int
sp_zgemm(char *transa, char *transb, int m, int n, int k,
doublecomplex alpha, SuperMatrix *A, doublecomplex *b, int ldb,
doublecomplex beta, doublecomplex *c, int ldc)
{
int incx = 1, incy = 1;
int j;
for (j = 0; j < n; ++j) {
sp_zgemv(transa, alpha, A, &b[ldb*j], incx, beta, &c[ldc*j], incy);
}
return 0;
}
int main(int argc, char *argv[])
{
void zmatvec_mult(doublecomplex alpha, doublecomplex x[], doublecomplex beta, doublecomplex y[]);
void zpsolve(int n, doublecomplex x[], doublecomplex y[]);
extern int zfgmr( int n,
void (*matvec_mult)(doublecomplex, doublecomplex [], doublecomplex, doublecomplex []),
void (*psolve)(int n, doublecomplex [], doublecomplex[]),
doublecomplex *rhs, doublecomplex *sol, double tol, int restrt, int *itmax,
FILE *fits);
extern int zfill_diag(int n, NCformat *Astore);
char equed[1] = {'B'};
yes_no_t equil;
trans_t trans;
SuperMatrix A, L, U;
SuperMatrix B, X;
NCformat *Astore;
NCformat *Ustore;
SCformat *Lstore;
doublecomplex *a;
int *asub, *xa;
int *etree;
int *perm_c; /* column permutation vector */
int *perm_r; /* row permutations from partial pivoting */
int nrhs, ldx, lwork, info, m, n, nnz;
doublecomplex *rhsb, *rhsx, *xact;
doublecomplex *work = NULL;
double *R, *C;
double u, rpg, rcond;
doublecomplex zero = {0.0, 0.0};
doublecomplex one = {1.0, 0.0};
doublecomplex none = {-1.0, 0.0};
mem_usage_t mem_usage;
superlu_options_t options;
SuperLUStat_t stat;
int restrt, iter, maxit, i;
double resid;
doublecomplex *x, *b;
#ifdef DEBUG
extern int num_drop_L, num_drop_U;
#endif
#if ( DEBUGlevel>=1 )
CHECK_MALLOC("Enter main()");
#endif
/* Defaults */
lwork = 0;
nrhs = 1;
trans = NOTRANS;
/* Set the default input options:
options.Fact = DOFACT;
options.Equil = YES;
options.ColPerm = COLAMD;
options.DiagPivotThresh = 0.1; //different from complete LU
options.Trans = NOTRANS;
options.IterRefine = NOREFINE;
options.SymmetricMode = NO;
options.PivotGrowth = NO;
options.ConditionNumber = NO;
options.PrintStat = YES;
options.RowPerm = LargeDiag;
options.ILU_DropTol = 1e-4;
options.ILU_FillTol = 1e-2;
options.ILU_FillFactor = 10.0;
options.ILU_DropRule = DROP_BASIC | DROP_AREA;
options.ILU_Norm = INF_NORM;
options.ILU_MILU = SILU;
*/
ilu_set_default_options(&options);
/* Modify the defaults. */
options.PivotGrowth = YES; /* Compute reciprocal pivot growth */
options.ConditionNumber = YES;/* Compute reciprocal condition number */
if ( lwork > 0 ) {
work = SUPERLU_MALLOC(lwork);
if ( !work ) ABORT("Malloc fails for work[].");
}
/* Read matrix A from a file in Harwell-Boeing format.*/
if (argc < 2)
{
printf("Usage:\n%s [OPTION] < [INPUT] > [OUTPUT]\nOPTION:\n"
"-h -hb:\n\t[INPUT] is a Harwell-Boeing format matrix.\n"
"-r -rb:\n\t[INPUT] is a Rutherford-Boeing format matrix.\n"
"-t -triplet:\n\t[INPUT] is a triplet format matrix.\n",
argv[0]);
return 0;
}
else
{
switch (argv[1][1])
{
case 'H':
case 'h':
printf("Input a Harwell-Boeing format matrix:\n");
zreadhb(&m, &n, &nnz, &a, &asub, &xa);
break;
case 'R':
case 'r':
printf("Input a Rutherford-Boeing format matrix:\n");
zreadrb(&m, &n, &nnz, &a, &asub, &xa);
break;
case 'T':
case 't':
printf("Input a triplet format matrix:\n");
zreadtriple(&m, &n, &nnz, &a, &asub, &xa);
break;
default:
printf("Unrecognized format.\n");
return 0;
}
}
zCreate_CompCol_Matrix(&A, m, n, nnz, a, asub, xa,
SLU_NC, SLU_Z, SLU_GE);
Astore = A.Store;
zfill_diag(n, Astore);
printf("Dimension %dx%d; # nonzeros %d\n", A.nrow, A.ncol, Astore->nnz);
fflush(stdout);
/* Generate the right-hand side */
if ( !(rhsb = doublecomplexMalloc(m * nrhs)) ) ABORT("Malloc fails for rhsb[].");
if ( !(rhsx = doublecomplexMalloc(m * nrhs)) ) ABORT("Malloc fails for rhsx[].");
zCreate_Dense_Matrix(&B, m, nrhs, rhsb, m, SLU_DN, SLU_Z, SLU_GE);
zCreate_Dense_Matrix(&X, m, nrhs, rhsx, m, SLU_DN, SLU_Z, SLU_GE);
xact = doublecomplexMalloc(n * nrhs);
ldx = n;
zGenXtrue(n, nrhs, xact, ldx);
zFillRHS(trans, nrhs, xact, ldx, &A, &B);
if ( !(etree = intMalloc(n)) ) ABORT("Malloc fails for etree[].");
if ( !(perm_r = intMalloc(m)) ) ABORT("Malloc fails for perm_r[].");
if ( !(perm_c = intMalloc(n)) ) ABORT("Malloc fails for perm_c[].");
if ( !(R = (double *) SUPERLU_MALLOC(A.nrow * sizeof(double))) )
ABORT("SUPERLU_MALLOC fails for R[].");
if ( !(C = (double *) SUPERLU_MALLOC(A.ncol * sizeof(double))) )
ABORT("SUPERLU_MALLOC fails for C[].");
info = 0;
#ifdef DEBUG
num_drop_L = 0;
