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321 lines
10 KiB
321 lines
10 KiB
2 years ago
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*> \brief \b ZLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate.
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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 ZLATDF + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlatdf.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/zlatdf.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/zlatdf.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 ZLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
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* JPIV )
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*
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* .. Scalar Arguments ..
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* INTEGER IJOB, LDZ, N
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* DOUBLE PRECISION RDSCAL, RDSUM
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* ..
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* .. Array Arguments ..
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* INTEGER IPIV( * ), JPIV( * )
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* COMPLEX*16 RHS( * ), Z( LDZ, * )
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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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*> ZLATDF computes the contribution to the reciprocal Dif-estimate
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*> by solving for x in Z * x = b, where b is chosen such that the norm
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*> of x is as large as possible. It is assumed that LU decomposition
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*> of Z has been computed by ZGETC2. On entry RHS = f holds the
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*> contribution from earlier solved sub-systems, and on return RHS = x.
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*>
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*> The factorization of Z returned by ZGETC2 has the form
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*> Z = P * L * U * Q, where P and Q are permutation matrices. L is lower
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*> triangular with unit diagonal elements and U is upper triangular.
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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] IJOB
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*> \verbatim
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*> IJOB is INTEGER
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*> IJOB = 2: First compute an approximative null-vector e
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*> of Z using ZGECON, e is normalized and solve for
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*> Zx = +-e - f with the sign giving the greater value of
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*> 2-norm(x). About 5 times as expensive as Default.
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*> IJOB .ne. 2: Local look ahead strategy where
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*> all entries of the r.h.s. b is chosen as either +1 or
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*> -1. Default.
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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 columns of the matrix Z.
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*> \endverbatim
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*>
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*> \param[in] Z
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*> \verbatim
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*> Z is COMPLEX*16 array, dimension (LDZ, N)
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*> On entry, the LU part of the factorization of the n-by-n
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*> matrix Z computed by ZGETC2: Z = P * L * U * Q
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*> \endverbatim
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*>
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*> \param[in] LDZ
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*> \verbatim
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*> LDZ is INTEGER
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*> The leading dimension of the array Z. LDA >= max(1, N).
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*> \endverbatim
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*>
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*> \param[in,out] RHS
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*> \verbatim
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*> RHS is COMPLEX*16 array, dimension (N).
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*> On entry, RHS contains contributions from other subsystems.
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*> On exit, RHS contains the solution of the subsystem with
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*> entries according to the value of IJOB (see above).
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*> \endverbatim
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*>
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*> \param[in,out] RDSUM
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*> \verbatim
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*> RDSUM is DOUBLE PRECISION
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*> On entry, the sum of squares of computed contributions to
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*> the Dif-estimate under computation by ZTGSYL, where the
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*> scaling factor RDSCAL (see below) has been factored out.
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*> On exit, the corresponding sum of squares updated with the
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*> contributions from the current sub-system.
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*> If TRANS = 'T' RDSUM is not touched.
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*> NOTE: RDSUM only makes sense when ZTGSY2 is called by CTGSYL.
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*> \endverbatim
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*>
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*> \param[in,out] RDSCAL
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*> \verbatim
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*> RDSCAL is DOUBLE PRECISION
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*> On entry, scaling factor used to prevent overflow in RDSUM.
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*> On exit, RDSCAL is updated w.r.t. the current contributions
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*> in RDSUM.
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*> If TRANS = 'T', RDSCAL is not touched.
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*> NOTE: RDSCAL only makes sense when ZTGSY2 is called by
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*> ZTGSYL.
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*> \endverbatim
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*>
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*> \param[in] IPIV
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*> \verbatim
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*> IPIV is INTEGER array, dimension (N).
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*> The pivot indices; for 1 <= i <= N, row i of the
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*> matrix has been interchanged with row IPIV(i).
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*> \endverbatim
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*>
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*> \param[in] JPIV
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*> \verbatim
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*> JPIV is INTEGER array, dimension (N).
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*> The pivot indices; for 1 <= j <= N, column j of the
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*> matrix has been interchanged with column JPIV(j).
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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 complex16OTHERauxiliary
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*
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*> \par Further Details:
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* =====================
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*>
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*> This routine is a further developed implementation of algorithm
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*> BSOLVE in [1] using complete pivoting in the LU factorization.
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*
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*> \par Contributors:
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* ==================
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*>
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*> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
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*> Umea University, S-901 87 Umea, Sweden.
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*
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*> \par References:
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* ================
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*>
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*> [1] Bo Kagstrom and Lars Westin,
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*> Generalized Schur Methods with Condition Estimators for
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*> Solving the Generalized Sylvester Equation, IEEE Transactions
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*> on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
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*>\n
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*> [2] Peter Poromaa,
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*> On Efficient and Robust Estimators for the Separation
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*> between two Regular Matrix Pairs with Applications in
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*> Condition Estimation. Report UMINF-95.05, Department of
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*> Computing Science, Umea University, S-901 87 Umea, Sweden,
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*> 1995.
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*
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* =====================================================================
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SUBROUTINE ZLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
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$ JPIV )
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*
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* -- LAPACK auxiliary 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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INTEGER IJOB, LDZ, N
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DOUBLE PRECISION RDSCAL, RDSUM
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* ..
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* .. Array Arguments ..
