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797 lines
30 KiB
797 lines
30 KiB
*> \brief <b> ZGBSVXX computes the solution to system of linear equations A * X = B for GB 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 ZGBSVXX + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgbsvxx.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/zgbsvxx.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/zgbsvxx.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 ZGBSVXX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
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* LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
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* RCOND, RPVGRW, BERR, N_ERR_BNDS,
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* ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
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* WORK, RWORK, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER EQUED, FACT, TRANS
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* INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS,
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* $ N_ERR_BNDS
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* DOUBLE PRECISION RCOND, RPVGRW
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* ..
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* .. Array Arguments ..
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* INTEGER IPIV( * )
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* COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
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* $ X( LDX , * ),WORK( * )
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* DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ),
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* $ ERR_BNDS_NORM( NRHS, * ),
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* $ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
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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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*> ZGBSVXX uses the LU factorization to compute the solution to a
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*> complex*16 system of linear equations A * X = B, where A is an
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*> N-by-N matrix and X and B are N-by-NRHS matrices.
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*>
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*> If requested, both normwise and maximum componentwise error bounds
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*> are returned. ZGBSVXX will return a solution with a tiny
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*> guaranteed error (O(eps) where eps is the working machine
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*> precision) unless the matrix is very ill-conditioned, in which
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*> case a warning is returned. Relevant condition numbers also are
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*> calculated and returned.
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*>
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*> ZGBSVXX accepts user-provided factorizations and equilibration
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*> factors; see the definitions of the FACT and EQUED options.
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*> Solving with refinement and using a factorization from a previous
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*> ZGBSVXX call will also produce a solution with either O(eps)
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*> errors or warnings, but we cannot make that claim for general
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*> user-provided factorizations and equilibration factors if they
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*> differ from what ZGBSVXX would itself produce.
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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 = 'E', double precision scaling factors are computed to equilibrate
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*> the system:
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*>
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*> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
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*> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
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*> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
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*>
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*> Whether or not the system will be equilibrated depends on the
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*> scaling of the matrix A, but if equilibration is used, A is
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*> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
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*> or diag(C)*B (if TRANS = 'T' or 'C').
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*>
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*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor
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*> the matrix A (after equilibration if FACT = 'E') as
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*>
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*> A = P * L * U,
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*>
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*> where P is a permutation matrix, L is a unit lower triangular
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*> matrix, and U is upper triangular.
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*>
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*> 3. If some U(i,i)=0, so that U is exactly singular, then the
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*> routine returns with INFO = i. Otherwise, the factored form of A
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*> is used to estimate the condition number of the matrix A (see
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*> argument RCOND). If the reciprocal of the condition number is less
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*> than machine precision, the routine still goes on to solve for X
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*> and compute error bounds as described below.
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*>
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*> 4. 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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*> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
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*> the routine will use iterative refinement to try to get a small
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*> error and error bounds. Refinement calculates the residual to at
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*> least twice the working precision.
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*>
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*> 6. If equilibration was used, the matrix X is premultiplied by
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*> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
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*> that it solves the original system before equilibration.
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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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*> \verbatim
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*> Some optional parameters are bundled in the PARAMS array. These
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*> settings determine how refinement is performed, but often the
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*> defaults are acceptable. If the defaults are acceptable, users
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*> can pass NPARAMS = 0 which prevents the source code from accessing
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*> the PARAMS argument.
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*> \endverbatim
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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 A is
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*> supplied on entry, and if not, whether the matrix A should be
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*> equilibrated before it is factored.
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*> = 'F': On entry, AF and IPIV contain the factored form of A.
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*> If EQUED is not 'N', the matrix A has been
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*> equilibrated with scaling factors given by R and C.
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*> A, AF, and IPIV are not modified.
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*> = 'N': The matrix A will be copied to AF and factored.
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*> = 'E': The matrix A will be equilibrated if necessary, then
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*> copied to AF 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 number of linear equations, i.e., the order of the
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*> matrix A. N >= 0.