num_drop_U = 0;
#endif
/* Initialize the statistics variables. */
StatInit(&stat);
/* Compute the incomplete factorization and compute the condition number
and pivot growth using dgsisx. */
B.ncol = 0; /* not to perform triangular solution */
zgsisx(&options, &A, perm_c, perm_r, etree, equed, R, C, &L, &U, work,
lwork, &B, &X, &rpg, &rcond, &mem_usage, &stat, &info);
/* Set RHS for GMRES. */
if (!(b = doublecomplexMalloc(m))) ABORT("Malloc fails for b[].");
if (*equed == 'R' || *equed == 'B') {
for (i = 0; i < n; ++i) zd_mult(&b[i], &rhsb[i], R[i]);
} else {
for (i = 0; i < m; i++) b[i] = rhsb[i];
}
printf("zgsisx(): info %d, equed %c\n", info, equed[0]);
if (info > 0 || rcond < 1e-8 || rpg > 1e8)
printf("WARNING: This preconditioner might be unstable.\n");
if ( info == 0 || info == n+1 ) {
if ( options.PivotGrowth == YES )
printf("Recip. pivot growth = %e\n", rpg);
if ( options.ConditionNumber == YES )
printf("Recip. condition number = %e\n", rcond);
} else if ( info > 0 && lwork == -1 ) {
printf("** Estimated memory: %d bytes\n", info - n);
}
Lstore = (SCformat *) L.Store;
Ustore = (NCformat *) U.Store;
printf("n(A) = %d, nnz(A) = %d\n", n, Astore->nnz);
printf("No of nonzeros in factor L = %d\n", Lstore->nnz);
printf("No of nonzeros in factor U = %d\n", Ustore->nnz);
printf("No of nonzeros in L+U = %d\n", Lstore->nnz + Ustore->nnz - n);
printf("Fill ratio: nnz(F)/nnz(A) = %.3f\n",
((double)(Lstore->nnz) + (double)(Ustore->nnz) - (double)n)
/ (double)Astore->nnz);
printf("L\\U MB %.3f\ttotal MB needed %.3f\n",
mem_usage.for_lu/1e6, mem_usage.total_needed/1e6);
fflush(stdout);
/* Set the global variables. */
GLOBAL_A = &A;
GLOBAL_L = &L;
GLOBAL_U = &U;
GLOBAL_STAT = &stat;
GLOBAL_PERM_C = perm_c;
GLOBAL_PERM_R = perm_r;
GLOBAL_OPTIONS = &options;
GLOBAL_R = R;
GLOBAL_C = C;
GLOBAL_MEM_USAGE = &mem_usage;
/* Set the options to do solve-only. */
options.Fact = FACTORED;
options.PivotGrowth = NO;
options.ConditionNumber = NO;
/* Set the variables used by GMRES. */
restrt = SUPERLU_MIN(n / 3 + 1, 50);
maxit = 1000;
iter = maxit;
resid = 1e-8;
if (!(x = doublecomplexMalloc(n))) ABORT("Malloc fails for x[].");
if (info <= n + 1)
{
int i_1 = 1;
double maxferr = 0.0, nrmA, nrmB, res, t;
doublecomplex temp;
extern double dznrm2_(int *, doublecomplex [], int *);
extern void zaxpy_(int *, doublecomplex *, doublecomplex [], int *, doublecomplex [], int *);
/* Initial guess */
for (i = 0; i < n; i++) x[i] = zero;
t = SuperLU_timer_();
/* Call GMRES */
zfgmr(n, zmatvec_mult, zpsolve, b, x, resid, restrt, &iter, stdout);
t = SuperLU_timer_() - t;
/* Output the result. */
nrmA = dznrm2_(&(Astore->nnz), (doublecomplex *)((DNformat *)A.Store)->nzval,
&i_1);
nrmB = dznrm2_(&m, b, &i_1);
sp_zgemv("N", none, &A, x, 1, one, b, 1);
res = dznrm2_(&m, b, &i_1);
resid = res / nrmB;
printf("||A||_F = %.1e, ||B||_2 = %.1e, ||B-A*X||_2 = %.1e, "
"relres = %.1e\n", nrmA, nrmB, res, resid);
if (iter >= maxit)
{
if (resid >= 1.0) iter = -180;
else if (resid > 1e-8) iter = -111;
}
printf("iteration: %d\nresidual: %.1e\nGMRES time: %.2f seconds.\n",
iter, resid, t);
/* Scale the solution back if equilibration was performed. */
if (*equed == 'C' || *equed == 'B')
for (i = 0; i < n; i++) zd_mult(&x[i], &x[i], C[i]);
for (i = 0; i < m; i++) {
z_sub(&temp, &x[i], &xact[i]);
maxferr = SUPERLU_MAX(maxferr, z_abs1(&temp));
}
printf("||X-X_true||_oo = %.1e\n", maxferr);
}
#ifdef DEBUG
printf("%d entries in L and %d entries in U dropped.\n",
num_drop_L, num_drop_U);
#endif
fflush(stdout);
if ( options.PrintStat ) StatPrint(&stat);
StatFree(&stat);
SUPERLU_FREE (rhsb);
SUPERLU_FREE (rhsx);
SUPERLU_FREE (xact);
SUPERLU_FREE (etree);
SUPERLU_FREE (perm_r);
SUPERLU_FREE (perm_c);
SUPERLU_FREE (R);
SUPERLU_FREE (C);
Destroy_CompCol_Matrix(&A);
Destroy_SuperMatrix_Store(&B);
Destroy_SuperMatrix_Store(&X);
if ( lwork >= 0 ) {
Destroy_SuperNode_Matrix(&L);
Destroy_CompCol_Matrix(&U);
}
SUPERLU_FREE(b);
SUPERLU_FREE(x);
#if ( DEBUGlevel>=1 )
CHECK_MALLOC("Exit main()");
#endif
return 0;
}
void zmatvec_mult(doublecomplex alpha, doublecomplex x[], doublecomplex beta, doublecomplex y[])
{
SuperMatrix *A = GLOBAL_A;
sp_zgemv("N", alpha, A, x, 1, beta, y, 1);
}
/*! \brief
*
* <pre>
* Purpose
* =======
*
* ZGSRFS improves the computed solution to a system of linear
* equations and provides error bounds and backward error estimates for
* the solution.
