definition of 'SuperLUStat_t'.
*
* info (output) int*
* = 0: successful exit
* < 0: if info = -i, the i-th argument had an illegal value
* > 0: if info = i, and i is
* <= A->ncol: U(i,i) is exactly zero. The factorization has
* been completed, but the factor U is exactly
* singular, so the solution and error bounds
* could not be computed.
* = A->ncol+1: U is nonsingular, but RCOND is less than machine
* precision, meaning that the matrix is singular to
* working precision. Nevertheless, the solution and
* error bounds are computed because there are a number
* of situations where the computed solution can be more
* accurate than the value of RCOND would suggest.
* > A->ncol+1: number of bytes allocated when memory allocation
* failure occurred, plus A->ncol.
*
*/
DNformat *Bstore, *Xstore;
double *Bmat, *Xmat;
int ldb, ldx, nrhs;
SuperMatrix *AA;/* A in SLU_NC format used by the factorization routine.*/
SuperMatrix AC; /* Matrix postmultiplied by Pc */
int colequ, equil, nofact, notran, rowequ, permc_spec;
trans_t trant;
char norm[1];
int i, j, info1;
double amax, anorm, bignum, smlnum, colcnd, rowcnd, rcmax, rcmin;
int relax, panel_size;
double drop_tol;
double t0; /* temporary time */
double *utime;
/* External functions */
extern double dlangs(char *, SuperMatrix *);
extern double hypre_F90_NAME_LAPACK(dlamch,DLAMCH)(const char *);
Bstore = (DNformat*) B->Store;
Xstore = (DNformat*) X->Store;
Bmat = ( double*) Bstore->nzval;
Xmat = ( double*) Xstore->nzval;
ldb = Bstore->lda;
ldx = Xstore->lda;
nrhs = B->ncol;
*info = 0;
nofact = (options->Fact != FACTORED);
equil = (options->Equil == YES);
notran = (options->Trans == NOTRANS);
if ( nofact ) {
*(unsigned char *)equed = 'N';
rowequ = FALSE;
colequ = FALSE;
} else {
rowequ = superlu_lsame(equed, "R") || superlu_lsame(equed, "B");
colequ = superlu_lsame(equed, "C") || superlu_lsame(equed, "B");
smlnum = hypre_F90_NAME_LAPACK(dlamch,DLAMCH)("Safe minimum");
bignum = 1. / smlnum;
}
#if 0
printf("dgssvx: Fact=%4d, Trans=%4d, equed=%c\n",
options->Fact, options->Trans, *equed);
#endif
/* Test the input parameters */
if (!nofact && options->Fact != DOFACT && options->Fact != SamePattern &&
options->Fact != SamePattern_SameRowPerm &&
!notran && options->Trans != TRANS && options->Trans != CONJ &&
!equil && options->Equil != NO)
*info = -1;
else if ( A->nrow != A->ncol || A->nrow < 0 ||
(A->Stype != SLU_NC && A->Stype != SLU_NR) ||
A->Dtype != SLU_D || A->Mtype != SLU_GE )
*info = -2;
else if (options->Fact == FACTORED &&
!(rowequ || colequ || superlu_lsame(equed, "N")))
*info = -6;
else {
if (rowequ) {
rcmin = bignum;
rcmax = 0.;
for (j = 0; j < A->nrow; ++j) {
rcmin = SUPERLU_MIN(rcmin, R[j]);
rcmax = SUPERLU_MAX(rcmax, R[j]);
}
if (rcmin <= 0.) *info = -7;
else if ( A->nrow > 0)
rowcnd = SUPERLU_MAX(rcmin,smlnum) / SUPERLU_MIN(rcmax,bignum);
else rowcnd = 1.;
}
if (colequ && *info == 0) {
rcmin = bignum;
rcmax = 0.;
for (j = 0; j < A->nrow; ++j) {
rcmin = SUPERLU_MIN(rcmin, C[j]);
