version 1.3, 2010/08/06 15:28:38
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version 1.19, 2023/08/07 08:38:52
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*> \brief <b> DGTSVX computes the solution to system of linear equations A * X = B for GT matrices </b> |
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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 DGTSVX + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgtsvx.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/dgtsvx.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/dgtsvx.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 DGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, |
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* DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, |
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* WORK, IWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER FACT, TRANS |
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* INTEGER INFO, LDB, LDX, N, NRHS |
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* DOUBLE PRECISION RCOND |
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* .. |
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* .. Array Arguments .. |
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* INTEGER IPIV( * ), IWORK( * ) |
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* DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ), |
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* $ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ), |
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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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*> DGTSVX uses the LU factorization to compute the solution to a real |
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*> system of linear equations A * X = B or A**T * X = B, |
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*> where A is a tridiagonal matrix of order N and X and B are N-by-NRHS |
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*> matrices. |
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*> |
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*> Error bounds on the solution and a condition estimate are also |
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*> provided. |
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*> \endverbatim |
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* |
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*> \par Description: |
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* ================= |
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*> |
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*> \verbatim |
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*> |
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*> The following steps are performed: |
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*> |
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*> 1. If FACT = 'N', the LU decomposition is used to factor the matrix A |
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*> as A = L * U, where L is a product of permutation and unit lower |
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*> bidiagonal matrices and U is upper triangular with nonzeros in |
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*> only the main diagonal and first two superdiagonals. |
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*> |
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*> 2. If some U(i,i)=0, so that U is exactly singular, then the routine |
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*> returns with INFO = i. Otherwise, the factored form of A is used |
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*> to estimate the condition number of the matrix A. If the |
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*> reciprocal of the condition number is less than machine precision, |
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*> INFO = N+1 is returned as a warning, but the routine still goes on |
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*> to solve for X and compute error bounds as described below. |
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*> |
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*> 3. The system of equations is solved for X using the factored form |
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*> of A. |
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*> |
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*> 4. Iterative refinement is applied to improve the computed solution |
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*> matrix and calculate error bounds and backward error estimates |
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*> for it. |
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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] FACT |
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*> \verbatim |
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*> FACT is CHARACTER*1 |
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*> Specifies whether or not the factored form of A has been |
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*> supplied on entry. |
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*> = 'F': DLF, DF, DUF, DU2, and IPIV contain the factored |
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*> form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV |
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*> will not be modified. |
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*> = 'N': The matrix will be copied to DLF, DF, and DUF |
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*> and factored. |
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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] 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] DL |
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*> \verbatim |
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*> DL is DOUBLE PRECISION array, dimension (N-1) |
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*> The (n-1) subdiagonal elements of A. |
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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 A. |
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*> \endverbatim |
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*> |
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*> \param[in] DU |
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*> \verbatim |
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*> DU is DOUBLE PRECISION array, dimension (N-1) |
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*> The (n-1) superdiagonal elements of A. |
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*> \endverbatim |
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*> |
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*> \param[in,out] DLF |
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*> \verbatim |
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*> DLF is DOUBLE PRECISION array, dimension (N-1) |
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*> If FACT = 'F', then DLF is an input argument and on entry |
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*> contains the (n-1) multipliers that define the matrix L from |
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*> the LU factorization of A as computed by DGTTRF. |
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*> |
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*> If FACT = 'N', then DLF is an output argument and on exit |
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*> contains the (n-1) multipliers that define the matrix L from |
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*> the LU factorization of A. |
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*> \endverbatim |
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*> |
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*> \param[in,out] DF |
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*> \verbatim |
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*> DF is DOUBLE PRECISION array, dimension (N) |
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*> If FACT = 'F', then DF is an input argument and on entry |
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*> contains the n diagonal elements of the upper triangular |
