version 1.3, 2010/08/06 15:28:49
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version 1.18, 2023/08/07 08:39:12
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*> \brief \b DTBRFS |
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* |
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* =========== DOCUMENTATION =========== |
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* |
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* Online html documentation available at |
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* http://www.netlib.org/lapack/explore-html/ |
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* |
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*> \htmlonly |
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*> Download DTBRFS + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtbrfs.f"> |
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*> [TGZ]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtbrfs.f"> |
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*> [ZIP]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtbrfs.f"> |
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*> [TXT]</a> |
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*> \endhtmlonly |
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* |
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* Definition: |
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* =========== |
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* |
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* SUBROUTINE DTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, |
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* LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER DIAG, TRANS, UPLO |
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* INTEGER INFO, KD, LDAB, LDB, LDX, N, NRHS |
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* .. |
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* .. Array Arguments .. |
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* INTEGER IWORK( * ) |
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* DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ), |
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* $ FERR( * ), WORK( * ), X( LDX, * ) |
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* .. |
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* |
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* |
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*> \par Purpose: |
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* ============= |
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*> |
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*> \verbatim |
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*> |
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*> DTBRFS provides error bounds and backward error estimates for the |
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*> solution to a system of linear equations with a triangular band |
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*> coefficient matrix. |
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*> |
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*> The solution matrix X must be computed by DTBTRS or some other |
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*> means before entering this routine. DTBRFS does not do iterative |
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*> refinement because doing so cannot improve the backward error. |
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*> \endverbatim |
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* |
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* Arguments: |
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* ========== |
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* |
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*> \param[in] UPLO |
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*> \verbatim |
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*> UPLO is CHARACTER*1 |
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*> = 'U': A is upper triangular; |
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*> = 'L': A is lower triangular. |
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*> \endverbatim |
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*> |
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*> \param[in] TRANS |
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*> \verbatim |
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*> TRANS is CHARACTER*1 |
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*> Specifies the form of the system of equations: |
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*> = 'N': A * X = B (No transpose) |
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*> = 'T': A**T * X = B (Transpose) |
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*> = 'C': A**H * X = B (Conjugate transpose = Transpose) |
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*> \endverbatim |
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*> |
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*> \param[in] DIAG |
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*> \verbatim |
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*> DIAG is CHARACTER*1 |
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*> = 'N': A is non-unit triangular; |
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*> = 'U': A is unit triangular. |
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*> \endverbatim |
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*> |
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*> \param[in] N |
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*> \verbatim |
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*> N is INTEGER |
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*> The order of the matrix A. N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] KD |
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*> \verbatim |
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*> KD is INTEGER |
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*> The number of superdiagonals or subdiagonals of the |
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*> triangular band matrix A. KD >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] NRHS |
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*> \verbatim |
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*> NRHS is INTEGER |
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*> The number of right hand sides, i.e., the number of columns |
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*> of the matrices B and X. NRHS >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] AB |
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*> \verbatim |
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*> AB is DOUBLE PRECISION array, dimension (LDAB,N) |
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*> The upper or lower triangular band matrix A, stored in the |
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*> first kd+1 rows of the array. The j-th column of A is stored |
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*> in the j-th column of the array AB as follows: |
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*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; |
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*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). |
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*> If DIAG = 'U', the diagonal elements of A are not referenced |
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*> and are assumed to be 1. |
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*> \endverbatim |
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*> |
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*> \param[in] LDAB |
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*> \verbatim |
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*> LDAB is INTEGER |
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*> The leading dimension of the array AB. LDAB >= KD+1. |
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*> \endverbatim |
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*> |
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*> \param[in] B |
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*> \verbatim |
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*> B is DOUBLE PRECISION array, dimension (LDB,NRHS) |
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*> The right hand side matrix B. |
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*> \endverbatim |
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*> |
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*> \param[in] LDB |
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*> \verbatim |
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*> LDB is INTEGER |
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*> The leading dimension of the array B. LDB >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[in] X |
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*> \verbatim |
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*> X is DOUBLE PRECISION array, dimension (LDX,NRHS) |
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*> The solution matrix X. |
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*> \endverbatim |
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*> |
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*> \param[in] LDX |
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*> \verbatim |
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*> LDX is INTEGER |
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*> The leading dimension of the array X. LDX >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[out] FERR |
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*> \verbatim |
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*> FERR is DOUBLE PRECISION array, dimension (NRHS) |
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*> The estimated forward error bound for each solution vector |
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*> X(j) (the j-th column of the solution matrix X). |
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*> If XTRUE is the true solution corresponding to X(j), FERR(j) |
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*> is an estimated upper bound for the magnitude of the largest |
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*> element in (X(j) - XTRUE) divided by the magnitude of the |
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*> largest element in X(j). The estimate is as reliable as |
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*> the estimate for RCOND, and is almost always a slight |
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*> overestimate of the true error. |
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*> \endverbatim |
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*> |
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*> \param[out] BERR |
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*> \verbatim |
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*> BERR is DOUBLE PRECISION array, dimension (NRHS) |
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*> The componentwise relative backward error of each solution |
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*> vector X(j) (i.e., the smallest relative change in |