rcmax = SUPERLU_MAX(rcmax, C[j]);
}
if (rcmin <= 0.) *info = -8;
else if (A->nrow > 0)
colcnd = SUPERLU_MAX(rcmin,smlnum) / SUPERLU_MIN(rcmax,bignum);
else colcnd = 1.;
}
if (*info == 0) {
if ( lwork < -1 ) *info = -12;
else if ( B->ncol < 0 || Bstore->lda < SUPERLU_MAX(0, A->nrow) ||
B->Stype != SLU_DN || B->Dtype != SLU_D ||
B->Mtype != SLU_GE )
*info = -13;
else if ( X->ncol < 0 || Xstore->lda < SUPERLU_MAX(0, A->nrow) ||
(B->ncol != 0 && B->ncol != X->ncol) ||
X->Stype != SLU_DN ||
X->Dtype != SLU_D || X->Mtype != SLU_GE )
*info = -14;
}
}
if (*info != 0) {
i = -(*info);
superlu_xerbla("dgssvx", &i);
return;
}
/* Initialization for factor parameters */
panel_size = sp_ienv(1);
relax = sp_ienv(2);
drop_tol = 0.0;
utime = stat->utime;
/* Convert A to SLU_NC format when necessary. */
if ( A->Stype == SLU_NR ) {
NRformat *Astore = (NRformat*) A->Store;
AA = (SuperMatrix *) SUPERLU_MALLOC( sizeof(SuperMatrix) );
dCreate_CompCol_Matrix(AA, A->ncol, A->nrow, Astore->nnz,
(double*) Astore->nzval, Astore->colind, Astore->rowptr,
SLU_NC, A->Dtype, A->Mtype);
if ( notran ) { /* Reverse the transpose argument. */
trant = TRANS;
notran = 0;
} else {
trant = NOTRANS;
notran = 1;
}
} else { /* A->Stype == SLU_NC */
trant = options->Trans;
AA = A;
}
if ( nofact && equil ) {
t0 = SuperLU_timer_();
/* Compute row and column scalings to equilibrate the matrix A. */
dgsequ(AA, R, C, &rowcnd, &colcnd, &amax, &info1);
if ( info1 == 0 ) {
/* Equilibrate matrix A. */
dlaqgs(AA, R, C, rowcnd, colcnd, amax, equed);
rowequ = superlu_lsame(equed, "R") || superlu_lsame(equed, "B");
colequ = superlu_lsame(equed, "C") || superlu_lsame(equed, "B");
}
utime[EQUIL] = SuperLU_timer_() - t0;
}
if ( nrhs > 0 ) {
/* Scale the right hand side if equilibration was performed. */
if ( notran ) {
if ( rowequ ) {
for (j = 0; j < nrhs; ++j)
for (i = 0; i < A->nrow; ++i) {
Bmat[i + j*ldb] *= R[i];
}
}
} else if ( colequ ) {
for (j = 0; j < nrhs; ++j)
for (i = 0; i < A->nrow; ++i) {
Bmat[i + j*ldb] *= C[i];
}
}
}
if ( nofact ) {
t0 = SuperLU_timer_();
/*
* Gnet column permutation vector perm_c[], according to permc_spec:
* permc_spec = NATURAL: natural ordering
* permc_spec = MMD_AT_PLUS_A: minimum degree on structure of A'+A
* permc_spec = MMD_ATA: minimum degree on structure of A'*A
* permc_spec = COLAMD: approximate minimum degree column ordering
* permc_spec = MY_PERMC: the ordering already supplied in perm_c[]
*/
permc_spec = options->ColPerm;
if ( permc_spec != MY_PERMC && options->Fact == DOFACT )
get_perm_c(permc_spec, AA, perm_c);
utime[COLPERM] = SuperLU_timer_() - t0;
t0 = SuperLU_timer_();
sp_preorder(options, AA, perm_c, etree, &AC);
utime[ETREE] = SuperLU_timer_() - t0;
/* printf("Factor PA = LU ... relax %d\tw %d\tmaxsuper %d\trowblk %d\n",
relax, panel_size, sp_ienv(3), sp_ienv(4));
fflush(stdout); */
/* Compute the LU factorization of A*Pc. */
t0 = SuperLU_timer_();
dgstrf(options, &AC, drop_tol, relax, panel_size,
etree, work, lwork, perm_c, perm_r, L, U, stat, info);
