version 1.3, 2010/08/06 15:28:46
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version 1.16, 2017/06/17 10:54:01
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*> \brief \b DPTRFS |
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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 DPTRFS + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dptrfs.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/dptrfs.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/dptrfs.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 DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, |
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* BERR, WORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* INTEGER INFO, LDB, LDX, N, NRHS |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ), |
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* $ E( * ), EF( * ), FERR( * ), WORK( * ), |
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* $ 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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*> DPTRFS improves the computed solution to a system of linear |
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*> equations when the coefficient matrix is symmetric positive definite |
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*> and tridiagonal, and provides error bounds and backward error |
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*> 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] 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] 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 matrix B. NRHS >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] D |
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*> \verbatim |
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*> D is DOUBLE PRECISION array, dimension (N) |
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*> The n diagonal elements of the tridiagonal matrix A. |
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*> \endverbatim |
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*> |
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*> \param[in] E |
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*> \verbatim |
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*> E is DOUBLE PRECISION array, dimension (N-1) |
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*> The (n-1) subdiagonal elements of the tridiagonal matrix A. |
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*> \endverbatim |
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*> |
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*> \param[in] DF |
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*> \verbatim |
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*> DF is DOUBLE PRECISION array, dimension (N) |
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*> The n diagonal elements of the diagonal matrix D from the |
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*> factorization computed by DPTTRF. |
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*> \endverbatim |
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*> |
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*> \param[in] EF |
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*> \verbatim |
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*> EF is DOUBLE PRECISION array, dimension (N-1) |
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*> The (n-1) subdiagonal elements of the unit bidiagonal factor |
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*> L from the factorization computed by DPTTRF. |
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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 DPTTRS. |
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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 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). |
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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 (2*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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*> \date December 2016 |
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* |
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*> \ingroup doublePTcomputational |
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* |
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* ===================================================================== |
SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, |
SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, |
$ BERR, WORK, INFO ) |
$ BERR, WORK, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational routine (version 3.7.0) -- |
* -- 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 |
* December 2016 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER INFO, LDB, LDX, N, NRHS |
INTEGER INFO, LDB, LDX, N, NRHS |
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$ X( LDX, * ) |
$ X( LDX, * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DPTRFS improves the computed solution to a system of linear |
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* equations when the coefficient matrix is symmetric positive definite |
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* and tridiagonal, and provides error bounds and backward error |
|
* estimates for the solution. |
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* |
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* Arguments |
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* ========= |
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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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* 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 matrix B. NRHS >= 0. |
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* |
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* D (input) DOUBLE PRECISION array, dimension (N) |
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* The n diagonal elements of the tridiagonal matrix A. |
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* |
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* E (input) DOUBLE PRECISION array, dimension (N-1) |
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* The (n-1) subdiagonal elements of the tridiagonal matrix A. |
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* |
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* DF (input) DOUBLE PRECISION array, dimension (N) |
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* The n diagonal elements of the diagonal matrix D from the |
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* factorization computed by DPTTRF. |
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* |
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* EF (input) DOUBLE PRECISION array, dimension (N-1) |
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* The (n-1) subdiagonal elements of the unit bidiagonal factor |
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* L from the factorization computed by DPTTRF. |
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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 DPTTRS. |
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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 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). |
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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 (2*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 .. |
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* m(i,j) = abs(A(i,j)), i = j, |
* m(i,j) = abs(A(i,j)), i = j, |
* m(i,j) = -abs(A(i,j)), i .ne. j, |
* m(i,j) = -abs(A(i,j)), i .ne. j, |
* |
* |
* and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)'. |
* and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**T. |
* |
* |
* Solve M(L) * x = e. |
* Solve M(L) * x = e. |
* |
* |
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WORK( I ) = ONE + WORK( I-1 )*ABS( EF( I-1 ) ) |
WORK( I ) = ONE + WORK( I-1 )*ABS( EF( I-1 ) ) |
60 CONTINUE |
60 CONTINUE |
* |
* |
* Solve D * M(L)' * x = b. |
* Solve D * M(L)**T * x = b. |
* |
* |
WORK( N ) = WORK( N ) / DF( N ) |
WORK( N ) = WORK( N ) / DF( N ) |
DO 70 I = N - 1, 1, -1 |
DO 70 I = N - 1, 1, -1 |