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*> \endverbatim
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*>
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*> \param[in] KL
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*> \verbatim
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*> KL is INTEGER
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*> The number of subdiagonals within the band of A. KL >= 0.
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*> \endverbatim
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*>
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*> \param[in] KU
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*> \verbatim
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*> KU is INTEGER
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*> The number of superdiagonals within the band of A. KU >= 0.
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*> \endverbatim
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*>
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*> \param[in] NRHS
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*> \verbatim
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*> NRHS is INTEGER
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*> The number of right hand sides, i.e., the number of columns
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*> of the matrices B and X. NRHS >= 0.
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*> \endverbatim
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*>
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*> \param[in,out] AB
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*> \verbatim
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*> AB is COMPLEX*16 array, dimension (LDAB,N)
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*> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
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*> The j-th column of A is stored in the j-th column of the
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*> array AB as follows:
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*> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
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*>
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*> If FACT = 'F' and EQUED is not 'N', then AB must have been
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*> equilibrated by the scaling factors in R and/or C. AB is not
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*> modified if FACT = 'F' or 'N', or if FACT = 'E' and
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*> EQUED = 'N' on exit.
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*>
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*> On exit, if EQUED .ne. 'N', A is scaled as follows:
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*> EQUED = 'R': A := diag(R) * A
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*> EQUED = 'C': A := A * diag(C)
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*> EQUED = 'B': A := diag(R) * A * diag(C).
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*> \endverbatim
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*>
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*> \param[in] LDAB
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*> \verbatim
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*> LDAB is INTEGER
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*> The leading dimension of the array AB. LDAB >= KL+KU+1.
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*> \endverbatim
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*>
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*> \param[in,out] AFB
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*> \verbatim
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*> AFB is COMPLEX*16 array, dimension (LDAFB,N)
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*> If FACT = 'F', then AFB is an input argument and on entry
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*> contains details of the LU factorization of the band matrix
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*> A, as computed by ZGBTRF. U is stored as an upper triangular
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*> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
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*> and the multipliers used during the factorization are stored
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*> in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is
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*> the factored form of the equilibrated matrix A.
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*>
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*> If FACT = 'N', then AF is an output argument and on exit
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*> returns the factors L and U from the factorization A = P*L*U
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*> of the original matrix A.
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*>
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*> If FACT = 'E', then AF is an output argument and on exit
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*> returns the factors L and U from the factorization A = P*L*U
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*> of the equilibrated matrix A (see the description of A for
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*> the form of the equilibrated matrix).
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*> \endverbatim
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*>
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*> \param[in] LDAFB
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*> \verbatim
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*> LDAFB is INTEGER
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*> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
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*> \endverbatim
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*>
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*> \param[in,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 factorization A = P*L*U
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*> as computed by ZGETRF; row i of the matrix was interchanged
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*> with row IPIV(i).
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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 factorization A = P*L*U
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*> of the original matrix A.
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*>
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*> If FACT = 'E', then IPIV is an output argument and on exit
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*> contains the pivot indices from the factorization A = P*L*U
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*> of the equilibrated matrix A.
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*> \endverbatim
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*>
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*> \param[in,out] EQUED
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*> \verbatim
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*> EQUED is CHARACTER*1
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*> Specifies the form of equilibration that was done.
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*> = 'N': No equilibration (always true if FACT = 'N').
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*> = 'R': Row equilibration, i.e., A has been premultiplied by
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*> diag(R).
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*> = 'C': Column equilibration, i.e., A has been postmultiplied
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*> by diag(C).
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*> = 'B': Both row and column equilibration, i.e., A has been
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*> replaced by diag(R) * A * diag(C).
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*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
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*> output argument.
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*> \endverbatim
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*>
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*> \param[in,out] R
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*> \verbatim
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*> R is DOUBLE PRECISION array, dimension (N)
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*> The row scale factors for A. If EQUED = 'R' or 'B', A is
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*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
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*> is not accessed. R is an input argument if FACT = 'F';
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*> otherwise, R is an output argument. If FACT = 'F' and
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*> EQUED = 'R' or 'B', each element of R must be positive.
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*> If R is output, each element of R is a power of the radix.
