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341 lines
11 KiB
341 lines
11 KiB
2 years ago
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*> \brief <b> ZPTSVX computes the solution to system of linear equations A * X = B for PT 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 ZPTSVX + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zptsvx.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/zptsvx.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/zptsvx.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 ZPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
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* RCOND, FERR, BERR, WORK, RWORK, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER FACT
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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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* DOUBLE PRECISION BERR( * ), D( * ), DF( * ), FERR( * ),
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* $ RWORK( * )
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* COMPLEX*16 B( LDB, * ), E( * ), EF( * ), 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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*> ZPTSVX uses the factorization A = L*D*L**H to compute the solution
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*> to a complex system of linear equations A*X = B, where A is an
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*> N-by-N Hermitian positive definite tridiagonal matrix and X and B
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*> are N-by-NRHS 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 matrix A is factored as A = L*D*L**H, where L
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*> is a unit lower bidiagonal matrix and D is diagonal. The
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*> factorization can also be regarded as having the form
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*> A = U**H*D*U.
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*>
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*> 2. If the leading principal minor of order i is not positive,
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*> then the routine returns with INFO = i. Otherwise, the factored
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*> form of A is used to estimate the condition number of the matrix
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*> A. If the reciprocal of the condition number is less than machine
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*> precision, INFO = N+1 is returned as a warning, but the routine
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*> still goes on to solve for X and compute error bounds as
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*> 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 the matrix
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*> A is supplied on entry.
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*> = 'F': On entry, DF and EF contain the factored form of A.
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*> D, E, DF, and EF will not be modified.
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*> = 'N': The matrix A will be copied to DF and EF and
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*> factored.
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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 matrices B and X. 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 COMPLEX*16 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,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 diagonal matrix D
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*> from the L*D*L**H factorization of A.
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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 diagonal matrix D
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*> from the L*D*L**H factorization of A.
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*> \endverbatim
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*>
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*> \param[in,out] EF
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*> \verbatim
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*> EF is COMPLEX*16 array, dimension (N-1)
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*> If FACT = 'F', then EF is an input argument and on entry
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*> contains the (n-1) subdiagonal elements of the unit
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*> bidiagonal factor L from the L*D*L**H factorization of A.
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*> If FACT = 'N', then EF is an output argument and on exit
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*> contains the (n-1) subdiagonal elements of the unit
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*> bidiagonal factor L from the L*D*L**H factorization of A.
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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*16 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*16 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 reciprocal condition number of the matrix A. If RCOND
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*> is less than the machine precision (in particular, if
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*> RCOND = 0), the matrix is singular to working precision.
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*> This condition is indicated by a return code of 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 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 any
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*> 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*16 array, dimension (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 DOUBLE PRECISION 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: the leading principal minor of order i of A
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*> is not positive, so the factorization could not
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*> be completed, and the solution has not been
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*> computed. 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 complex16PTsolve
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*
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* =====================================================================
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SUBROUTINE ZPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
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$ RCOND, FERR, BERR, 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
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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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DOUBLE PRECISION BERR( * ), D( * ), DF( * ), FERR( * ),
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$ RWORK( * )
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COMPLEX*16 B( LDB, * ), E( * ), EF( * ), WORK( * ),
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$ 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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DOUBLE PRECISION ZERO
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PARAMETER ( ZERO = 0.0D+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL NOFACT
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DOUBLE PRECISION ANORM
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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DOUBLE PRECISION DLAMCH, ZLANHT
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EXTERNAL LSAME, DLAMCH, ZLANHT
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* ..
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* .. External Subroutines ..
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EXTERNAL DCOPY, XERBLA, ZCOPY, ZLACPY, ZPTCON, ZPTRFS,
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$ ZPTTRF, ZPTTRS
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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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* Test the input parameters.
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*
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INFO = 0
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NOFACT = LSAME( FACT, '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( N.LT.0 ) THEN
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INFO = -2
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ELSE IF( NRHS.LT.0 ) THEN
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INFO = -3
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ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -9
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ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
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INFO = -11
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'ZPTSVX', -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 L*D*L**H (or U**H*D*U) factorization of A.
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*
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CALL DCOPY( N, D, 1, DF, 1 )
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IF( N.GT.1 )
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$ CALL ZCOPY( N-1, E, 1, EF, 1 )
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CALL ZPTTRF( N, DF, EF, 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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ANORM = ZLANHT( '1', N, D, E )
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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 ZPTCON( N, DF, EF, ANORM, RCOND, RWORK, INFO )
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*
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* Compute the solution vectors X.
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*
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CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
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CALL ZPTTRS( 'Lower', N, NRHS, DF, EF, X, LDX, 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 ZPTRFS( 'Lower', N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
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$ 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.DLAMCH( '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 ZPTSVX
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*
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END
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