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413 lines
14 KiB
413 lines
14 KiB
*> \brief <b> CGTSVX 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 CGTSVX + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgtsvx.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/cgtsvx.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/cgtsvx.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 CGTSVX( 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, RWORK, 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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* REAL RCOND
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* ..
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* .. Array Arguments ..
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* INTEGER IPIV( * )
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* REAL BERR( * ), FERR( * ), RWORK( * )
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* COMPLEX B( LDB, * ), D( * ), DF( * ), DL( * ),
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* $ DLF( * ), DU( * ), DU2( * ), DUF( * ),
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* $ 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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*> CGTSVX uses the LU factorization to compute the solution to a complex
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*> system of linear equations A * X = B, A**T * X = B, or A**H * 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 form
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*> of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV will not
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*> 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)
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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 COMPLEX 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 COMPLEX 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 COMPLEX 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 COMPLEX 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 CGTTRF.
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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 COMPLEX 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 COMPLEX 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 COMPLEX 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 CGTTRF.
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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 COMPLEX 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 COMPLEX 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 REAL
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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 REAL 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 REAL 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 COMPLEX array, dimension (2*N)
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*> \endverbatim
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*>
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*> \param[out] RWORK
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*> \verbatim
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*> RWORK is REAL 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 complexGTsolve
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*
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* =====================================================================
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SUBROUTINE CGTSVX( 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, RWORK, INFO )
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*
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* -- LAPACK driver routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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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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REAL RCOND
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* ..
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* .. Array Arguments ..
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INTEGER IPIV( * )
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REAL BERR( * ), FERR( * ), RWORK( * )
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COMPLEX B( LDB, * ), D( * ), DF( * ), DL( * ),
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$ DLF( * ), DU( * ), DU2( * ), DUF( * ),
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$ WORK( * ), X( LDX, * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ZERO
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PARAMETER ( ZERO = 0.0E+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL NOFACT, NOTRAN
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CHARACTER NORM
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REAL ANORM
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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REAL CLANGT, SLAMCH
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EXTERNAL LSAME, CLANGT, SLAMCH
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* ..
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* .. External Subroutines ..
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EXTERNAL CCOPY, CGTCON, CGTRFS, CGTTRF, CGTTRS, CLACPY,
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$ XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX
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* ..
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* .. Executable Statements ..
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*
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INFO = 0
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NOFACT = LSAME( FACT, 'N' )
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NOTRAN = LSAME( TRANS, 'N' )
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IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
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INFO = -1
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ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
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$ LSAME( TRANS, 'C' ) ) THEN
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INFO = -2
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ELSE IF( N.LT.0 ) THEN
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INFO = -3
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ELSE IF( NRHS.LT.0 ) THEN
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INFO = -4
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ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -14
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ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
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INFO = -16
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'CGTSVX', -INFO )
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RETURN
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END IF
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*
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IF( NOFACT ) THEN
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*
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* Compute the LU factorization of A.
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*
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CALL CCOPY( N, D, 1, DF, 1 )
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IF( N.GT.1 ) THEN
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CALL CCOPY( N-1, DL, 1, DLF, 1 )
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CALL CCOPY( N-1, DU, 1, DUF, 1 )
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END IF
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CALL CGTTRF( N, DLF, DF, DUF, DU2, IPIV, INFO )
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*
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* Return if INFO is non-zero.
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*
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IF( INFO.GT.0 )THEN
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RCOND = ZERO
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RETURN
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END IF
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END IF
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*
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* Compute the norm of the matrix A.
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*
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IF( NOTRAN ) THEN
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NORM = '1'
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ELSE
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NORM = 'I'
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END IF
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ANORM = CLANGT( NORM, N, DL, D, DU )
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*
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* Compute the reciprocal of the condition number of A.
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*
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CALL CGTCON( NORM, N, DLF, DF, DUF, DU2, IPIV, ANORM, RCOND, WORK,
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$ INFO )
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*
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* Compute the solution vectors X.
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*
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CALL CLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
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CALL CGTTRS( TRANS, N, NRHS, DLF, DF, DUF, DU2, IPIV, X, LDX,
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$ INFO )
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*
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* Use iterative refinement to improve the computed solutions and
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* compute error bounds and backward error estimates for them.
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*
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CALL CGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV,
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$ B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
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*
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* Set INFO = N+1 if the matrix is singular to working precision.
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*
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IF( RCOND.LT.SLAMCH( 'Epsilon' ) )
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$ INFO = N + 1
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*
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RETURN
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*
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* End of CGTSVX
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*
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END
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