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*> If R is input, each element of R should be a power of the radix
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*> to ensure a reliable solution and error estimates. Scaling by
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*> powers of the radix does not cause rounding errors unless the
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*> result underflows or overflows. Rounding errors during scaling
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*> lead to refining with a matrix that is not equivalent to the
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*> input matrix, producing error estimates that may not be
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*> reliable.
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*> \endverbatim
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*>
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*> \param[in,out] C
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*> \verbatim
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*> C is DOUBLE PRECISION array, dimension (N)
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*> The column scale factors for A. If EQUED = 'C' or 'B', A is
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*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
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*> is not accessed. C is an input argument if FACT = 'F';
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*> otherwise, C is an output argument. If FACT = 'F' and
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*> EQUED = 'C' or 'B', each element of C must be positive.
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*> If C is output, each element of C is a power of the radix.
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*> If C is input, each element of C should be a power of the radix
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*> to ensure a reliable solution and error estimates. Scaling by
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*> powers of the radix does not cause rounding errors unless the
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*> result underflows or overflows. Rounding errors during scaling
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*> lead to refining with a matrix that is not equivalent to the
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*> input matrix, producing error estimates that may not be
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*> reliable.
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*> \endverbatim
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*>
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*> \param[in,out] B
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*> \verbatim
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*> B is COMPLEX*16 array, dimension (LDB,NRHS)
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*> On entry, the N-by-NRHS right hand side matrix B.
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*> On exit,
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*> if EQUED = 'N', B is not modified;
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*> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
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*> diag(R)*B;
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*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
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*> overwritten by diag(C)*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, the N-by-NRHS solution matrix X to the original
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*> system of equations. Note that A and B are modified on exit
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*> if EQUED .ne. 'N', and the solution to the equilibrated system is
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*> inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or
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*> inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
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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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*> Reciprocal scaled condition number. This is an estimate of the
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*> reciprocal Skeel condition number of the matrix A after
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*> equilibration (if done). If this is less than the machine
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*> precision (in particular, if it is zero), the matrix is singular
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*> to working precision. Note that the error may still be small even
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*> if this number is very small and the matrix appears ill-
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*> conditioned.
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*> \endverbatim
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*>
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*> \param[out] RPVGRW
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*> \verbatim
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*> RPVGRW is DOUBLE PRECISION
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*> Reciprocal pivot growth. On exit, this contains the reciprocal
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*> pivot growth factor norm(A)/norm(U). The "max absolute element"
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*> norm is used. If this is much less than 1, then the stability of
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*> the LU factorization of the (equilibrated) matrix A could be poor.
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*> This also means that the solution X, estimated condition numbers,
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*> and error bounds could be unreliable. If factorization fails with
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*> 0<INFO<=N, then this contains the reciprocal pivot growth factor
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*> for the leading INFO columns of A. In ZGESVX, this quantity is
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*> returned in RWORK(1).
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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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*> Componentwise relative backward error. This is the
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*> componentwise relative backward error of each solution vector X(j)
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*> (i.e., the smallest relative change in any element of A or B that
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*> makes X(j) an exact solution).
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*> \endverbatim
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*>
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*> \param[in] N_ERR_BNDS
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*> \verbatim
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*> N_ERR_BNDS is INTEGER
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*> Number of error bounds to return for each right hand side
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*> and each type (normwise or componentwise). See ERR_BNDS_NORM and
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*> ERR_BNDS_COMP below.
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*> \endverbatim
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*>
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*> \param[out] ERR_BNDS_NORM
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*> \verbatim
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*> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
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*> For each right-hand side, this array contains information about
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*> various error bounds and condition numbers corresponding to the
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*> normwise relative error, which is defined as follows:
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*>
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*> Normwise relative error in the ith solution vector:
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*> max_j (abs(XTRUE(j,i) - X(j,i)))
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*> ------------------------------
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*> max_j abs(X(j,i))
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*>
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*> The array is indexed by the type of error information as described
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*> below. There currently are up to three pieces of information
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*> returned.
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*>
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*> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
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*> right-hand side.