*
* If equilibration was performed, the system becomes:
* (diag(R)*A_original*diag(C)) * X = diag(R)*B_original.
*
* See supermatrix.h for the definition of 'SuperMatrix' structure.
*
* Arguments
* =========
*
* trans (input) trans_t
* Specifies the form of the system of equations:
* = NOTRANS: A * X = B (No transpose)
* = TRANS: A'* X = B (Transpose)
* = CONJ: A**H * X = B (Conjugate transpose)
*
* A (input) SuperMatrix*
* The original matrix A in the system, or the scaled A if
* equilibration was done. The type of A can be:
* Stype = SLU_NC, Dtype = SLU_Z, Mtype = SLU_GE.
*
* L (input) SuperMatrix*
* The factor L from the factorization Pr*A*Pc=L*U. Use
* compressed row subscripts storage for supernodes,
* i.e., L has types: Stype = SLU_SC, Dtype = SLU_Z, Mtype = SLU_TRLU.
*
* U (input) SuperMatrix*
* The factor U from the factorization Pr*A*Pc=L*U as computed by
* zgstrf(). Use column-wise storage scheme,
* i.e., U has types: Stype = SLU_NC, Dtype = SLU_Z, Mtype = SLU_TRU.
*
* perm_c (input) int*, dimension (A->ncol)
* Column permutation vector, which defines the
* permutation matrix Pc; perm_c[i] = j means column i of A is
* in position j in A*Pc.
*
* perm_r (input) int*, dimension (A->nrow)
* Row permutation vector, which defines the permutation matrix Pr;
* perm_r[i] = j means row i of A is in position j in Pr*A.
*
* equed (input) 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).
*
* R (input) double*, dimension (A->nrow)
* The row scale factors for A.
* If equed = 'R' or 'B', A is premultiplied by diag(R).
* If equed = 'N' or 'C', R is not accessed.
*
* C (input) double*, dimension (A->ncol)
* The column scale factors for A.
* If equed = 'C' or 'B', A is postmultiplied by diag(C).
* If equed = 'N' or 'R', C is not accessed.
*
* B (input) SuperMatrix*
* B has types: Stype = SLU_DN, Dtype = SLU_Z, Mtype = SLU_GE.
* The right hand side matrix B.
* if equed = 'R' or 'B', B is premultiplied by diag(R).
*
* X (input/output) SuperMatrix*
* X has types: Stype = SLU_DN, Dtype = SLU_Z, Mtype = SLU_GE.
* On entry, the solution matrix X, as computed by zgstrs().
* On exit, the improved solution matrix X.
* if *equed = 'C' or 'B', X should be premultiplied by diag(C)
* in order to obtain the solution to the original system.
*
* 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.
*
* 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).
*
* 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
*
* Internal Parameters
* ===================
*
* ITMAX is the maximum number of steps of iterative refinement.
*
* </pre>
*/
void
zgsrfs(trans_t trans, SuperMatrix *A, SuperMatrix *L, SuperMatrix *U,
int *perm_c, int *perm_r, char *equed, double *R, double *C,
SuperMatrix *B, SuperMatrix *X, double *ferr, double *berr,
SuperLUStat_t *stat, int *info)
{
#define ITMAX 5
/* Table of constant values */
int ione = 1;
doublecomplex ndone = {-1., 0.};
doublecomplex done = {1., 0.};
/* Local variables */
NCformat *Astore;
doublecomplex *Aval;
SuperMatrix Bjcol;
DNformat *Bstore, *Xstore, *Bjcol_store;
doublecomplex *Bmat, *Xmat, *Bptr, *Xptr;
int kase;
double safe1, safe2;
int i, j, k, irow, nz, count, notran, rowequ, colequ;
int ldb, ldx, nrhs;
double s, xk, lstres, eps, safmin;
char transc[1];
trans_t transt;
doublecomplex *work;
double *rwork;
int *iwork;
int isave[3];
extern int zlacon2_(int *, doublecomplex *, doublecomplex *, double *, int *, int []);
#ifdef _CRAY
extern int CCOPY(int *, doublecomplex *, int *, doublecomplex *, int *);
extern int CSAXPY(int *, doublecomplex *, doublecomplex *, int *, doublecomplex *, int *);
#else
extern int zcopy_(int *, doublecomplex *, int *, doublecomplex *, int *);
extern int zaxpy_(int *, doublecomplex *, doublecomplex *, int *, doublecomplex *, int *);
#endif
Astore = A->Store;
Aval = Astore->nzval;
Bstore = B->Store;
Xstore = X->Store;
Bmat = Bstore->nzval;
Xmat = Xstore->nzval;
ldb = Bstore->lda;
ldx = Xstore->lda;
nrhs = B->ncol;
/* Test the input parameters */
*info = 0;
notran = (trans == NOTRANS);
if ( !notran && trans != TRANS && trans != CONJ ) *info = -1;
else if ( A->nrow != A->ncol || A->nrow < 0 ||
A->Stype != SLU_NC || A->Dtype != SLU_Z || A->Mtype != SLU_GE )
*info = -2;
else if ( L->nrow != L->ncol || L->nrow < 0 ||
L->Stype != SLU_SC || L->Dtype != SLU_Z || L->Mtype != SLU_TRLU )
*info = -3;
else if ( U->nrow != U->ncol || U->nrow < 0 ||
U->Stype != SLU_NC || U->Dtype != SLU_Z || U->Mtype != SLU_TRU )
*info = -4;
else if ( ldb < SUPERLU_MAX(0, A->nrow) ||
B->Stype != SLU_DN || B->Dtype != SLU_Z || B->Mtype != SLU_GE )
*info = -10;
else if ( ldx < SUPERLU_MAX(0, A->nrow) ||