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*> matrix U from the LU factorization of A. |
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*> |
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*> If FACT = 'N', then DF is an output argument and on exit |
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*> contains the n diagonal elements of the upper triangular |
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*> matrix U from the LU factorization of A. |
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*> \endverbatim |
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*> |
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*> \param[in,out] DUF |
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*> \verbatim |
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*> DUF is DOUBLE PRECISION array, dimension (N-1) |
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*> If FACT = 'F', then DUF is an input argument and on entry |
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*> contains the (n-1) elements of the first superdiagonal of U. |
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*> |
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*> If FACT = 'N', then DUF is an output argument and on exit |
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*> contains the (n-1) elements of the first superdiagonal of U. |
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*> \endverbatim |
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*> |
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*> \param[in,out] DU2 |
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*> \verbatim |
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*> DU2 is DOUBLE PRECISION array, dimension (N-2) |
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*> If FACT = 'F', then DU2 is an input argument and on entry |
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*> contains the (n-2) elements of the second superdiagonal of |
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*> U. |
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*> |
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*> If FACT = 'N', then DU2 is an output argument and on exit |
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*> contains the (n-2) elements of the second superdiagonal of |
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*> U. |
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*> \endverbatim |
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*> |
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*> \param[in,out] IPIV |
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*> \verbatim |
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*> IPIV is INTEGER array, dimension (N) |
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*> If FACT = 'F', then IPIV is an input argument and on entry |
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*> contains the pivot indices from the LU factorization of A as |
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*> computed by DGTTRF. |
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*> |
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*> If FACT = 'N', then IPIV is an output argument and on exit |
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*> contains the pivot indices from the LU factorization of A; |
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*> row i of the matrix was interchanged with row IPIV(i). |
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*> IPIV(i) will always be either i or i+1; IPIV(i) = i indicates |
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*> a row interchange was not required. |
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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 N-by-NRHS 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[out] X |
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*> \verbatim |
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*> X is DOUBLE PRECISION array, dimension (LDX,NRHS) |
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*> If INFO = 0 or INFO = N+1, the N-by-NRHS 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] RCOND |
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*> \verbatim |
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*> RCOND is DOUBLE PRECISION |
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*> The estimate of the reciprocal condition number of the matrix |
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*> A. If RCOND is less than the machine precision (in |
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*> particular, if RCOND = 0), the matrix is singular to working |
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*> precision. This condition is indicated by a return code of |
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*> INFO > 0. |
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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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*> > 0: if INFO = i, and i is |
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*> <= N: U(i,i) is exactly zero. The factorization |
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*> has not been completed unless i = N, but the |
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*> factor U is exactly singular, so the solution |
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*> and error bounds could not be computed. |
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*> RCOND = 0 is returned. |
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*> = N+1: U is nonsingular, but RCOND is less than machine |
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*> precision, meaning that the matrix is singular |
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*> to working precision. Nevertheless, the |
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*> solution and error bounds are computed because |
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*> there are a number of situations where the |
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*> computed solution can be more accurate than the |
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*> value of RCOND would suggest. |
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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 doubleGTsolve |
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* |
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* ===================================================================== |
SUBROUTINE DGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, |
SUBROUTINE DGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, |
$ DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, |
$ DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, |
$ WORK, IWORK, INFO ) |
$ WORK, IWORK, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK driver 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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* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER FACT, TRANS |
CHARACTER FACT, TRANS |
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$ FERR( * ), WORK( * ), X( LDX, * ) |
$ FERR( * ), WORK( * ), X( LDX, * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
|
* |
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* DGTSVX uses the LU factorization to compute the solution to a real |
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* system of linear equations A * X = B or A**T * X = B, |
|
* where A is a tridiagonal matrix of order N and X and B are N-by-NRHS |
|
* matrices. |
|
* |
|
* Error bounds on the solution and a condition estimate are also |
|
* provided. |
|
* |
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* Description |
|
* =========== |
|
* |
|
* The following steps are performed: |
|
* |
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* 1. If FACT = 'N', the LU decomposition is used to factor the matrix A |
|
* as A = L * U, where L is a product of permutation and unit lower |
|
* bidiagonal matrices and U is upper triangular with nonzeros in |
|
* only the main diagonal and first two superdiagonals. |
|
* |
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* 2. If some U(i,i)=0, so that U is exactly singular, then the routine |
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* returns with INFO = i. Otherwise, the factored form of A is used |
|
* to estimate the condition number of the matrix A. If the |
|