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*>
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*> The second index in ERR_BNDS_NORM(:,err) contains the following
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*> three fields:
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*> err = 1 "Trust/don't trust" boolean. Trust the answer if the
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*> reciprocal condition number is less than the threshold
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*> sqrt(n) * dlamch('Epsilon').
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*>
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*> err = 2 "Guaranteed" error bound: The estimated forward error,
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*> almost certainly within a factor of 10 of the true error
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*> so long as the next entry is greater than the threshold
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*> sqrt(n) * dlamch('Epsilon'). This error bound should only
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*> be trusted if the previous boolean is true.
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*>
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*> err = 3 Reciprocal condition number: Estimated normwise
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*> reciprocal condition number. Compared with the threshold
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*> sqrt(n) * dlamch('Epsilon') to determine if the error
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*> estimate is "guaranteed". These reciprocal condition
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*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
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*> appropriately scaled matrix Z.
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*> Let Z = S*A, where S scales each row by a power of the
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*> radix so all absolute row sums of Z are approximately 1.
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*>
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*> See Lapack Working Note 165 for further details and extra
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*> cautions.
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*> \endverbatim
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*>
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*> \param[out] ERR_BNDS_COMP
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*> \verbatim
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*> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
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*> For each right-hand side, this array contains information about
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*> various error bounds and condition numbers corresponding to the
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*> componentwise relative error, which is defined as follows:
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*>
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*> Componentwise relative error in the ith solution vector:
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*> abs(XTRUE(j,i) - X(j,i))
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*> max_j ----------------------
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*> abs(X(j,i))
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*>
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*> The array is indexed by the right-hand side i (on which the
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*> componentwise relative error depends), and the type of error
|
|
*> information as described below. There currently are up to three
|
|
*> pieces of information returned for each right-hand side. If
|
|
*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
|
|
*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
|
|
*> the first (:,N_ERR_BNDS) entries are returned.
|
|
*>
|
|
*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
|
|
*> right-hand side.
|
|
*>
|
|
*> The second index in ERR_BNDS_COMP(:,err) contains the following
|
|
*> three fields:
|
|
*> err = 1 "Trust/don't trust" boolean. Trust the answer if the
|
|
*> reciprocal condition number is less than the threshold
|
|
*> sqrt(n) * dlamch('Epsilon').
|
|
*>
|
|
*> err = 2 "Guaranteed" error bound: The estimated forward error,
|
|
*> almost certainly within a factor of 10 of the true error
|
|
*> so long as the next entry is greater than the threshold
|
|
*> sqrt(n) * dlamch('Epsilon'). This error bound should only
|
|
*> be trusted if the previous boolean is true.
|
|
*>
|
|
*> err = 3 Reciprocal condition number: Estimated componentwise
|
|
*> reciprocal condition number. Compared with the threshold
|
|
*> sqrt(n) * dlamch('Epsilon') to determine if the error
|
|
*> estimate is "guaranteed". These reciprocal condition
|
|
*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
|
|
*> appropriately scaled matrix Z.
|
|
*> Let Z = S*(A*diag(x)), where x is the solution for the
|
|
*> current right-hand side and S scales each row of
|
|
*> A*diag(x) by a power of the radix so all absolute row
|
|
*> sums of Z are approximately 1.
|
|
*>
|
|
*> See Lapack Working Note 165 for further details and extra
|
|
*> cautions.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] NPARAMS
|
|
*> \verbatim
|
|
*> NPARAMS is INTEGER
|
|
*> Specifies the number of parameters set in PARAMS. If <= 0, the
|
|
*> PARAMS array is never referenced and default values are used.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in,out] PARAMS
|
|
*> \verbatim
|
|
*> PARAMS is DOUBLE PRECISION array, dimension NPARAMS
|
|
*> Specifies algorithm parameters. If an entry is < 0.0, then
|
|
*> that entry will be filled with default value used for that
|
|
*> parameter. Only positions up to NPARAMS are accessed; defaults
|
|
*> are used for higher-numbered parameters.
|
|
*>
|
|
*> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
|
|
*> refinement or not.
