--- rpl/lapack/lapack/dtrrfs.f 2010/04/21 13:45:27 1.2 +++ rpl/lapack/lapack/dtrrfs.f 2023/08/07 08:39:13 1.18 @@ -1,12 +1,188 @@ +*> \brief \b DTRRFS +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download DTRRFS + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE DTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X, +* LDX, FERR, BERR, WORK, IWORK, INFO ) +* +* .. Scalar Arguments .. +* CHARACTER DIAG, TRANS, UPLO +* INTEGER INFO, LDA, LDB, LDX, N, NRHS +* .. +* .. Array Arguments .. +* INTEGER IWORK( * ) +* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ), +* $ WORK( * ), X( LDX, * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> DTRRFS provides error bounds and backward error estimates for the +*> solution to a system of linear equations with a triangular +*> coefficient matrix. +*> +*> The solution matrix X must be computed by DTRTRS or some other +*> means before entering this routine. DTRRFS does not do iterative +*> refinement because doing so cannot improve the backward error. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] UPLO +*> \verbatim +*> UPLO is CHARACTER*1 +*> = 'U': A is upper triangular; +*> = 'L': A is lower triangular. +*> \endverbatim +*> +*> \param[in] TRANS +*> \verbatim +*> TRANS is CHARACTER*1 +*> 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) +*> \endverbatim +*> +*> \param[in] DIAG +*> \verbatim +*> DIAG is CHARACTER*1 +*> = 'N': A is non-unit triangular; +*> = 'U': A is unit triangular. +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The order of the matrix A. N >= 0. +*> \endverbatim +*> +*> \param[in] NRHS +*> \verbatim +*> NRHS is INTEGER +*> The number of right hand sides, i.e., the number of columns +*> of the matrices B and X. NRHS >= 0. +*> \endverbatim +*> +*> \param[in] A +*> \verbatim +*> A is DOUBLE PRECISION array, dimension (LDA,N) +*> The triangular matrix A. If UPLO = 'U', the leading N-by-N +*> upper triangular part of the array A contains the upper +*> triangular matrix, and the strictly lower triangular part of +*> A is not referenced. If UPLO = 'L', the leading N-by-N lower +*> triangular part of the array A contains the lower triangular +*> matrix, and the strictly upper triangular part of A is not +*> referenced. If DIAG = 'U', the diagonal elements of A are +*> also not referenced and are assumed to be 1. +*> \endverbatim +*> +*> \param[in] LDA +*> \verbatim +*> LDA is INTEGER +*> The leading dimension of the array A. LDA >= max(1,N). +*> \endverbatim +*> +*> \param[in] B +*> \verbatim +*> B is DOUBLE PRECISION array, dimension (LDB,NRHS) +*> The right hand side matrix B. +*> \endverbatim +*> +*> \param[in] LDB +*> \verbatim +*> LDB is INTEGER +*> The leading dimension of the array B. LDB >= max(1,N). +*> \endverbatim +*> +*> \param[in] X +*> \verbatim +*> X is DOUBLE PRECISION array, dimension (LDX,NRHS) +*> The solution matrix X. +*> \endverbatim +*> +*> \param[in] LDX +*> \verbatim +*> LDX is INTEGER +*> The leading dimension of the array X. LDX >= max(1,N). +*> \endverbatim +*> +*> \param[out] FERR +*> \verbatim +*> FERR is DOUBLE PRECISION array, dimension (NRHS) +*> 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. +*> \endverbatim +*> +*> \param[out] BERR +*> \verbatim +*> BERR is DOUBLE PRECISION array, dimension (NRHS) +*> 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). +*> \endverbatim +*> +*> \param[out] WORK +*> \verbatim +*> WORK is DOUBLE PRECISION array, dimension (3*N) +*> \endverbatim +*> +*> \param[out] IWORK +*> \verbatim +*> IWORK is INTEGER array, dimension (N) +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> = 0: successful exit +*> < 0: if INFO = -i, the i-th argument had an illegal value +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup doubleOTHERcomputational +* +* ===================================================================== SUBROUTINE DTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X, $ LDX, FERR, BERR, WORK, IWORK, INFO ) * -* -- LAPACK routine (version 3.2) -- +* -- LAPACK computational routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- -* November 2006 -* -* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. * * .. Scalar Arguments .. CHARACTER DIAG, TRANS, UPLO @@ -18,89 +194,6 @@ $ WORK( * ), X( LDX, * ) * .. * -* Purpose -* ======= -* -* DTRRFS provides error bounds and backward error estimates for the -* solution to a system of linear equations with a triangular -* coefficient matrix. -* -* The solution matrix X must be computed by DTRTRS or some other -* means before entering this routine. DTRRFS does not do iterative -* refinement because doing so cannot improve the backward error. -* -* Arguments -* ========= -* -* UPLO (input) CHARACTER*1 -* = 'U': A is upper triangular; -* = 'L': A is lower triangular. -* -* TRANS (input) CHARACTER*1 -* 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) -* -* DIAG (input) CHARACTER*1 -* = 'N': A is non-unit triangular; -* = 'U': A is unit triangular. -* -* N (input) INTEGER -* The order of the matrix A. N >= 0. -* -* NRHS (input) INTEGER -* The number of right hand sides, i.e., the number of columns -* of the matrices B and X. NRHS >= 0. -* -* A (input) DOUBLE PRECISION array, dimension (LDA,N) -* The triangular matrix A. If UPLO = 'U', the leading N-by-N -* upper triangular part of the array A contains the upper -* triangular matrix, and the strictly lower triangular part of -* A is not referenced. If UPLO = 'L', the leading N-by-N lower -* triangular part of the array A contains the lower triangular -* matrix, and the strictly upper triangular part of A is not -* referenced. If DIAG = 'U', the diagonal elements of A are -* also not referenced and are assumed to be 1. -* -* LDA (input) INTEGER -* The leading dimension of the array A. LDA >= max(1,N). -* -* B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) -* The right hand side matrix B. -* -* LDB (input) INTEGER -* The leading dimension of the array B. LDB >= max(1,N). -* -* X (input) DOUBLE PRECISION array, dimension (LDX,NRHS) -* The solution matrix X. -* -* LDX (input) INTEGER -* The leading dimension of the array X. LDX >= max(1,N). -* -* FERR (output) DOUBLE PRECISION array, dimension (NRHS) -* 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 PRECISION array, dimension (NRHS) -* 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). -* -* WORK (workspace) DOUBLE PRECISION array, dimension (3*N) -* -* IWORK (workspace) INTEGER array, dimension (N) -* -* INFO (output) INTEGER -* = 0: successful exit -* < 0: if INFO = -i, the i-th argument had an illegal value -* * ===================================================================== * * .. Parameters .. @@ -190,7 +283,7 @@ DO 250 J = 1, NRHS * * Compute residual R = B - op(A) * X, -* where op(A) = A or A', depending on TRANS. +* where op(A) = A or A**T, depending on TRANS. * CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 ) CALL DTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK( N+1 ), 1 ) @@ -250,7 +343,7 @@ END IF ELSE * -* Compute abs(A')*abs(X) + abs(B). +* Compute abs(A**T)*abs(X) + abs(B). * IF( UPPER ) THEN IF( NOUNIT ) THEN @@ -338,7 +431,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(op(A)'). +* Multiply by diag(W)*inv(op(A)**T). * CALL DTRSV( UPLO, TRANST, DIAG, N, A, LDA, WORK( N+1 ), $ 1 )