X->Stype != SLU_DN || X->Dtype != SLU_Z || X->Mtype != SLU_GE )
*info = -11;
if (*info != 0) {
i = -(*info);
input_error("zgsrfs", &i);
return;
}
/* Quick return if possible */
if ( A->nrow == 0 || nrhs == 0) {
for (j = 0; j < nrhs; ++j) {
ferr[j] = 0.;
berr[j] = 0.;
}
return;
}
rowequ = strncmp(equed, "R", 1)==0 || strncmp(equed, "B", 1)==0;
colequ = strncmp(equed, "C", 1)==0 || strncmp(equed, "B", 1)==0;
/* Allocate working space */
work = doublecomplexMalloc(2*A->nrow);
rwork = (double *) SUPERLU_MALLOC( A->nrow * sizeof(double) );
iwork = intMalloc(A->nrow);
if ( !work || !rwork || !iwork )
ABORT("Malloc fails for work/rwork/iwork.");
if ( notran ) {
*(unsigned char *)transc = 'N';
transt = TRANS;
} else if ( trans == TRANS ) {
*(unsigned char *)transc = 'T';
transt = NOTRANS;
} else if ( trans == CONJ ) {
*(unsigned char *)transc = 'C';
transt = NOTRANS;
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = A->ncol + 1;
eps = dmach("Epsilon");
safmin = dmach("Safe minimum");
/* Set SAFE1 essentially to be the underflow threshold times the
number of additions in each row. */
safe1 = nz * safmin;
safe2 = safe1 / eps;
/* Compute the number of nonzeros in each row (or column) of A */
for (i = 0; i < A->nrow; ++i) iwork[i] = 0;
if ( notran ) {
for (k = 0; k < A->ncol; ++k)
for (i = Astore->colptr[k]; i < Astore->colptr[k+1]; ++i)
++iwork[Astore->rowind[i]];
} else {
for (k = 0; k < A->ncol; ++k)
iwork[k] = Astore->colptr[k+1] - Astore->colptr[k];
}
/* Copy one column of RHS B into Bjcol. */
Bjcol.Stype = B->Stype;
Bjcol.Dtype = B->Dtype;
Bjcol.Mtype = B->Mtype;
Bjcol.nrow = B->nrow;
Bjcol.ncol = 1;
Bjcol.Store = (void *) SUPERLU_MALLOC( sizeof(DNformat) );
if ( !Bjcol.Store ) ABORT("SUPERLU_MALLOC fails for Bjcol.Store");
Bjcol_store = Bjcol.Store;
Bjcol_store->lda = ldb;
Bjcol_store->nzval = work; /* address aliasing */
/* Do for each right hand side ... */
for (j = 0; j < nrhs; ++j) {
count = 0;
lstres = 3.;
Bptr = &Bmat[j*ldb];
Xptr = &Xmat[j*ldx];
while (1) { /* Loop until stopping criterion is satisfied. */
/* Compute residual R = B - op(A) * X,
where op(A) = A, A**T, or A**H, depending on TRANS. */
#ifdef _CRAY
CCOPY(&A->nrow, Bptr, &ione, work, &ione);
#else
zcopy_(&A->nrow, Bptr, &ione, work, &ione);
#endif
sp_zgemv(transc, ndone, A, Xptr, ione, done, work, ione);
/* Compute componentwise relative backward error from formula
max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
where abs(Z) is the componentwise absolute value of the matrix
or vector Z. If the i-th component of the denominator is less
than SAFE2, then SAFE1 is added to the i-th component of the
numerator before dividing. */
for (i = 0; i < A->nrow; ++i) rwork[i] = z_abs1( &Bptr[i] );
/* Compute abs(op(A))*abs(X) + abs(B). */
if ( notran ) {
for (k = 0; k < A->ncol; ++k) {
xk = z_abs1( &Xptr[k] );
for (i = Astore->colptr[k]; i < Astore->colptr[k+1]; ++i)
rwork[Astore->rowind[i]] += z_abs1(&Aval[i]) * xk;
}
} else { /* trans = TRANS or CONJ */
for (k = 0; k < A->ncol; ++k) {
s = 0.;
for (i = Astore->colptr[k]; i < Astore->colptr[k+1]; ++i) {
irow = Astore->rowind[i];
s += z_abs1(&Aval[i]) * z_abs1(&Xptr[irow]);
}
rwork[k] += s;
}
}
s = 0.;
for (i = 0; i < A->nrow; ++i) {
if (rwork[i] > safe2) {
s = SUPERLU_MAX( s, z_abs1(&work[i]) / rwork[i] );
} else if ( rwork[i] != 0.0 ) {
s = SUPERLU_MAX( s, (z_abs1(&work[i]) + safe1) / rwork[i] );
}
/* If rwork[i] is exactly 0.0, then we know the true
residual also must be exactly 0.0. */
}
berr[j] = s;
/* Test stopping criterion. Continue iterating if
1) The residual BERR(J) is larger than machine epsilon, and
2) BERR(J) decreased by at least a factor of 2 during the
last iteration, and
3) At most ITMAX iterations tried. */
if (berr[j] > eps && berr[j] * 2. <= lstres && count < ITMAX) {
/* Update solution and try again. */
zgstrs (trans, L, U, perm_c, perm_r, &Bjcol, stat, info);
#ifdef _CRAY
CAXPY(&A->nrow, &done, work, &ione,
&Xmat[j*ldx], &ione);
#else
zaxpy_(&A->nrow, &done, work, &ione,
&Xmat[j*ldx], &ione);
#endif
lstres = berr[j];
++count;
} else {
break;
}
} /* end while */
stat->RefineSteps = count;
/* Bound error from formula:
norm(X - XTRUE) / norm(X) .le. FERR = norm( abs(inv(op(A)))*
( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
where
norm(Z) is the magnitude of the largest component of Z
inv(op(A)) is the inverse of op(A)
abs(Z) is the componentwise absolute value of the matrix or
vector Z
NZ is the maximum number of nonzeros in any row of A, plus 1
EPS is machine epsilon
The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
is incremented by SAFE1 if the i-th component of
abs(op(A))*abs(X) + abs(B) is less than SAFE2.