|
|
*> Default: 1.0D+0
|
|
*> = 0.0: No refinement is performed, and no error bounds are
|
|
*> computed.
|
|
*> = 1.0: Use the extra-precise refinement algorithm.
|
|
*> (other values are reserved for future use)
|
|
*>
|
|
*> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
|
|
*> computations allowed for refinement.
|
|
*> Default: 10
|
|
*> Aggressive: Set to 100 to permit convergence using approximate
|
|
*> factorizations or factorizations other than LU. If
|
|
*> the factorization uses a technique other than
|
|
*> Gaussian elimination, the guarantees in
|
|
*> err_bnds_norm and err_bnds_comp may no longer be
|
|
*> trustworthy.
|
|
*>
|
|
*> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
|
|
*> will attempt to find a solution with small componentwise
|
|
*> relative error in the double-precision algorithm. Positive
|
|
*> is true, 0.0 is false.
|
|
*> Default: 1.0 (attempt componentwise convergence)
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] WORK
|
|
*> \verbatim
|
|
*> WORK is COMPLEX*16 array, dimension (2*N)
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] RWORK
|
|
*> \verbatim
|
|
*> RWORK is DOUBLE PRECISION array, dimension (2*N)
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] INFO
|
|
*> \verbatim
|
|
*> INFO is INTEGER
|
|
*> = 0: Successful exit. The solution to every right-hand side is
|
|
*> guaranteed.
|
|
*> < 0: If INFO = -i, the i-th argument had an illegal value
|
|
*> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
|
|
*> has been completed, but the factor U is exactly singular, so
|
|
*> the solution and error bounds could not be computed. RCOND = 0
|
|
*> is returned.
|
|
*> = N+J: The solution corresponding to the Jth right-hand side is
|
|
*> not guaranteed. The solutions corresponding to other right-
|
|
*> hand sides K with K > J may not be guaranteed as well, but
|
|
*> only the first such right-hand side is reported. If a small
|
|
*> componentwise error is not requested (PARAMS(3) = 0.0) then
|
|
*> the Jth right-hand side is the first with a normwise error
|
|
*> bound that is not guaranteed (the smallest J such
|
|
*> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
|
|
*> the Jth right-hand side is the first with either a normwise or
|
|
*> componentwise error bound that is not guaranteed (the smallest
|
|
*> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
|
|
*> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
|
|
*> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
|
|
*> about all of the right-hand sides check ERR_BNDS_NORM or
|
|
*> ERR_BNDS_COMP.
|
|
*> \endverbatim
|
|
*
|
|
* Authors:
|
|
* ========
|
|
*
|
|
*> \author Univ. of Tennessee
|
|
*> \author Univ. of California Berkeley
|
|
*> \author Univ. of Colorado Denver
|
|
*> \author NAG Ltd.
|
|
*
|
|
*> \ingroup complex16GBsolve
|
|
*
|
|
* =====================================================================
|
|
SUBROUTINE ZGBSVXX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
|
|
$ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
|
|
$ RCOND, RPVGRW, BERR, N_ERR_BNDS,
|
|
$ ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
|
|
$ WORK, RWORK, INFO )
|
|
*
|
|
* -- LAPACK driver routine --
|
|
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
|
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
|
*
|
|
* .. Scalar Arguments ..
|
|
CHARACTER EQUED, FACT, TRANS
|
|
INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS,
|
|
$ N_ERR_BNDS
|
|
DOUBLE PRECISION RCOND, RPVGRW
|
|
* ..
|
|
* .. Array Arguments ..
|
|
INTEGER IPIV( * )
|
|
COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
|
|
$ X( LDX , * ),WORK( * )
|
|
DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ),
|
|
$ ERR_BNDS_NORM( NRHS, * ),
|
|
$ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
|
|
* ..
|
|
*
|
|
* ==================================================================
|
|
*
|
|
* .. Parameters ..
