/*! \file Copyright (c) 2003, The Regents of the University of California, through Lawrence Berkeley National Laboratory (subject to receipt of any required approvals from U.S. Dept. of Energy) All rights reserved. The source code is distributed under BSD license, see the file License.txt at the top-level directory. */ #include #include "slu_mt_ddefs.h" int_t pdgst07(trans_t *trans, int_t n, int_t nrhs, SuperMatrix *A, double *b, int_t ldb, double *x, int_t ldx, double *xact, int_t ldxact, double *ferr, double *berr, double *reslts) { /* * -- SuperLU MT routine (version 2.0) -- * Lawrence Berkeley National Lab, Univ. of California Berkeley, * and Xerox Palo Alto Research Center. * September 10, 2007 * * Purpose * ======= * * pdgst07() tests the error bounds from iterative refinement for the * computed solution to a system of equations op(A)*X = B, where A is a * general n by n matrix and op(A) = A or A**T, depending on TRANS. * * RESLTS(1) = test of the error bound * = norm(X - XACT) / ( norm(X) * FERR ) * A large value is returned if this ratio is not less than one. * * RESLTS(2) = residual from the iterative refinement routine * = the maximum of BERR / ( (n+1)*EPS + (*) ), where * (*) = (n+1)*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i ) * * Arguments * ========= * * TRANS (input) trans_t * Specifies the form of the system of equations. * = NOTRANS: A * X = B (No transpose) * = TRANS: A**T * X = B (Transpose) * = CONJ: A**H * X = B (Conjugate transpose = Transpose) * * N (input) INT_T * The number of rows of the matrices X and XACT. N >= 0. * * NRHS (input) INT_T * The number of columns of the matrices X and XACT. NRHS >= 0. * * A (input) SuperMatrix *, dimension (A->nrow, A->ncol) * The original n by n matrix A. * * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) * The right hand side vectors for the system of linear * equations. * * LDB (input) INT_T * The leading dimension of the array B. LDB >= max(1,N). * * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS) * The computed solution vectors. Each vector is stored as a * column of the matrix X. * * LDX (input) INT_T * The leading dimension of the array X. LDX >= max(1,N). * * XACT (input) DOUBLE PRECISION array, dimension (LDX,NRHS) * The exact solution vectors. Each vector is stored as a * column of the matrix XACT. * * LDXACT (input) INT_T * The leading dimension of the array XACT. LDXACT >= max(1,N). * * * FERR (input) DOUBLE PRECISION array, dimension (NRHS) * The estimated forward error bounds for each solution vector * X. If XTRUE is the true solution, FERR bounds the magnitude * of the largest entry in (X - XTRUE) divided by the magnitude * of the largest entry in X. * * BERR (input) DOUBLE PRECISION array, dimension (NRHS) * The componentwise relative backward error of each solution * vector (i.e., the smallest relative change in any entry of A * * or B that makes X an exact solution). * * RESLTS (output) DOUBLE PRECISION array, dimension (2) * The maximum over the NRHS solution vectors of the ratios: * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) * RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) * * ===================================================================== */ /* Table of constant values */ int c__1 = 1; /* System generated locals */ double d__1, d__2; /* Local variables */ double diff, axbi; int_t imax, irow; int_t i, j, k; int n__1; double unfl, ovfl; double xnorm; double errbnd; int_t notran; double eps, tmp; double *rwork; double *Aval; NCformat *Astore; /* Function prototypes */ extern int lsame_(char *, char *); extern int idamax_(int *, double *, int *); extern double dlamch_(char *); /* Quick exit if N = 0 or NRHS = 0. */ if ( n <= 0 || nrhs <= 0 ) { reslts[0] = 0.; reslts[1] = 0.; return 0; } eps = dlamch_("Epsilon"); unfl = dlamch_("Safe minimum"); ovfl = 1. / unfl; notran = (trans == NOTRANS); rwork = (double *) SUPERLU_MALLOC(n*sizeof(double)); if ( !rwork ) SUPERLU_ABORT("SUPERLU_MALLOC fails for rwork"); Astore = A->Store; Aval = (double *) Astore->nzval; /* Test 1: Compute the maximum of norm(X - XACT) / ( norm(X) * FERR ) over all the vectors X and XACT using the infinity-norm. */ errbnd = 0.; for (j = 0; j < nrhs; ++j) { n__1 = n; imax = idamax_(&n__1, &x[j*ldx], &c__1); d__1 = fabs(x[imax-1 + j*ldx]); xnorm = SUPERLU_MAX(d__1,unfl); diff = 0.; for (i = 0; i < n; ++i) { d__1 = fabs(x[i+j*ldx] - xact[i+j*ldxact]); diff = SUPERLU_MAX(diff, d__1); } if (xnorm > 1.) { goto L20; } else if (diff <= ovfl * xnorm) { goto L20; } else { errbnd = 1. / eps; goto L30; } L20: #if 0 if (diff / xnorm <= ferr[j]) { d__1 = diff / xnorm / ferr[j]; errbnd = SUPERLU_MAX(errbnd,d__1); } else { errbnd = 1. / eps; } #endif d__1 = diff / xnorm / ferr[j]; errbnd = SUPERLU_MAX(errbnd,d__1); /*printf("Ferr: %f\n", errbnd);*/ L30: ; } reslts[0] = errbnd; /* Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where (*) = (n+1)*UNFL / (min_i (abs(op(A))*abs(X) + abs(b))_i ) */ for (k = 0; k < nrhs; ++k) { for (i = 0; i < n; ++i) rwork[i] = fabs( b[i + k*ldb] ); if ( notran ) { for (j = 0; j < n; ++j) { tmp = fabs( x[j + k*ldx] ); for (i = Astore->colptr[j]; i < Astore->colptr[j+1]; ++i) { rwork[Astore->rowind[i]] += fabs(Aval[i]) * tmp; } } } else { for (j = 0; j < n; ++j) { tmp = 0.; for (i = Astore->colptr[j]; i < Astore->colptr[j+1]; ++i) { irow = Astore->rowind[i]; d__1 = fabs( x[irow + k*ldx] ); tmp += fabs(Aval[i]) * d__1; } rwork[j] += tmp; } } axbi = rwork[0]; for (i = 1; i < n; ++i) axbi = SUPERLU_MIN(axbi, rwork[i]); /* Computing MAX */ d__1 = axbi, d__2 = (n + 1) * unfl; tmp = berr[k] / ((n + 1) * eps + (n + 1) * unfl / SUPERLU_MAX(d__1,d__2)); if (k == 0) { reslts[1] = tmp; } else { reslts[1] = SUPERLU_MAX(reslts[1], tmp); } } SUPERLU_FREE(rwork); return 0; } /* pdgst07 */