Use ZLACON2 to estimate the infinity-norm of the matrix
inv(op(A)) * diag(W),
where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) */
for (i = 0; i < A->nrow; ++i) rwork[i] = z_abs1( &Bptr[i] );
/* Compute abs(op(A))*abs(X) + abs(B). */
if ( notran ) {
for (k = 0; k < A->ncol; ++k) {
xk = z_abs1( &Xptr[k] );
for (i = Astore->colptr[k]; i < Astore->colptr[k+1]; ++i)
rwork[Astore->rowind[i]] += z_abs1(&Aval[i]) * xk;
}
} else { /* trans == TRANS or CONJ */
for (k = 0; k < A->ncol; ++k) {
s = 0.;
for (i = Astore->colptr[k]; i < Astore->colptr[k+1]; ++i) {
irow = Astore->rowind[i];
xk = z_abs1( &Xptr[irow] );
s += z_abs1(&Aval[i]) * xk;
}
rwork[k] += s;
}
}
for (i = 0; i < A->nrow; ++i)
if (rwork[i] > safe2)
rwork[i] = z_abs(&work[i]) + (iwork[i]+1)*eps*rwork[i];
else
rwork[i] = z_abs(&work[i])+(iwork[i]+1)*eps*rwork[i]+safe1;
kase = 0;
do {
zlacon2_(&A->nrow, &work[A->nrow], work, &ferr[j], &kase, isave);
if (kase == 0) break;
if (kase == 1) {
/* Multiply by diag(W)*inv(op(A)**T)*(diag(C) or diag(R)). */
if ( notran && colequ )
for (i = 0; i < A->ncol; ++i) {
zd_mult(&work[i], &work[i], C[i]);
}
else if ( !notran && rowequ )
for (i = 0; i < A->nrow; ++i) {
zd_mult(&work[i], &work[i], R[i]);
}
zgstrs (transt, L, U, perm_c, perm_r, &Bjcol, stat, info);
for (i = 0; i < A->nrow; ++i) {
zd_mult(&work[i], &work[i], rwork[i]);
}
} else {
/* Multiply by (diag(C) or diag(R))*inv(op(A))*diag(W). */
for (i = 0; i < A->nrow; ++i) {
zd_mult(&work[i], &work[i], rwork[i]);
}
zgstrs (trans, L, U, perm_c, perm_r, &Bjcol, stat, info);
if ( notran && colequ )
for (i = 0; i < A->ncol; ++i) {
zd_mult(&work[i], &work[i], C[i]);
}
else if ( !notran && rowequ )
for (i = 0; i < A->ncol; ++i) {
zd_mult(&work[i], &work[i], R[i]);
}
}
} while ( kase != 0 );
/* Normalize error. */
lstres = 0.;
if ( notran && colequ ) {
for (i = 0; i < A->nrow; ++i)
lstres = SUPERLU_MAX( lstres, C[i] * z_abs1( &Xptr[i]) );
} else if ( !notran && rowequ ) {
for (i = 0; i < A->nrow; ++i)
lstres = SUPERLU_MAX( lstres, R[i] * z_abs1( &Xptr[i]) );
} else {
for (i = 0; i < A->nrow; ++i)
lstres = SUPERLU_MAX( lstres, z_abs1( &Xptr[i]) );
}
if ( lstres != 0. )
ferr[j] /= lstres;
} /* for each RHS j ... */
SUPERLU_FREE(work);
SUPERLU_FREE(rwork);
SUPERLU_FREE(iwork);
SUPERLU_FREE(Bjcol.Store);
return;
} /* zgsrfs */
void
zgsrfs(char *trans, SuperMatrix *A, SuperMatrix *L, SuperMatrix *U,
int *perm_r, int *perm_c, char *equed, double *R, double *C,
SuperMatrix *B, SuperMatrix *X,
double *ferr, double *berr, int *info)
{
/*
* Purpose
* =======
*
* ZGSRFS improves the computed solution to a system of linear
* equations and provides error bounds and backward error estimates for
* the solution.
*
* If equilibration was performed, the system becomes:
* (diag(R)*A_original*diag(C)) * X = diag(R)*B_original.
*
* See supermatrix.h for the definition of 'SuperMatrix' structure.