|
|
DOUBLE PRECISION ZERO, ONE
|
|
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
|
|
INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
|
|
INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
|
|
INTEGER CMP_ERR_I, PIV_GROWTH_I
|
|
PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
|
|
$ BERR_I = 3 )
|
|
PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
|
|
PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
|
|
$ PIV_GROWTH_I = 9 )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
|
|
INTEGER INFEQU, I, J, KL, KU
|
|
DOUBLE PRECISION AMAX, BIGNUM, COLCND, RCMAX, RCMIN,
|
|
$ ROWCND, SMLNUM
|
|
* ..
|
|
* .. External Functions ..
|
|
EXTERNAL LSAME, DLAMCH, ZLA_GBRPVGRW
|
|
LOGICAL LSAME
|
|
DOUBLE PRECISION DLAMCH, ZLA_GBRPVGRW
|
|
* ..
|
|
* .. External Subroutines ..
|
|
EXTERNAL ZGBEQUB, ZGBTRF, ZGBTRS, ZLACPY, ZLAQGB,
|
|
$ XERBLA, ZLASCL2, ZGBRFSX
|
|
* ..
|
|
* .. Intrinsic Functions ..
|
|
INTRINSIC MAX, MIN
|
|
* ..
|
|
* .. Executable Statements ..
|
|
*
|
|
INFO = 0
|
|
NOFACT = LSAME( FACT, 'N' )
|
|
EQUIL = LSAME( FACT, 'E' )
|
|
NOTRAN = LSAME( TRANS, 'N' )
|
|
SMLNUM = DLAMCH( 'Safe minimum' )
|
|
BIGNUM = ONE / SMLNUM
|
|
IF( NOFACT .OR. EQUIL ) THEN
|
|
EQUED = 'N'
|
|
ROWEQU = .FALSE.
|
|
COLEQU = .FALSE.
|
|
ELSE
|
|
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
|
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
|
END IF
|
|
*
|
|
* Default is failure. If an input parameter is wrong or
|
|
* factorization fails, make everything look horrible. Only the
|
|
* pivot growth is set here, the rest is initialized in ZGBRFSX.
|
|
*
|
|
RPVGRW = ZERO
|
|
*
|
|
* Test the input parameters. PARAMS is not tested until ZGERFSX.
|
|
*
|
|
IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
|
|
$ LSAME( FACT, 'F' ) ) THEN
|
|
INFO = -1
|
|
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
|
|
$ LSAME( TRANS, 'C' ) ) THEN
|
|
INFO = -2
|
|
ELSE IF( N.LT.0 ) THEN
|
|
INFO = -3
|
|
ELSE IF( KL.LT.0 ) THEN
|
|
INFO = -4
|
|
ELSE IF( KU.LT.0 ) THEN
|
|
INFO = -5
|
|
ELSE IF( NRHS.LT.0 ) THEN
|
|
INFO = -6
|
|
ELSE IF( LDAB.LT.KL+KU+1 ) THEN
|
|
INFO = -8
|
|
ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
|
|
INFO = -10
|
|
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
|
|
$ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
|
|
INFO = -12
|
|
ELSE
|
|
IF( ROWEQU ) THEN
|
|
RCMIN = BIGNUM
|
|
RCMAX = ZERO
|
|
DO 10 J = 1, N
|
|
RCMIN = MIN( RCMIN, R( J ) )
|
|
RCMAX = MAX( RCMAX, R( J ) )
|
|
10 CONTINUE
|
|
IF( RCMIN.LE.ZERO ) THEN
|
|
INFO = -13
|
|
ELSE IF( N.GT.0 ) THEN
|
|
ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
|
|
ELSE
|
|
ROWCND = ONE
|
|
END IF
|
|
END IF
|
|
IF( COLEQU .AND. INFO.EQ.0 ) THEN
|
|
RCMIN = BIGNUM
|
|
RCMAX = ZERO
|
|
DO 20 J = 1, N
|
|
RCMIN = MIN( RCMIN, C( J ) )
|
|
RCMAX = MAX( RCMAX, C( J ) )
|
|
20 CONTINUE
|
|
IF( RCMIN.LE.ZERO ) THEN
|
|
INFO = -14
|
|
ELSE IF( N.GT.0 ) THEN
|
|
COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
|
|
ELSE
|
|
COLCND = ONE
|
|
END IF
|
|
END IF
|
|
IF( INFO.EQ.0 ) THEN
|
|
IF( LDB.LT.MAX( 1, N ) ) THEN
|
|
INFO = -15
|
|
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
|
|
INFO = -16
|
|
END IF
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'ZGBSVXX', -INFO )
|
|
RETURN
|
|
END IF
|
|
*
|
|
IF( EQUIL ) THEN
|
|
*
|
|
* Compute row and column scalings to equilibrate the matrix A.