utime[FACT] = SuperLU_timer_() - t0;
if ( lwork == -1 ) {
mem_usage->total_needed = *info - A->ncol;
return;
}
}
if ( options->PivotGrowth ) {
if ( *info > 0 ) {
if ( *info <= A->ncol ) {
/* Compute the reciprocal pivot growth factor of the leading
rank-deficient *info columns of A. */
*recip_pivot_growth = dPivotGrowth(*info, AA, perm_c, L, U);
}
return;
}
/* Compute the reciprocal pivot growth factor *recip_pivot_growth. */
*recip_pivot_growth = dPivotGrowth(A->ncol, AA, perm_c, L, U);
}
if ( options->ConditionNumber ) {
/* Estimate the reciprocal of the condition number of A. */
t0 = SuperLU_timer_();
if ( notran ) {
*(unsigned char *)norm = '1';
} else {
*(unsigned char *)norm = 'I';
}
anorm = dlangs(norm, AA);
dgscon(norm, L, U, anorm, rcond, stat, info);
utime[RCOND] = SuperLU_timer_() - t0;
}
if ( nrhs > 0 ) {
/* Compute the solution matrix X. */
for (j = 0; j < nrhs; j++) /* Save a copy of the right hand sides */
for (i = 0; i < B->nrow; i++)
Xmat[i + j*ldx] = Bmat[i + j*ldb];
t0 = SuperLU_timer_();
dgstrs (trant, L, U, perm_c, perm_r, X, stat, info);
utime[SOLVE] = SuperLU_timer_() - t0;
/* Use iterative refinement to improve the computed solution and compute
error bounds and backward error estimates for it. */
t0 = SuperLU_timer_();
if ( options->IterRefine != NOREFINE ) {
dgsrfs(trant, AA, L, U, perm_c, perm_r, equed, R, C, B,
X, ferr, berr, stat, info);
} else {
for (j = 0; j < nrhs; ++j) ferr[j] = berr[j] = 1.0;
}
utime[REFINE] = SuperLU_timer_() - t0;
/* Transform the solution matrix X to a solution of the original system. */
if ( notran ) {
if ( colequ ) {
for (j = 0; j < nrhs; ++j)
for (i = 0; i < A->nrow; ++i) {
Xmat[i + j*ldx] *= C[i];
}
}
} else if ( rowequ ) {
for (j = 0; j < nrhs; ++j)
for (i = 0; i < A->nrow; ++i) {
Xmat[i + j*ldx] *= R[i];
}
}
} /* end if nrhs > 0 */
if ( options->ConditionNumber ) {
/* Set INFO = A->ncol+1 if the matrix is singular to working precision. */
if (*rcond < hypre_F90_NAME_LAPACK(dlamch,DLAMCH)("E")) *info=A->ncol+1;
}
if ( nofact ) {
dQuerySpace(L, U, mem_usage);
Destroy_CompCol_Permuted(&AC);
}
if ( A->Stype == SLU_NR ) {
Destroy_SuperMatrix_Store(AA);
SUPERLU_FREE(AA);
}
}
void
dlaqgs(SuperMatrix *A, double *r, double *c,
double rowcnd, double colcnd, double amax, char *equed)
{
/*
Purpose
=======
DLAQGS equilibrates a general sparse M by N matrix A using the row and
scaling factors in the vectors R and C.
See supermatrix.h for the definition of 'SuperMatrix' structure.
Arguments
=========
A (input/output) SuperMatrix*
On exit, the equilibrated matrix. See EQUED for the form of
the equilibrated matrix. The type of A can be:
Stype = NC; Dtype = SLU_D; Mtype = GE.
R (input) double*, dimension (A->nrow)
The row scale factors for A.
C (input) double*, dimension (A->ncol)
The column scale factors for A.
ROWCND (input) double
Ratio of the smallest R(i) to the largest R(i).
COLCND (input) double
Ratio of the smallest C(i) to the largest C(i).
AMAX (input) double
Absolute value of largest matrix entry.