*
* Arguments
* =========
*
* trans (input) char*
* Specifies the form of the system of equations:
* = 'N': A * X = B (No transpose)
* = 'T': A**T * X = B (Transpose)
* = 'C': A**H * X = B (Conjugate transpose = Transpose)
*
* A (input) SuperMatrix*
* The original matrix A in the system, or the scaled A if
* equilibration was done. The type of A can be:
* Stype = NC, Dtype = _Z, Mtype = GE.
*
* L (input) SuperMatrix*
* The factor L from the factorization Pr*A*Pc=L*U. Use
* compressed row subscripts storage for supernodes,
* i.e., L has types: Stype = SC, Dtype = _Z, Mtype = TRLU.
*
* U (input) SuperMatrix*
* The factor U from the factorization Pr*A*Pc=L*U as computed by
* zgstrf(). Use column-wise storage scheme,
* i.e., U has types: Stype = NC, Dtype = _Z, Mtype = TRU.
*
* perm_r (input) int*, dimension (A->nrow)
* Row permutation vector, which defines the permutation matrix Pr;
* perm_r[i] = j means row i of A is in position j in Pr*A.
*
* perm_c (input) int*, dimension (A->ncol)
* Column permutation vector, which defines the
* permutation matrix Pc; perm_c[i] = j means column i of A is
* in position j in A*Pc.
*
* equed (input) 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).
*
* R (input) double*, dimension (A->nrow)
* The row scale factors for A.
* If equed = 'R' or 'B', A is premultiplied by diag(R).
* If equed = 'N' or 'C', R is not accessed.
*
* C (input) double*, dimension (A->ncol)
* The column scale factors for A.
* If equed = 'C' or 'B', A is postmultiplied by diag(C).
* If equed = 'N' or 'R', C is not accessed.
*
* B (input) SuperMatrix*
* B has types: Stype = DN, Dtype = _Z, Mtype = GE.
* The right hand side matrix B.
* if equed = 'R' or 'B', B is premultiplied by diag(R).
*
* X (input/output) SuperMatrix*
* X has types: Stype = DN, Dtype = _Z, Mtype = GE.
* On entry, the solution matrix X, as computed by zgstrs().
* On exit, the improved solution matrix X.
* if *equed = 'C' or 'B', X should be premultiplied by diag(C)
* in order to obtain the solution to the original system.
*
* 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.
*
* 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).
*
* info (output) int*
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Internal Parameters
* ===================
*
* ITMAX is the maximum number of steps of iterative refinement.
*
*/
#define ITMAX 5
/* Table of constant values */
int ione = 1;
doublecomplex ndone = {-1., 0.};
doublecomplex done = {1., 0.};
/* Local variables */
NCformat *Astore;
doublecomplex *Aval;
SuperMatrix Bjcol;
DNformat *Bstore, *Xstore, *Bjcol_store;
doublecomplex *Bmat, *Xmat, *Bptr, *Xptr;
int kase;
double safe1, safe2;
int i, j, k, irow, nz, count, notran, rowequ, colequ;
int ldb, ldx, nrhs;
double s, xk, lstres, eps, safmin;
char transt[1];
doublecomplex *work;
double *rwork;
int *iwork;
extern double dlamch_(char *);
extern int zlacon_(int *, doublecomplex *, doublecomplex *, double *, int *);
#ifdef _CRAY
extern int CCOPY(int *, doublecomplex *, int *, doublecomplex *, int *);
extern int CSAXPY(int *, doublecomplex *, doublecomplex *, int *, doublecomplex *, int *);
#else
extern int zcopy_(int *, doublecomplex *, int *, doublecomplex *, int *);
extern int zaxpy_(int *, doublecomplex *, doublecomplex *, int *, doublecomplex *, int *);
#endif
Astore = A->Store;
Aval = Astore->nzval;
Bstore = B->Store;
Xstore = X->Store;
Bmat = Bstore->nzval;
Xmat = Xstore->nzval;
ldb = Bstore->lda;
ldx = Xstore->lda;
nrhs = B->ncol;
/* Test the input parameters */
*info = 0;
notran = lsame_(trans, "N");
if ( !notran && !lsame_(trans, "T") && !lsame_(trans, "C")) *info = -1;
else if ( A->nrow != A->ncol || A->nrow < 0 ||
A->Stype != NC || A->Dtype != _Z || A->Mtype != GE )
*info = -2;
else if ( L->nrow != L->ncol || L->nrow < 0 ||
L->Stype != SC || L->Dtype != _Z || L->Mtype != TRLU )
*info = -3;
else if ( U->nrow != U->ncol || U->nrow < 0 ||
U->Stype != NC || U->Dtype != _Z || U->Mtype != TRU )
*info = -4;
else if ( ldb < MAX(0, A->nrow) ||
B->Stype != DN || B->Dtype != _Z || B->Mtype != GE )
*info = -10;
else if ( ldx < MAX(0, A->nrow) ||
X->Stype != DN || X->Dtype != _Z || X->Mtype != GE )