|
|
*
|
|
CALL ZGBEQUB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
|
|
$ AMAX, INFEQU )
|
|
IF( INFEQU.EQ.0 ) THEN
|
|
*
|
|
* Equilibrate the matrix.
|
|
*
|
|
CALL ZLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
|
|
$ AMAX, EQUED )
|
|
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
|
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
|
END IF
|
|
*
|
|
* If the scaling factors are not applied, set them to 1.0.
|
|
*
|
|
IF ( .NOT.ROWEQU ) THEN
|
|
DO J = 1, N
|
|
R( J ) = 1.0D+0
|
|
END DO
|
|
END IF
|
|
IF ( .NOT.COLEQU ) THEN
|
|
DO J = 1, N
|
|
C( J ) = 1.0D+0
|
|
END DO
|
|
END IF
|
|
END IF
|
|
*
|
|
* Scale the right-hand side.
|
|
*
|
|
IF( NOTRAN ) THEN
|
|
IF( ROWEQU ) CALL ZLASCL2( N, NRHS, R, B, LDB )
|
|
ELSE
|
|
IF( COLEQU ) CALL ZLASCL2( N, NRHS, C, B, LDB )
|
|
END IF
|
|
*
|
|
IF( NOFACT .OR. EQUIL ) THEN
|
|
*
|
|
* Compute the LU factorization of A.
|
|
*
|
|
DO 40, J = 1, N
|
|
DO 30, I = KL+1, 2*KL+KU+1
|
|
AFB( I, J ) = AB( I-KL, J )
|
|
30 CONTINUE
|
|
40 CONTINUE
|
|
CALL ZGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
|
|
*
|
|
* Return if INFO is non-zero.
|
|
*
|
|
IF( INFO.GT.0 ) THEN
|
|
*
|
|
* Pivot in column INFO is exactly 0
|
|
* Compute the reciprocal pivot growth factor of the
|
|
* leading rank-deficient INFO columns of A.
|
|
*
|
|
RPVGRW = ZLA_GBRPVGRW( N, KL, KU, INFO, AB, LDAB, AFB,
|
|
$ LDAFB )
|
|
RETURN
|
|
END IF
|
|
END IF
|
|
*
|
|
* Compute the reciprocal pivot growth factor RPVGRW.
|
|
*
|
|
RPVGRW = ZLA_GBRPVGRW( N, KL, KU, N, AB, LDAB, AFB, LDAFB )
|
|
*
|
|
* Compute the solution matrix X.
|
|
*
|
|
CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
|
|
CALL ZGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
|
|
$ INFO )
|
|
*
|
|
* Use iterative refinement to improve the computed solution and
|
|
* compute error bounds and backward error estimates for it.
|
|
*
|
|
CALL ZGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
|
|
$ IPIV, R, C, B, LDB, X, LDX, RCOND, BERR,
|
|
$ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
|
|
$ WORK, RWORK, INFO )
|
|
|
|
*
|
|
* Scale solutions.
|
|
*
|
|
IF ( COLEQU .AND. NOTRAN ) THEN
|
|
CALL ZLASCL2( N, NRHS, C, X, LDX )
|
|
ELSE IF ( ROWEQU .AND. .NOT.NOTRAN ) THEN
|
|
CALL ZLASCL2( N, NRHS, R, X, LDX )
|
|
END IF
|
|
*
|
|
RETURN
|
|
*
|
|
* End of ZGBSVXX
|
|
*
|
|
END
|
|
|