EQUED (output) char*
Specifies the form of equilibration that was done.
= 'N': No equilibration
= 'R': Row equilibration, i.e., A has been premultiplied by
diag(R).
= 'C': Column equilibration, i.e., A has been postmultiplied
by diag(C).
= 'B': Both row and column equilibration, i.e., A has been
replaced by diag(R) * A * diag(C).
Internal Parameters
===================
THRESH is a threshold value used to decide if row or column scaling
should be done based on the ratio of the row or column scaling
factors. If ROWCND < THRESH, row scaling is done, and if
COLCND < THRESH, column scaling is done.
LARGE and SMALL are threshold values used to decide if row scaling
should be done based on the absolute size of the largest matrix
element. If AMAX > LARGE or AMAX < SMALL, row scaling is done.
=====================================================================
*/
#define THRESH (0.1)
/* Local variables */
NCformat *Astore;
double *Aval;
int i, j, irow;
double large, small, cj;
extern double hypre_F90_NAME_LAPACK(dlamch,DLAMCH)(char *);
/* Quick return if possible */
if (A->nrow <= 0 || A->ncol <= 0) {
*(unsigned char *)equed = 'N';
return;
}
Astore = A->Store;
Aval = Astore->nzval;
/* Initialize LARGE and SMALL. */
small = hypre_F90_NAME_LAPACK(dlamch,DLAMCH)("Safe minimum") / hypre_F90_NAME_LAPACK(dlamch,DLAMCH)("Precision");
large = 1. / small;
if (rowcnd >= THRESH && amax >= small && amax <= large) {
if (colcnd >= THRESH)
*(unsigned char *)equed = 'N';
else {
/* Column scaling */
for (j = 0; j < A->ncol; ++j) {
cj = c[j];
for (i = Astore->colptr[j]; i < Astore->colptr[j+1]; ++i) {
Aval[i] *= cj;
}
}
*(unsigned char *)equed = 'C';
}
} else if (colcnd >= THRESH) {
/* Row scaling, no column scaling */
for (j = 0; j < A->ncol; ++j)
for (i = Astore->colptr[j]; i < Astore->colptr[j+1]; ++i) {
irow = Astore->rowind[i];
Aval[i] *= r[irow];
}
*(unsigned char *)equed = 'R';
} else {
/* Row and column scaling */
for (j = 0; j < A->ncol; ++j) {
cj = c[j];
for (i = Astore->colptr[j]; i < Astore->colptr[j+1]; ++i) {
irow = Astore->rowind[i];
Aval[i] *= cj * r[irow];
}
}
*(unsigned char *)equed = 'B';
}
return;
} /* dlaqgs */
void
dgsequ(SuperMatrix *A, double *r, double *c, double *rowcnd,
double *colcnd, double *amax, int *info)
{
/*
Purpose
=======
DGSEQU computes row and column scalings intended to equilibrate an
M-by-N sparse matrix A and reduce its condition number. R returns the row
scale factors and C the column scale factors, chosen to try to make
the largest element in each row and column of the matrix B with
elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.
R(i) and C(j) are restricted to be between SMLNUM = smallest safe
number and BIGNUM = largest safe number. Use of these scaling
factors is not guaranteed to reduce the condition number of A but
works well in practice.
See supermatrix.h for the definition of 'SuperMatrix' structure.
Arguments
=========
A (input) SuperMatrix*
The matrix of dimension (A->nrow, A->ncol) whose equilibration
factors are to be computed. The type of A can be:
Stype = SLU_NC; Dtype = SLU_D; Mtype = SLU_GE.
R (output) double*, size A->nrow
If INFO = 0 or INFO > M, R contains the row scale factors
for A.
C (output) double*, size A->ncol
If INFO = 0, C contains the column scale factors for A.
ROWCND (output) double*
If INFO = 0 or INFO > M, ROWCND contains the ratio of the
smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
AMAX is neither too large nor too small, it is not worth
scaling by R.
COLCND (output) double*
If INFO = 0, COLCND contains the ratio of the smallest
C(i) to the largest C(i). If COLCND >= 0.1, it is not
worth scaling by C.