*info = -11;
if (*info != 0) {
i = -(*info);
xerbla_("zgsrfs", &i);
return;
}
/* Quick return if possible */
if ( A->nrow == 0 || nrhs == 0) {
for (j = 0; j < nrhs; ++j) {
ferr[j] = 0.;
berr[j] = 0.;
}
return;
}
rowequ = lsame_(equed, "R") || lsame_(equed, "B");
colequ = lsame_(equed, "C") || lsame_(equed, "B");
/* Allocate working space */
work = doublecomplexMalloc(2*A->nrow);
rwork = (double *) SUPERLU_MALLOC( A->nrow * sizeof(double) );
iwork = intMalloc(A->nrow);
if ( !work || !rwork || !iwork )
ABORT("Malloc fails for work/rwork/iwork.");
if ( notran ) {
*(unsigned char *)transt = 'T';
} else {
*(unsigned char *)transt = 'N';
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = A->ncol + 1;
eps = dlamch_("Epsilon");
safmin = dlamch_("Safe minimum");
safe1 = nz * safmin;
safe2 = safe1 / eps;
/* Compute the number of nonzeros in each row (or column) of A */
for (i = 0; i < A->nrow; ++i) iwork[i] = 0;
if ( notran ) {
for (k = 0; k < A->ncol; ++k)
for (i = Astore->colptr[k]; i < Astore->colptr[k+1]; ++i)
++iwork[Astore->rowind[i]];
} else {
for (k = 0; k < A->ncol; ++k)
iwork[k] = Astore->colptr[k+1] - Astore->colptr[k];
}
/* Copy one column of RHS B into Bjcol. */
Bjcol.Stype = B->Stype;
Bjcol.Dtype = B->Dtype;
Bjcol.Mtype = B->Mtype;
Bjcol.nrow = B->nrow;
Bjcol.ncol = 1;
Bjcol.Store = (void *) SUPERLU_MALLOC( sizeof(DNformat) );
if ( !Bjcol.Store ) ABORT("SUPERLU_MALLOC fails for Bjcol.Store");
Bjcol_store = Bjcol.Store;
Bjcol_store->lda = ldb;
Bjcol_store->nzval = work; /* address aliasing */
/* Do for each right hand side ... */
for (j = 0; j < nrhs; ++j) {
count = 0;
lstres = 3.;
Bptr = &Bmat[j*ldb];
Xptr = &Xmat[j*ldx];
while (1) { /* Loop until stopping criterion is satisfied. */
/* Compute residual R = B - op(A) * X,
where op(A) = A, A**T, or A**H, depending on TRANS. */
#ifdef _CRAY
CCOPY(&A->nrow, Bptr, &ione, work, &ione);
#else
zcopy_(&A->nrow, Bptr, &ione, work, &ione);
#endif
sp_zgemv(trans, ndone, A, Xptr, ione, done, work, ione);
/* Compute componentwise relative backward error from formula
max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
where abs(Z) is the componentwise absolute value of the matrix
or vector Z. If the i-th component of the denominator is less
than SAFE2, then SAFE1 is added to the i-th component of the
numerator and denominator before dividing. */
for (i = 0; i < A->nrow; ++i) rwork[i] = z_abs1( &Bptr[i] );
/* Compute abs(op(A))*abs(X) + abs(B). */
if (notran) {
for (k = 0; k < A->ncol; ++k) {
xk = z_abs1( &Xptr[k] );
for (i = Astore->colptr[k]; i < Astore->colptr[k+1]; ++i)
rwork[Astore->rowind[i]] += z_abs1(&Aval[i]) * xk;
}
} else {
for (k = 0; k < A->ncol; ++k) {
s = 0.;
for (i = Astore->colptr[k]; i < Astore->colptr[k+1]; ++i) {
irow = Astore->rowind[i];
s += z_abs1(&Aval[i]) * z_abs1(&Xptr[irow]);
}
rwork[k] += s;
}
}
s = 0.;
for (i = 0; i < A->nrow; ++i) {
if (rwork[i] > safe2)
s = MAX( s, z_abs1(&work[i]) / rwork[i] );
else
s = MAX( s, (z_abs1(&work[i]) + safe1) /
(rwork[i] + safe1) );
}
berr[j] = s;
/* Test stopping criterion. Continue iterating if
1) The residual BERR(J) is larger than machine epsilon, and
2) BERR(J) decreased by at least a factor of 2 during the
last iteration, and
3) At most ITMAX iterations tried. */
if (berr[j] > eps && berr[j] * 2. <= lstres && count < ITMAX) {
/* Update solution and try again. */
zgstrs (trans, L, U, perm_r, perm_c, &Bjcol, info);
#ifdef _CRAY
CAXPY(&A->nrow, &done, work, &ione,
&Xmat[j*ldx], &ione);
#else
zaxpy_(&A->nrow, &done, work, &ione,
&Xmat[j*ldx], &ione);
#endif
lstres = berr[j];
++count;
} else {
break;
}
} /* end while */
/* Bound error from formula:
norm(X - XTRUE) / norm(X) .le. FERR = norm( abs(inv(op(A)))*
( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
where
norm(Z) is the magnitude of the largest component of Z
inv(op(A)) is the inverse of op(A)
abs(Z) is the componentwise absolute value of the matrix or
vector Z
NZ is the maximum number of nonzeros in any row of A, plus 1
EPS is machine epsilon
The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
is incremented by SAFE1 if the i-th component of
abs(op(A))*abs(X) + abs(B) is less than SAFE2.