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INTEGER IPIV( * ), JPIV( * )
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COMPLEX*16 RHS( * ), Z( LDZ, * )
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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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INTEGER MAXDIM
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PARAMETER ( MAXDIM = 2 )
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DOUBLE PRECISION ZERO, ONE
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
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COMPLEX*16 CONE
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PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
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* ..
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* .. Local Scalars ..
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INTEGER I, INFO, J, K
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DOUBLE PRECISION RTEMP, SCALE, SMINU, SPLUS
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COMPLEX*16 BM, BP, PMONE, TEMP
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* ..
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* .. Local Arrays ..
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DOUBLE PRECISION RWORK( MAXDIM )
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COMPLEX*16 WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
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* ..
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* .. External Subroutines ..
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EXTERNAL ZAXPY, ZCOPY, ZGECON, ZGESC2, ZLASSQ, ZLASWP,
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$ ZSCAL
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* ..
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* .. External Functions ..
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DOUBLE PRECISION DZASUM
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COMPLEX*16 ZDOTC
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EXTERNAL DZASUM, ZDOTC
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, DBLE, SQRT
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* ..
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* .. Executable Statements ..
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*
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IF( IJOB.NE.2 ) THEN
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*
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* Apply permutations IPIV to RHS
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*
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CALL ZLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
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*
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* Solve for L-part choosing RHS either to +1 or -1.
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*
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PMONE = -CONE
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DO 10 J = 1, N - 1
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BP = RHS( J ) + CONE
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BM = RHS( J ) - CONE
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SPLUS = ONE
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*
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* Look-ahead for L- part RHS(1:N-1) = +-1
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* SPLUS and SMIN computed more efficiently than in BSOLVE[1].
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*
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SPLUS = SPLUS + DBLE( ZDOTC( N-J, Z( J+1, J ), 1, Z( J+1,
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$ J ), 1 ) )
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SMINU = DBLE( ZDOTC( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 ) )
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SPLUS = SPLUS*DBLE( RHS( J ) )
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IF( SPLUS.GT.SMINU ) THEN
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RHS( J ) = BP
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ELSE IF( SMINU.GT.SPLUS ) THEN
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RHS( J ) = BM
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ELSE
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*
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* In this case the updating sums are equal and we can
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* choose RHS(J) +1 or -1. The first time this happens we
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* choose -1, thereafter +1. This is a simple way to get
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* good estimates of matrices like Byers well-known example
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* (see [1]). (Not done in BSOLVE.)
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*
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RHS( J ) = RHS( J ) + PMONE
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PMONE = CONE
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END IF
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*
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* Compute the remaining r.h.s.
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*
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TEMP = -RHS( J )
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CALL ZAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
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10 CONTINUE
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*
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* Solve for U- part, lockahead for RHS(N) = +-1. This is not done
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* In BSOLVE and will hopefully give us a better estimate because
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* any ill-conditioning of the original matrix is transferred to U
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* and not to L. U(N, N) is an approximation to sigma_min(LU).
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*
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CALL ZCOPY( N-1, RHS, 1, WORK, 1 )
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WORK( N ) = RHS( N ) + CONE
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RHS( N ) = RHS( N ) - CONE
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SPLUS = ZERO
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SMINU = ZERO
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DO 30 I = N, 1, -1
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TEMP = CONE / Z( I, I )
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WORK( I ) = WORK( I )*TEMP
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RHS( I ) = RHS( I )*TEMP
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DO 20 K = I + 1, N
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WORK( I ) = WORK( I ) - WORK( K )*( Z( I, K )*TEMP )
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RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
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20 CONTINUE
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SPLUS = SPLUS + ABS( WORK( I ) )
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SMINU = SMINU + ABS( RHS( I ) )
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30 CONTINUE
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IF( SPLUS.GT.SMINU )
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$ CALL ZCOPY( N, WORK, 1, RHS, 1 )
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*
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* Apply the permutations JPIV to the computed solution (RHS)
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*
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CALL ZLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
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*
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* Compute the sum of squares
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*
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CALL ZLASSQ( N, RHS, 1, RDSCAL, RDSUM )
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RETURN
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END IF
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*
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* ENTRY IJOB = 2
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*
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* Compute approximate nullvector XM of Z
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*
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CALL ZGECON( 'I', N, Z, LDZ, ONE, RTEMP, WORK, RWORK, INFO )
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CALL ZCOPY( N, WORK( N+1 ), 1, XM, 1 )
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*
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* Compute RHS
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*
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CALL ZLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
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TEMP = CONE / SQRT( ZDOTC( N, XM, 1, XM, 1 ) )
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CALL ZSCAL( N, TEMP, XM, 1 )
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CALL ZCOPY( N, XM, 1, XP, 1 )
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CALL ZAXPY( N, CONE, RHS, 1, XP, 1 )
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CALL ZAXPY( N, -CONE, XM, 1, RHS, 1 )
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CALL ZGESC2( N, Z, LDZ, RHS, IPIV, JPIV, SCALE )
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CALL ZGESC2( N, Z, LDZ, XP, IPIV, JPIV, SCALE )
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IF( DZASUM( N, XP, 1 ).GT.DZASUM( N, RHS, 1 ) )
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$ CALL ZCOPY( N, XP, 1, RHS, 1 )
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*
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* Compute the sum of squares
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*
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CALL ZLASSQ( N, RHS, 1, RDSCAL, RDSUM )
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RETURN
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*
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* End of ZLATDF
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*
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END
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