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*> any element of A or B that makes X(j) an exact solution). |
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*> \endverbatim |
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*> |
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*> \param[out] WORK |
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*> \verbatim |
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*> WORK is DOUBLE PRECISION array, dimension (3*N) |
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*> \endverbatim |
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*> |
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*> \param[out] IWORK |
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*> \verbatim |
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*> IWORK is INTEGER array, dimension (N) |
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*> \endverbatim |
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*> |
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*> \param[out] INFO |
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*> \verbatim |
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*> INFO is INTEGER |
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*> = 0: successful exit |
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*> < 0: if INFO = -i, the i-th argument had an illegal value |
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*> \endverbatim |
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* |
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* Authors: |
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* ======== |
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* |
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*> \author Univ. of Tennessee |
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*> \author Univ. of California Berkeley |
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*> \author Univ. of Colorado Denver |
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*> \author NAG Ltd. |
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* |
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*> \ingroup doubleOTHERcomputational |
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* |
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* ===================================================================== |
SUBROUTINE DTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, |
SUBROUTINE DTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, |
$ LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO ) |
$ 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, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* November 2006 |
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* |
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* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. |
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* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER DIAG, TRANS, UPLO |
CHARACTER DIAG, TRANS, UPLO |
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$ FERR( * ), WORK( * ), X( LDX, * ) |
$ FERR( * ), WORK( * ), X( LDX, * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DTBRFS provides error bounds and backward error estimates for the |
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* solution to a system of linear equations with a triangular band |
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* coefficient matrix. |
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* |
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* The solution matrix X must be computed by DTBTRS or some other |
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* means before entering this routine. DTBRFS does not do iterative |
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* refinement because doing so cannot improve the backward error. |
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* |
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* Arguments |
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* ========= |
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* |
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* UPLO (input) CHARACTER*1 |
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* = 'U': A is upper triangular; |
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* = 'L': A is lower triangular. |
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* |
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* TRANS (input) CHARACTER*1 |
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* Specifies the form of the system of equations: |
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* = 'N': A * X = B (No transpose) |
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* = 'T': A**T * X = B (Transpose) |
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* = 'C': A**H * X = B (Conjugate transpose = Transpose) |
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* |
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* DIAG (input) CHARACTER*1 |
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* = 'N': A is non-unit triangular; |
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* = 'U': A is unit triangular. |
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* |
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* N (input) INTEGER |
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* The order of the matrix A. N >= 0. |
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* |
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* KD (input) INTEGER |
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* The number of superdiagonals or subdiagonals of the |
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* triangular band matrix A. KD >= 0. |
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* |
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* NRHS (input) INTEGER |
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* The number of right hand sides, i.e., the number of columns |
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* of the matrices B and X. NRHS >= 0. |
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* |
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* AB (input) DOUBLE PRECISION array, dimension (LDAB,N) |
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* The upper or lower triangular band matrix A, stored in the |
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* first kd+1 rows of the array. The j-th column of A is stored |
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* in the j-th column of the array AB as follows: |
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* if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; |
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* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). |
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* If DIAG = 'U', the diagonal elements of A are not referenced |
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* and are assumed to be 1. |
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* |
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* LDAB (input) INTEGER |
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* The leading dimension of the array AB. LDAB >= KD+1. |
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* |
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* B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) |
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* The right hand side matrix B. |
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* |
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* LDB (input) INTEGER |
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* The leading dimension of the array B. LDB >= max(1,N). |
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* |
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* X (input) DOUBLE PRECISION array, dimension (LDX,NRHS) |
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* The solution matrix X. |
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* |
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* LDX (input) INTEGER |
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* The leading dimension of the array X. LDX >= max(1,N). |
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* |
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* FERR (output) DOUBLE PRECISION array, dimension (NRHS) |
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* The estimated forward error bound for each solution vector |
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* X(j) (the j-th column of the solution matrix X). |
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* If XTRUE is the true solution corresponding to X(j), FERR(j) |
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* is an estimated upper bound for the magnitude of the largest |
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* element in (X(j) - XTRUE) divided by the magnitude of the |
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* largest element in X(j). The estimate is as reliable as |
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* the estimate for RCOND, and is almost always a slight |
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* overestimate of the true error. |
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* |
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* BERR (output) DOUBLE PRECISION array, dimension (NRHS) |
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* The componentwise relative backward error of each solution |
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* vector X(j) (i.e., the smallest relative change in |
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* any element of A or B that makes X(j) an exact solution). |
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* |
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* WORK (workspace) DOUBLE PRECISION array, dimension (3*N) |
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* |
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* IWORK (workspace) INTEGER array, dimension (N) |
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* |
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* INFO (output) INTEGER |
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* = 0: successful exit |
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* < 0: if INFO = -i, the i-th argument had an illegal value |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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DO 250 J = 1, NRHS |
DO 250 J = 1, NRHS |
* |
* |
* Compute residual R = B - op(A) * X, |
* 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 DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 ) |
CALL DTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK( N+1 ), |
CALL DTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK( N+1 ), |
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END IF |
END IF |
ELSE |
ELSE |
* |
* |
* Compute abs(A')*abs(X) + abs(B). |
* Compute abs(A**T)*abs(X) + abs(B). |
* |
* |
IF( UPPER ) THEN |
IF( UPPER ) THEN |
IF( NOUNIT ) THEN |
IF( NOUNIT ) THEN |
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IF( KASE.NE.0 ) THEN |
IF( KASE.NE.0 ) THEN |
IF( KASE.EQ.1 ) THEN |
IF( KASE.EQ.1 ) THEN |
* |
* |
* Multiply by diag(W)*inv(op(A)'). |
* Multiply by diag(W)*inv(op(A)**T). |
* |
* |
CALL DTBSV( UPLO, TRANST, DIAG, N, KD, AB, LDAB, |
CALL DTBSV( UPLO, TRANST, DIAG, N, KD, AB, LDAB, |
$ WORK( N+1 ), 1 ) |
$ WORK( N+1 ), 1 ) |