Use ZLACON to estimate the infinity-norm of the matrix
inv(op(A)) * diag(W),
where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) */
for (i = 0; i < A->nrow; ++i) rwork[i] = z_abs1( &Bptr[i] );
/* Compute abs(op(A))*abs(X) + abs(B). */
if ( notran ) {
for (k = 0; k < A->ncol; ++k) {
xk = z_abs1( &Xptr[k] );
for (i = Astore->colptr[k]; i < Astore->colptr[k+1]; ++i)
rwork[Astore->rowind[i]] += z_abs1(&Aval[i]) * xk;
}
} else {
for (k = 0; k < A->ncol; ++k) {
s = 0.;
for (i = Astore->colptr[k]; i < Astore->colptr[k+1]; ++i) {
irow = Astore->rowind[i];
xk = z_abs1( &Xptr[irow] );
s += z_abs1(&Aval[i]) * xk;
}
rwork[k] += s;
}
}
for (i = 0; i < A->nrow; ++i)
if (rwork[i] > safe2)
rwork[i] = z_abs(&work[i]) + (iwork[i]+1)*eps*rwork[i];
else
rwork[i] = z_abs(&work[i])+(iwork[i]+1)*eps*rwork[i]+safe1;
kase = 0;
do {
zlacon_(&A->nrow, &work[A->nrow], work,
&ferr[j], &kase);
if (kase == 0) break;
if (kase == 1) {
/* Multiply by diag(W)*inv(op(A)**T)*(diag(C) or diag(R)). */
if ( notran && colequ )
for (i = 0; i < A->ncol; ++i) {
zd_mult(&work[i], &work[i], C[i]);
}
else if ( !notran && rowequ )
for (i = 0; i < A->nrow; ++i) {
zd_mult(&work[i], &work[i], R[i]);
}
zgstrs (transt, L, U, perm_r, perm_c, &Bjcol, info);
for (i = 0; i < A->nrow; ++i) {
zd_mult(&work[i], &work[i], rwork[i]);
}
} else {
/* Multiply by (diag(C) or diag(R))*inv(op(A))*diag(W). */
for (i = 0; i < A->nrow; ++i) {
zd_mult(&work[i], &work[i], rwork[i]);
}
zgstrs (trans, L, U, perm_r, perm_c, &Bjcol, info);
if ( notran && colequ )
for (i = 0; i < A->ncol; ++i) {
zd_mult(&work[i], &work[i], C[i]);
}
else if ( !notran && rowequ )
for (i = 0; i < A->ncol; ++i) {
zd_mult(&work[i], &work[i], R[i]);
}
}
} while ( kase != 0 );
/* Normalize error. */
lstres = 0.;
if ( notran && colequ ) {
for (i = 0; i < A->nrow; ++i)
lstres = MAX( lstres, C[i] * z_abs1( &Xptr[i]) );
} else if ( !notran && rowequ ) {
for (i = 0; i < A->nrow; ++i)
lstres = MAX( lstres, R[i] * z_abs1( &Xptr[i]) );
} else {
for (i = 0; i < A->nrow; ++i)
lstres = MAX( lstres, z_abs1( &Xptr[i]) );
}
if ( lstres != 0. )
ferr[j] /= lstres;
} /* for each RHS j ... */
SUPERLU_FREE(work);
SUPERLU_FREE(rwork);
SUPERLU_FREE(iwork);
SUPERLU_FREE(Bjcol.Store);
return;
} /* zgsrfs */
int
sp_zgemm(char *transa, char *transb, int m, int n, int k,
doublecomplex alpha, SuperMatrix *A, doublecomplex *b, int ldb,
doublecomplex beta, doublecomplex *c, int ldc)
{
/* Purpose
=======
sp_z performs one of the matrix-matrix operations
C := alpha*op( A )*op( B ) + beta*C,
where op( X ) is one of
op( X ) = X or op( X ) = X' or op( X ) = conjg( X' ),
alpha and beta are scalars, and A, B and C are matrices, with op( A )
an m by k matrix, op( B ) a k by n matrix and C an m by n matrix.
Parameters
==========
TRANSA - (input) char*
On entry, TRANSA specifies the form of op( A ) to be used in
the matrix multiplication as follows:
TRANSA = 'N' or 'n', op( A ) = A.
TRANSA = 'T' or 't', op( A ) = A'.
TRANSA = 'C' or 'c', op( A ) = conjg( A' ).
Unchanged on exit.
TRANSB - (input) char*
On entry, TRANSB specifies the form of op( B ) to be used in
the matrix multiplication as follows:
TRANSB = 'N' or 'n', op( B ) = B.
TRANSB = 'T' or 't', op( B ) = B'.
TRANSB = 'C' or 'c', op( B ) = conjg( B' ).
Unchanged on exit.
M - (input) int
On entry, M specifies the number of rows of the matrix
op( A ) and of the matrix C. M must be at least zero.
Unchanged on exit.
N - (input) int
On entry, N specifies the number of columns of the matrix
op( B ) and the number of columns of the matrix C. N must be
at least zero.
Unchanged on exit.
K - (input) int
On entry, K specifies the number of columns of the matrix
op( A ) and the number of rows of the matrix op( B ). K must
be at least zero.
Unchanged on exit.
ALPHA - (input) doublecomplex
On entry, ALPHA specifies the scalar alpha.
A - (input) SuperMatrix*
Matrix A with a sparse format, of dimension (A->nrow, A->ncol).
Currently, the type of A can be:
Stype = NC or NCP; Dtype = _Z; Mtype = GE.
In the future, more general A can be handled.
B - DOUBLE COMPLEX PRECISION array of DIMENSION ( LDB, kb ), where kb is
n when TRANSB = 'N' or 'n', and is k otherwise.
Before entry with TRANSB = 'N' or 'n', the leading k by n
part of the array B must contain the matrix B, otherwise
the leading n by k part of the array B must contain the
matrix B.
Unchanged on exit.
LDB - (input) int
On entry, LDB specifies the first dimension of B as declared
in the calling (sub) program. LDB must be at least max( 1, n ).
Unchanged on exit.
BETA - (input) doublecomplex
On entry, BETA specifies the scalar beta. When BETA is
supplied as zero then C need not be set on input.
C - DOUBLE COMPLEX PRECISION array of DIMENSION ( LDC, n ).
Before entry, the leading m by n part of the array C must
contain the matrix C, except when beta is zero, in which
case C need not be set on entry.
On exit, the array C is overwritten by the m by n matrix
( alpha*op( A )*B + beta*C ).
LDC - (input) int
On entry, LDC specifies the first dimension of C as declared
in the calling (sub)program. LDC must be at least max(1,m).
Unchanged on exit.
==== Sparse Level 3 Blas routine.
*/
int incx = 1, incy = 1;
int j;
for (j = 0; j < n; ++j) {
sp_zgemv(transa, alpha, A, &b[ldb*j], incx, beta, &c[ldc*j], incy);
}
return 0;
}