AMAX (output) double*
Absolute value of largest matrix element. If AMAX is very
close to overflow or very close to underflow, the matrix
should be scaled.
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->nrow: the i-th row of A is exactly zero
> A->ncol: the (i-M)-th column of A is exactly zero
=====================================================================
*/
/* Local variables */
NCformat *Astore;
double *Aval;
int i, j, irow;
double rcmin, rcmax;
double bignum, smlnum;
extern double hypre_F90_NAME_LAPACK(dlamch,DLAMCH)(const char *);
/* Test the input parameters. */
*info = 0;
if ( A->nrow < 0 || A->ncol < 0 ||
A->Stype != SLU_NC || A->Dtype != SLU_D || A->Mtype != SLU_GE )
*info = -1;
if (*info != 0) {
i = -(*info);
superlu_xerbla("dgsequ", &i);
return;
}
/* Quick return if possible */
if ( A->nrow == 0 || A->ncol == 0 ) {
*rowcnd = 1.;
*colcnd = 1.;
*amax = 0.;
return;
}
Astore = (NCformat*) A->Store;
Aval = (double*) Astore->nzval;
/* Get machine constants. */
smlnum = hypre_F90_NAME_LAPACK(dlamch,DLAMCH)("S");
bignum = 1. / smlnum;
/* Compute row scale factors. */
for (i = 0; i < A->nrow; ++i) r[i] = 0.;
/* Find the maximum element in each row. */
for (j = 0; j < A->ncol; ++j)
for (i = Astore->colptr[j]; i < Astore->colptr[j+1]; ++i) {
irow = Astore->rowind[i];
r[irow] = SUPERLU_MAX( r[irow], fabs(Aval[i]) );
}
/* Find the maximum and minimum scale factors. */
rcmin = bignum;
rcmax = 0.;
for (i = 0; i < A->nrow; ++i) {
rcmax = SUPERLU_MAX(rcmax, r[i]);
rcmin = SUPERLU_MIN(rcmin, r[i]);
}
*amax = rcmax;
if (rcmin == 0.) {
/* Find the first zero scale factor and return an error code. */
for (i = 0; i < A->nrow; ++i)
if (r[i] == 0.) {
*info = i + 1;
return;
}
} else {
/* Invert the scale factors. */
for (i = 0; i < A->nrow; ++i)
r[i] = 1. / SUPERLU_MIN( SUPERLU_MAX( r[i], smlnum ), bignum );
/* Compute ROWCND = min(R(I)) / max(R(I)) */
*rowcnd = SUPERLU_MAX( rcmin, smlnum ) / SUPERLU_MIN( rcmax, bignum );
}
/* Compute column scale factors */
for (j = 0; j < A->ncol; ++j) c[j] = 0.;
/* Find the maximum element in each column, assuming the row
scalings computed above. */
for (j = 0; j < A->ncol; ++j)
for (i = Astore->colptr[j]; i < Astore->colptr[j+1]; ++i) {
irow = Astore->rowind[i];
c[j] = SUPERLU_MAX( c[j], fabs(Aval[i]) * r[irow] );
}
/* Find the maximum and minimum scale factors. */
rcmin = bignum;
rcmax = 0.;
for (j = 0; j < A->ncol; ++j) {
rcmax = SUPERLU_MAX(rcmax, c[j]);
rcmin = SUPERLU_MIN(rcmin, c[j]);
}
if (rcmin == 0.) {
/* Find the first zero scale factor and return an error code. */
for (j = 0; j < A->ncol; ++j)
if ( c[j] == 0. ) {
*info = A->nrow + j + 1;
return;
}
} else {
/* Invert the scale factors. */
for (j = 0; j < A->ncol; ++j)
c[j] = 1. / SUPERLU_MIN( SUPERLU_MAX( c[j], smlnum ), bignum);
/* Compute COLCND = min(C(J)) / max(C(J)) */
*colcnd = SUPERLU_MAX( rcmin, smlnum ) / SUPERLU_MIN( rcmax, bignum );
}
return;
} /* dgsequ */