* reciprocal of the condition number is less than machine precision, |
|
* INFO = N+1 is returned as a warning, but the routine still goes on |
|
* to solve for X and compute error bounds as described below. |
|
* |
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* 3. The system of equations is solved for X using the factored form |
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* of A. |
|
* |
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* 4. Iterative refinement is applied to improve the computed solution |
|
* matrix and calculate error bounds and backward error estimates |
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* for it. |
|
* |
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* Arguments |
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* ========= |
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* |
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* FACT (input) CHARACTER*1 |
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* Specifies whether or not the factored form of A has been |
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* supplied on entry. |
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* = 'F': DLF, DF, DUF, DU2, and IPIV contain the factored |
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* form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV |
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* will not be modified. |
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* = 'N': The matrix will be copied to DLF, DF, and DUF |
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* and factored. |
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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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* 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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* DL (input) DOUBLE PRECISION array, dimension (N-1) |
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* The (n-1) subdiagonal elements of A. |
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* |
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* D (input) DOUBLE PRECISION array, dimension (N) |
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* The n diagonal elements of A. |
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* |
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* DU (input) DOUBLE PRECISION array, dimension (N-1) |
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* The (n-1) superdiagonal elements of A. |
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* |
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* DLF (input or output) DOUBLE PRECISION array, dimension (N-1) |
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* If FACT = 'F', then DLF is an input argument and on entry |
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* contains the (n-1) multipliers that define the matrix L from |
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* the LU factorization of A as computed by DGTTRF. |
|
* |
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* If FACT = 'N', then DLF is an output argument and on exit |
|
* contains the (n-1) multipliers that define the matrix L from |
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* the LU factorization of A. |
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* |
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* DF (input or output) DOUBLE PRECISION array, dimension (N) |
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* If FACT = 'F', then DF is an input argument and on entry |
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* contains the n diagonal elements of the upper triangular |
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* matrix U from the LU factorization of A. |
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* |
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* If FACT = 'N', then DF is an output argument and on exit |
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* contains the n diagonal elements of the upper triangular |
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* matrix U from the LU factorization of A. |
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* |
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* DUF (input or output) DOUBLE PRECISION array, dimension (N-1) |
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* If FACT = 'F', then DUF is an input argument and on entry |
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* contains the (n-1) elements of the first superdiagonal of U. |
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* |
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* If FACT = 'N', then DUF is an output argument and on exit |
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* contains the (n-1) elements of the first superdiagonal of U. |
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* |
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* DU2 (input or output) DOUBLE PRECISION array, dimension (N-2) |
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* If FACT = 'F', then DU2 is an input argument and on entry |
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* contains the (n-2) elements of the second superdiagonal of |
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* U. |
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* |
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* If FACT = 'N', then DU2 is an output argument and on exit |
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* contains the (n-2) elements of the second superdiagonal of |
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* U. |
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* |
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* IPIV (input or output) INTEGER array, dimension (N) |
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* If FACT = 'F', then IPIV is an input argument and on entry |
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* contains the pivot indices from the LU factorization of A as |
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* computed by DGTTRF. |
|
* |
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* If FACT = 'N', then IPIV is an output argument and on exit |
|
* contains the pivot indices from the LU factorization of A; |
|
* row i of the matrix was interchanged with row IPIV(i). |
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* IPIV(i) will always be either i or i+1; IPIV(i) = i indicates |
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* a row interchange was not required. |
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* |
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* B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) |
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* The N-by-NRHS 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 (output) DOUBLE PRECISION array, dimension (LDX,NRHS) |
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* If INFO = 0 or INFO = N+1, the N-by-NRHS 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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* RCOND (output) DOUBLE PRECISION |
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* The estimate of the reciprocal condition number of the matrix |
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* A. If RCOND is less than the machine precision (in |
|
* particular, if RCOND = 0), the matrix is singular to working |
|
* precision. This condition is indicated by a return code of |
|
* INFO > 0. |
|
* |
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* 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). |
|
* |
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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 |
|
* > 0: if INFO = i, and i is |
|
* <= N: U(i,i) is exactly zero. The factorization |
|
* has not been completed unless i = N, but the |
|
* factor U is exactly singular, so the solution |
|
* and error bounds could not be computed. |
|
* RCOND = 0 is returned. |
|
* = N+1: U is nonsingular, but RCOND is less than machine |
|
* precision, meaning that the matrix is singular |
|
* to working precision. Nevertheless, the |
|
* solution and error bounds are computed because |
|
* there are a number of situations where the |
|
* computed solution can be more accurate than the |
|
* value of RCOND would suggest. |
|
* |
|
* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |