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version 1.17, 2023/08/07 08:38:47
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*> \brief \b DGBRFS |
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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 DGBRFS + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbrfs.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/dgbrfs.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/dgbrfs.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 DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, |
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* IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, |
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* INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER TRANS |
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* INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS |
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* .. |
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* .. Array Arguments .. |
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* INTEGER IPIV( * ), IWORK( * ) |
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* DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), |
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* $ BERR( * ), 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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*> DGBRFS improves the computed solution to a system of linear |
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*> equations when the coefficient matrix is banded, and provides |
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*> error bounds and backward error estimates for the solution. |
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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] 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] 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] KL |
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*> \verbatim |
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*> KL is INTEGER |
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*> The number of subdiagonals within the band of A. KL >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] KU |
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*> \verbatim |
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*> KU is INTEGER |
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*> The number of superdiagonals within the band of A. KU >= 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 original band matrix A, stored in rows 1 to KL+KU+1. |
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*> The j-th column of A is stored in the j-th column of the |
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*> array AB as follows: |
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*> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl). |
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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 >= KL+KU+1. |
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*> \endverbatim |
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*> |
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*> \param[in] AFB |
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*> \verbatim |
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*> AFB is DOUBLE PRECISION array, dimension (LDAFB,N) |
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*> Details of the LU factorization of the band matrix A, as |
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*> computed by DGBTRF. U is stored as an upper triangular band |
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*> matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and |
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*> the multipliers used during the factorization are stored in |
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*> rows KL+KU+2 to 2*KL+KU+1. |
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*> \endverbatim |
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*> |
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*> \param[in] LDAFB |
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*> \verbatim |
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*> LDAFB is INTEGER |
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*> The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1. |
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*> \endverbatim |
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*> |
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*> \param[in] IPIV |
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*> \verbatim |
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*> IPIV is INTEGER array, dimension (N) |
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*> The pivot indices from DGBTRF; for 1<=i<=N, row i of the |
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*> matrix was interchanged with row IPIV(i). |
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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,out] X |
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*> \verbatim |
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*> X is DOUBLE PRECISION array, dimension (LDX,NRHS) |
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*> On entry, the solution matrix X, as computed by DGBTRS. |
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*> On exit, the improved 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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*> \par Internal Parameters: |
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* ========================= |
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*> |
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*> \verbatim |
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*> ITMAX is the maximum number of steps of iterative refinement. |
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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 doubleGBcomputational |
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* |
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* ===================================================================== |
SUBROUTINE DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, |
SUBROUTINE DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, |
$ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, |
$ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, |
$ INFO ) |
$ 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 TRANS |
CHARACTER TRANS |
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$ BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) |
$ BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DGBRFS improves the computed solution to a system of linear |
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* equations when the coefficient matrix is banded, and provides |
|
* error bounds and backward error estimates for the solution. |
|
* |
|
* Arguments |
|
* ========= |
|
* |
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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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* N (input) INTEGER |
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* The order of the matrix A. N >= 0. |
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* |
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* KL (input) INTEGER |
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* The number of subdiagonals within the band of A. KL >= 0. |
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* |
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* KU (input) INTEGER |
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* The number of superdiagonals within the band of A. KU >= 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 original band matrix A, stored in rows 1 to KL+KU+1. |
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* The j-th column of A is stored in the j-th column of the |
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* array AB as follows: |
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* AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl). |
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* |
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* LDAB (input) INTEGER |
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* The leading dimension of the array AB. LDAB >= KL+KU+1. |
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* |
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* AFB (input) DOUBLE PRECISION array, dimension (LDAFB,N) |
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* Details of the LU factorization of the band matrix A, as |
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* computed by DGBTRF. U is stored as an upper triangular band |
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* matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and |
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* the multipliers used during the factorization are stored in |
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* rows KL+KU+2 to 2*KL+KU+1. |
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* |
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* LDAFB (input) INTEGER |
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* The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1. |
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* |
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* IPIV (input) INTEGER array, dimension (N) |
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* The pivot indices from DGBTRF; for 1<=i<=N, row i of the |
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* matrix was interchanged with row IPIV(i). |
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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/output) DOUBLE PRECISION array, dimension (LDX,NRHS) |
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* On entry, the solution matrix X, as computed by DGBTRS. |
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* On exit, the improved 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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* Internal Parameters |
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* =================== |
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* |
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* ITMAX is the maximum number of steps of iterative refinement. |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |