NAKATA Maho
ac82605ea9
Several LAPACK, BLAS, and CBLAS source files defined a local statement
function named ABS1 for the complex 1-norm approximation:
ABS1( X ) = ABS( REAL( X ) ) + ABS( AIMAG( X ) )
ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) )
The majority of the codebase already uses CABS1 for this identical
purpose. This commit renames ABS1 to CABS1 in all remaining files
(definition line, declaration line, and all call sites within the
same file) to make the naming consistent across the repository.
A small number of fixed-form lines required continuation-line splits
to stay within the 72-column limit after the rename.
No numerical change. Statement functions are file-local in Fortran,
so there is no ABI or interface impact.
This is a preparatory cleanup before inlining these statement
functions (see issue #1200).
809 lines
26 KiB
FortranFixed
809 lines
26 KiB
FortranFixed
*> \brief <b> CGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE 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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*> Download CGGEVX + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cggevx.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/cggevx.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/cggevx.f">
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*> [TXT]</a>
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE CGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
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* ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI,
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* LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV,
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* WORK, LWORK, RWORK, IWORK, BWORK, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER BALANC, JOBVL, JOBVR, SENSE
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* INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
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* REAL ABNRM, BBNRM
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* ..
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* .. Array Arguments ..
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* LOGICAL BWORK( * )
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* INTEGER IWORK( * )
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* REAL LSCALE( * ), RCONDE( * ), RCONDV( * ),
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* $ RSCALE( * ), RWORK( * )
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* COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
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* $ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
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* $ WORK( * )
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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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*> CGGEVX computes for a pair of N-by-N complex nonsymmetric matrices
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*> (A,B) the generalized eigenvalues, and optionally, the left and/or
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*> right generalized eigenvectors.
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*>
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*> Optionally, it also computes a balancing transformation to improve
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*> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
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*> LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
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*> the eigenvalues (RCONDE), and reciprocal condition numbers for the
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*> right eigenvectors (RCONDV).
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*>
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*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
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*> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
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*> singular. It is usually represented as the pair (alpha,beta), as
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*> there is a reasonable interpretation for beta=0, and even for both
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*> being zero.
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*>
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*> The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
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*> of (A,B) satisfies
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*> A * v(j) = lambda(j) * B * v(j) .
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*> The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
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*> of (A,B) satisfies
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*> u(j)**H * A = lambda(j) * u(j)**H * B.
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*> where u(j)**H is the conjugate-transpose of u(j).
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*>
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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] BALANC
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*> \verbatim
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*> BALANC is CHARACTER*1
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*> Specifies the balance option to be performed:
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*> = 'N': do not diagonally scale or permute;
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*> = 'P': permute only;
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*> = 'S': scale only;
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*> = 'B': both permute and scale.
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*> Computed reciprocal condition numbers will be for the
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*> matrices after permuting and/or balancing. Permuting does
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*> not change condition numbers (in exact arithmetic), but
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*> balancing does.
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*> \endverbatim
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*>
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*> \param[in] JOBVL
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*> \verbatim
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*> JOBVL is CHARACTER*1
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*> = 'N': do not compute the left generalized eigenvectors;
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*> = 'V': compute the left generalized eigenvectors.
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*> \endverbatim
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*>
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*> \param[in] JOBVR
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*> \verbatim
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*> JOBVR is CHARACTER*1
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*> = 'N': do not compute the right generalized eigenvectors;
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*> = 'V': compute the right generalized eigenvectors.
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*> \endverbatim
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*>
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*> \param[in] SENSE
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*> \verbatim
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*> SENSE is CHARACTER*1
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*> Determines which reciprocal condition numbers are computed.
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*> = 'N': none are computed;
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*> = 'E': computed for eigenvalues only;
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*> = 'V': computed for eigenvectors only;
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*> = 'B': computed for eigenvalues and eigenvectors.
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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 matrices A, B, VL, and VR. N >= 0.
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*> \endverbatim
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*>
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*> \param[in,out] A
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*> \verbatim
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*> A is COMPLEX array, dimension (LDA, N)
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*> On entry, the matrix A in the pair (A,B).
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*> On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
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*> or both, then A contains the first part of the complex Schur
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*> form of the "balanced" versions of the input A and B.
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of A. LDA >= max(1,N).
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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 array, dimension (LDB, N)
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*> On entry, the matrix B in the pair (A,B).
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*> On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
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*> or both, then B contains the second part of the complex
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*> Schur form of the "balanced" versions of the input A and 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 B. LDB >= max(1,N).
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*> \endverbatim
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*>
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*> \param[out] ALPHA
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*> \verbatim
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*> ALPHA is COMPLEX array, dimension (N)
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*> \endverbatim
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*>
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*> \param[out] BETA
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*> \verbatim
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*> BETA is COMPLEX array, dimension (N)
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*> On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized
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*> eigenvalues.
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*>
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*> Note: the quotient ALPHA(j)/BETA(j) ) may easily over- or
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*> underflow, and BETA(j) may even be zero. Thus, the user
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*> should avoid naively computing the ratio ALPHA/BETA.
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*> However, ALPHA will be always less than and usually
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*> comparable with norm(A) in magnitude, and BETA always less
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*> than and usually comparable with norm(B).
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*> \endverbatim
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*>
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*> \param[out] VL
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*> \verbatim
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*> VL is COMPLEX array, dimension (LDVL,N)
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*> If JOBVL = 'V', the left generalized eigenvectors u(j) are
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*> stored one after another in the columns of VL, in the same
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*> order as their eigenvalues.
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*> Each eigenvector will be scaled so the largest component
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*> will have abs(real part) + abs(imag. part) = 1.
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*> Not referenced if JOBVL = 'N'.
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*> \endverbatim
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*>
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*> \param[in] LDVL
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*> \verbatim
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*> LDVL is INTEGER
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*> The leading dimension of the matrix VL. LDVL >= 1, and
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*> if JOBVL = 'V', LDVL >= N.
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*> \endverbatim
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*>
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*> \param[out] VR
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*> \verbatim
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*> VR is COMPLEX array, dimension (LDVR,N)
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*> If JOBVR = 'V', the right generalized eigenvectors v(j) are
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*> stored one after another in the columns of VR, in the same
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*> order as their eigenvalues.
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*> Each eigenvector will be scaled so the largest component
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*> will have abs(real part) + abs(imag. part) = 1.
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*> Not referenced if JOBVR = 'N'.
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*> \endverbatim
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*>
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*> \param[in] LDVR
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*> \verbatim
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*> LDVR is INTEGER
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*> The leading dimension of the matrix VR. LDVR >= 1, and
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*> if JOBVR = 'V', LDVR >= N.
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*> \endverbatim
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*>
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*> \param[out] ILO
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*> \verbatim
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*> ILO is INTEGER
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*> \endverbatim
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*>
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*> \param[out] IHI
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*> \verbatim
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*> IHI is INTEGER
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*> ILO and IHI are integer values such that on exit
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*> A(i,j) = 0 and B(i,j) = 0 if i > j and
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*> j = 1,...,ILO-1 or i = IHI+1,...,N.
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*> If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
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*> \endverbatim
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*>
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*> \param[out] LSCALE
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*> \verbatim
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*> LSCALE is REAL array, dimension (N)
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*> Details of the permutations and scaling factors applied
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*> to the left side of A and B. If PL(j) is the index of the
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*> row interchanged with row j, and DL(j) is the scaling
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*> factor applied to row j, then
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*> LSCALE(j) = PL(j) for j = 1,...,ILO-1
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*> = DL(j) for j = ILO,...,IHI
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*> = PL(j) for j = IHI+1,...,N.
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*> The order in which the interchanges are made is N to IHI+1,
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*> then 1 to ILO-1.
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*> \endverbatim
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*>
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*> \param[out] RSCALE
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*> \verbatim
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*> RSCALE is REAL array, dimension (N)
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*> Details of the permutations and scaling factors applied
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*> to the right side of A and B. If PR(j) is the index of the
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*> column interchanged with column j, and DR(j) is the scaling
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*> factor applied to column j, then
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*> RSCALE(j) = PR(j) for j = 1,...,ILO-1
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*> = DR(j) for j = ILO,...,IHI
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*> = PR(j) for j = IHI+1,...,N
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*> The order in which the interchanges are made is N to IHI+1,
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*> then 1 to ILO-1.
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*> \endverbatim
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*>
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*> \param[out] ABNRM
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*> \verbatim
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*> ABNRM is REAL
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*> The one-norm of the balanced matrix A.
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*> \endverbatim
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*>
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*> \param[out] BBNRM
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*> \verbatim
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*> BBNRM is REAL
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*> The one-norm of the balanced matrix B.
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*> \endverbatim
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*>
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*> \param[out] RCONDE
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*> \verbatim
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*> RCONDE is REAL array, dimension (N)
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*> If SENSE = 'E' or 'B', the reciprocal condition numbers of
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*> the eigenvalues, stored in consecutive elements of the array.
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*> If SENSE = 'N' or 'V', RCONDE is not referenced.
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*> \endverbatim
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*>
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*> \param[out] RCONDV
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*> \verbatim
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*> RCONDV is REAL array, dimension (N)
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*> If SENSE = 'V' or 'B', the estimated reciprocal condition
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*> numbers of the eigenvectors, stored in consecutive elements
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*> of the array. If the eigenvalues cannot be reordered to
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*> compute RCONDV(j), RCONDV(j) is set to 0; this can only occur
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*> when the true value would be very small anyway.
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*> If SENSE = 'N' or 'E', RCONDV is not referenced.
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is COMPLEX array, dimension (MAX(1,LWORK))
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*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
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*> \endverbatim
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK >= max(1,2*N).
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*> If SENSE = 'E', LWORK >= max(1,4*N).
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*> If SENSE = 'V' or 'B', LWORK >= max(1,2*N*N+2*N).
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*>
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*> If LWORK = -1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK array, returns
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*> this value as the first entry of the WORK array, and no error
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*> message related to LWORK is issued by XERBLA.
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*> \endverbatim
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*>
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*> \param[out] RWORK
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*> \verbatim
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*> RWORK is REAL array, dimension (lrwork)
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*> lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B',
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*> and at least max(1,2*N) otherwise.
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*> Real workspace.
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*> \endverbatim
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*>
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*> \param[out] IWORK
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*> \verbatim
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*> IWORK is INTEGER array, dimension (N+2)
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*> If SENSE = 'E', IWORK is not referenced.
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*> \endverbatim
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*>
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*> \param[out] BWORK
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*> \verbatim
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*> BWORK is LOGICAL array, dimension (N)
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*> If SENSE = 'N', BWORK is not referenced.
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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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*> = 1,...,N:
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*> The QZ iteration failed. No eigenvectors have been
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*> calculated, but ALPHA(j) and BETA(j) should be correct
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*> for j=INFO+1,...,N.
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*> > N: =N+1: other than QZ iteration failed in CHGEQZ.
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*> =N+2: error return from CTGEVC.
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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 ggevx
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*
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*> \par Further Details:
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* =====================
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*>
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*> \verbatim
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*>
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*> Balancing a matrix pair (A,B) includes, first, permuting rows and
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*> columns to isolate eigenvalues, second, applying diagonal similarity
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*> transformation to the rows and columns to make the rows and columns
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*> as close in norm as possible. The computed reciprocal condition
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*> numbers correspond to the balanced matrix. Permuting rows and columns
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*> will not change the condition numbers (in exact arithmetic) but
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*> diagonal scaling will. For further explanation of balancing, see
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*> section 4.11.1.2 of LAPACK Users' Guide.
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*>
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*> An approximate error bound on the chordal distance between the i-th
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*> computed generalized eigenvalue w and the corresponding exact
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*> eigenvalue lambda is
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*>
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*> chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
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*>
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*> An approximate error bound for the angle between the i-th computed
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*> eigenvector VL(i) or VR(i) is given by
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*>
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*> EPS * norm(ABNRM, BBNRM) / DIF(i).
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*>
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*> For further explanation of the reciprocal condition numbers RCONDE
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*> and RCONDV, see section 4.11 of LAPACK User's Guide.
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*> \endverbatim
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*>
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* =====================================================================
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SUBROUTINE CGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B,
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$ LDB,
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$ ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI,
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$ LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV,
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$ WORK, LWORK, RWORK, IWORK, BWORK, INFO )
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IMPLICIT NONE
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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 BALANC, JOBVL, JOBVR, SENSE
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INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
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REAL ABNRM, BBNRM
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* ..
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* .. Array Arguments ..
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LOGICAL BWORK( * )
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INTEGER IWORK( * )
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REAL LSCALE( * ), RCONDE( * ), RCONDV( * ),
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$ RSCALE( * ), RWORK( * )
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COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
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$ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
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$ WORK( * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ZERO, ONE
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PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
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COMPLEX CZERO, CONE
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PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ),
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$ CONE = ( 1.0E+0, 0.0E+0 ) )
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* ..
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* .. Local Scalars ..
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LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY, NOSCL,
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$ WANTSB, WANTSE, WANTSN, WANTSV
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CHARACTER CHTEMP
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INTEGER I, ICOLS, IERR, IJOBVL, IJOBVR, IN, IROWS,
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$ ITAU, IWRK, IWRK1, J, JC, JR, M, MAXWRK, MINWRK
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REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
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$ SMLNUM, TEMP
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COMPLEX X
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* ..
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* .. Local Arrays ..
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LOGICAL LDUMMA( 1 )
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* ..
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* .. External Subroutines ..
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EXTERNAL CGEQRF, CGGBAK, CGGBAL, CGGHRD, CHGEQZ,
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$ CLACPY,
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$ CLASCL, CLASET, CTGEVC, CTGSNA, CUNGQR, CUNMQR,
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$ SLASCL, XERBLA
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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INTEGER ILAENV
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REAL CLANGE, SLAMCH, SROUNDUP_LWORK
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EXTERNAL LSAME, ILAENV, CLANGE, SLAMCH,
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$ SROUNDUP_LWORK
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, AIMAG, MAX, REAL, SQRT
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* ..
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* .. Statement Functions ..
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REAL CABS1
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* ..
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* .. Statement Function definitions ..
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CABS1( X ) = ABS( REAL( X ) ) + ABS( AIMAG( X ) )
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* ..
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* .. Executable Statements ..
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*
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* Decode the input arguments
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*
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IF( LSAME( JOBVL, 'N' ) ) THEN
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IJOBVL = 1
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ILVL = .FALSE.
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ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
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IJOBVL = 2
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ILVL = .TRUE.
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ELSE
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IJOBVL = -1
|
|
ILVL = .FALSE.
|
|
END IF
|
|
*
|
|
IF( LSAME( JOBVR, 'N' ) ) THEN
|
|
IJOBVR = 1
|
|
ILVR = .FALSE.
|
|
ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
|
|
IJOBVR = 2
|
|
ILVR = .TRUE.
|
|
ELSE
|
|
IJOBVR = -1
|
|
ILVR = .FALSE.
|
|
END IF
|
|
ILV = ILVL .OR. ILVR
|
|
*
|
|
NOSCL = LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'P' )
|
|
WANTSN = LSAME( SENSE, 'N' )
|
|
WANTSE = LSAME( SENSE, 'E' )
|
|
WANTSV = LSAME( SENSE, 'V' )
|
|
WANTSB = LSAME( SENSE, 'B' )
|
|
*
|
|
* Test the input arguments
|
|
*
|
|
INFO = 0
|
|
LQUERY = ( LWORK.EQ.-1 )
|
|
IF( .NOT.( NOSCL .OR. LSAME( BALANC,'S' ) .OR.
|
|
$ LSAME( BALANC, 'B' ) ) ) THEN
|
|
INFO = -1
|
|
ELSE IF( IJOBVL.LE.0 ) THEN
|
|
INFO = -2
|
|
ELSE IF( IJOBVR.LE.0 ) THEN
|
|
INFO = -3
|
|
ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSB .OR. WANTSV ) )
|
|
$ THEN
|
|
INFO = -4
|
|
ELSE IF( N.LT.0 ) THEN
|
|
INFO = -5
|
|
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
|
INFO = -7
|
|
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
|
|
INFO = -9
|
|
ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
|
|
INFO = -13
|
|
ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
|
|
INFO = -15
|
|
END IF
|
|
*
|
|
* Compute workspace
|
|
* (Note: Comments in the code beginning "Workspace:" describe the
|
|
* minimal amount of workspace needed at that point in the code,
|
|
* as well as the preferred amount for good performance.
|
|
* NB refers to the optimal block size for the immediately
|
|
* following subroutine, as returned by ILAENV. The workspace is
|
|
* computed assuming ILO = 1 and IHI = N, the worst case.)
|
|
*
|
|
IF( INFO.EQ.0 ) THEN
|
|
IF( N.EQ.0 ) THEN
|
|
MINWRK = 1
|
|
MAXWRK = 1
|
|
ELSE
|
|
MINWRK = 2*N
|
|
IF( WANTSE ) THEN
|
|
MINWRK = 4*N
|
|
ELSE IF( WANTSV .OR. WANTSB ) THEN
|
|
MINWRK = 2*N*( N + 1)
|
|
END IF
|
|
MAXWRK = MINWRK
|
|
MAXWRK = MAX( MAXWRK,
|
|
$ N + N*ILAENV( 1, 'CGEQRF', ' ', N, 1, N,
|
|
$ 0 ) )
|
|
MAXWRK = MAX( MAXWRK,
|
|
$ N + N*ILAENV( 1, 'CUNMQR', ' ', N, 1, N,
|
|
$ 0 ) )
|
|
IF( ILVL ) THEN
|
|
MAXWRK = MAX( MAXWRK, N +
|
|
$ N*ILAENV( 1, 'CUNGQR', ' ', N, 1, N,
|
|
$ 0 ) )
|
|
END IF
|
|
END IF
|
|
WORK( 1 ) = SROUNDUP_LWORK(MAXWRK)
|
|
*
|
|
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
|
|
INFO = -25
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'CGGEVX', -INFO )
|
|
RETURN
|
|
ELSE IF( LQUERY ) THEN
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Quick return if possible
|
|
*
|
|
IF( N.EQ.0 )
|
|
$ RETURN
|
|
*
|
|
* Get machine constants
|
|
*
|
|
EPS = SLAMCH( 'P' )
|
|
SMLNUM = SLAMCH( 'S' )
|
|
BIGNUM = ONE / SMLNUM
|
|
SMLNUM = SQRT( SMLNUM ) / EPS
|
|
BIGNUM = ONE / SMLNUM
|
|
*
|
|
* Scale A if max element outside range [SMLNUM,BIGNUM]
|
|
*
|
|
ANRM = CLANGE( 'M', N, N, A, LDA, RWORK )
|
|
ILASCL = .FALSE.
|
|
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
|
|
ANRMTO = SMLNUM
|
|
ILASCL = .TRUE.
|
|
ELSE IF( ANRM.GT.BIGNUM ) THEN
|
|
ANRMTO = BIGNUM
|
|
ILASCL = .TRUE.
|
|
END IF
|
|
IF( ILASCL )
|
|
$ CALL CLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
|
|
*
|
|
* Scale B if max element outside range [SMLNUM,BIGNUM]
|
|
*
|
|
BNRM = CLANGE( 'M', N, N, B, LDB, RWORK )
|
|
ILBSCL = .FALSE.
|
|
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
|
|
BNRMTO = SMLNUM
|
|
ILBSCL = .TRUE.
|
|
ELSE IF( BNRM.GT.BIGNUM ) THEN
|
|
BNRMTO = BIGNUM
|
|
ILBSCL = .TRUE.
|
|
END IF
|
|
IF( ILBSCL )
|
|
$ CALL CLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
|
|
*
|
|
* Permute and/or balance the matrix pair (A,B)
|
|
* (Real Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise)
|
|
*
|
|
CALL CGGBAL( BALANC, N, A, LDA, B, LDB, ILO, IHI, LSCALE,
|
|
$ RSCALE,
|
|
$ RWORK, IERR )
|
|
*
|
|
* Compute ABNRM and BBNRM
|
|
*
|
|
ABNRM = CLANGE( '1', N, N, A, LDA, RWORK( 1 ) )
|
|
IF( ILASCL ) THEN
|
|
RWORK( 1 ) = ABNRM
|
|
CALL SLASCL( 'G', 0, 0, ANRMTO, ANRM, 1, 1, RWORK( 1 ), 1,
|
|
$ IERR )
|
|
ABNRM = RWORK( 1 )
|
|
END IF
|
|
*
|
|
BBNRM = CLANGE( '1', N, N, B, LDB, RWORK( 1 ) )
|
|
IF( ILBSCL ) THEN
|
|
RWORK( 1 ) = BBNRM
|
|
CALL SLASCL( 'G', 0, 0, BNRMTO, BNRM, 1, 1, RWORK( 1 ), 1,
|
|
$ IERR )
|
|
BBNRM = RWORK( 1 )
|
|
END IF
|
|
*
|
|
* Reduce B to triangular form (QR decomposition of B)
|
|
* (Complex Workspace: need N, prefer N*NB )
|
|
*
|
|
IROWS = IHI + 1 - ILO
|
|
IF( ILV .OR. .NOT.WANTSN ) THEN
|
|
ICOLS = N + 1 - ILO
|
|
ELSE
|
|
ICOLS = IROWS
|
|
END IF
|
|
ITAU = 1
|
|
IWRK = ITAU + IROWS
|
|
CALL CGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
|
|
$ WORK( IWRK ), LWORK+1-IWRK, IERR )
|
|
*
|
|
* Apply the unitary transformation to A
|
|
* (Complex Workspace: need N, prefer N*NB)
|
|
*
|
|
CALL CUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
|
|
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
|
|
$ LWORK+1-IWRK, IERR )
|
|
*
|
|
* Initialize VL and/or VR
|
|
* (Workspace: need N, prefer N*NB)
|
|
*
|
|
IF( ILVL ) THEN
|
|
CALL CLASET( 'Full', N, N, CZERO, CONE, VL, LDVL )
|
|
IF( IROWS.GT.1 ) THEN
|
|
CALL CLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
|
|
$ VL( ILO+1, ILO ), LDVL )
|
|
END IF
|
|
CALL CUNGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
|
|
$ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
|
|
END IF
|
|
*
|
|
IF( ILVR )
|
|
$ CALL CLASET( 'Full', N, N, CZERO, CONE, VR, LDVR )
|
|
*
|
|
* Reduce to generalized Hessenberg form
|
|
* (Workspace: none needed)
|
|
*
|
|
IF( ILV .OR. .NOT.WANTSN ) THEN
|
|
*
|
|
* Eigenvectors requested -- work on whole matrix.
|
|
*
|
|
CALL CGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
|
|
$ LDVL, VR, LDVR, IERR )
|
|
ELSE
|
|
CALL CGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
|
|
$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR )
|
|
END IF
|
|
*
|
|
* Perform QZ algorithm (Compute eigenvalues, and optionally, the
|
|
* Schur forms and Schur vectors)
|
|
* (Complex Workspace: need N)
|
|
* (Real Workspace: need N)
|
|
*
|
|
IWRK = ITAU
|
|
IF( ILV .OR. .NOT.WANTSN ) THEN
|
|
CHTEMP = 'S'
|
|
ELSE
|
|
CHTEMP = 'E'
|
|
END IF
|
|
*
|
|
CALL CHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
|
|
$ ALPHA, BETA, VL, LDVL, VR, LDVR, WORK( IWRK ),
|
|
$ LWORK+1-IWRK, RWORK, IERR )
|
|
IF( IERR.NE.0 ) THEN
|
|
IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
|
|
INFO = IERR
|
|
ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
|
|
INFO = IERR - N
|
|
ELSE
|
|
INFO = N + 1
|
|
END IF
|
|
GO TO 90
|
|
END IF
|
|
*
|
|
* Compute Eigenvectors and estimate condition numbers if desired
|
|
* CTGEVC: (Complex Workspace: need 2*N )
|
|
* (Real Workspace: need 2*N )
|
|
* CTGSNA: (Complex Workspace: need 2*N*N if SENSE='V' or 'B')
|
|
* (Integer Workspace: need N+2 )
|
|
*
|
|
IF( ILV .OR. .NOT.WANTSN ) THEN
|
|
IF( ILV ) THEN
|
|
IF( ILVL ) THEN
|
|
IF( ILVR ) THEN
|
|
CHTEMP = 'B'
|
|
ELSE
|
|
CHTEMP = 'L'
|
|
END IF
|
|
ELSE
|
|
CHTEMP = 'R'
|
|
END IF
|
|
*
|
|
CALL CTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL,
|
|
$ LDVL, VR, LDVR, N, IN, WORK( IWRK ), RWORK,
|
|
$ IERR )
|
|
IF( IERR.NE.0 ) THEN
|
|
INFO = N + 2
|
|
GO TO 90
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( .NOT.WANTSN ) THEN
|
|
*
|
|
* compute eigenvectors (CTGEVC) and estimate condition
|
|
* numbers (CTGSNA). Note that the definition of the condition
|
|
* number is not invariant under transformation (u,v) to
|
|
* (Q*u, Z*v), where (u,v) are eigenvectors of the generalized
|
|
* Schur form (S,T), Q and Z are orthogonal matrices. In order
|
|
* to avoid using extra 2*N*N workspace, we have to
|
|
* re-calculate eigenvectors and estimate the condition numbers
|
|
* one at a time.
|
|
*
|
|
DO 20 I = 1, N
|
|
*
|
|
DO 10 J = 1, N
|
|
BWORK( J ) = .FALSE.
|
|
10 CONTINUE
|
|
BWORK( I ) = .TRUE.
|
|
*
|
|
IWRK = N + 1
|
|
IWRK1 = IWRK + N
|
|
*
|
|
IF( WANTSE .OR. WANTSB ) THEN
|
|
CALL CTGEVC( 'B', 'S', BWORK, N, A, LDA, B, LDB,
|
|
$ WORK( 1 ), N, WORK( IWRK ), N, 1, M,
|
|
$ WORK( IWRK1 ), RWORK, IERR )
|
|
IF( IERR.NE.0 ) THEN
|
|
INFO = N + 2
|
|
GO TO 90
|
|
END IF
|
|
END IF
|
|
*
|
|
CALL CTGSNA( SENSE, 'S', BWORK, N, A, LDA, B, LDB,
|
|
$ WORK( 1 ), N, WORK( IWRK ), N, RCONDE( I ),
|
|
$ RCONDV( I ), 1, M, WORK( IWRK1 ),
|
|
$ LWORK-IWRK1+1, IWORK, IERR )
|
|
*
|
|
20 CONTINUE
|
|
END IF
|
|
END IF
|
|
*
|
|
* Undo balancing on VL and VR and normalization
|
|
* (Workspace: none needed)
|
|
*
|
|
IF( ILVL ) THEN
|
|
CALL CGGBAK( BALANC, 'L', N, ILO, IHI, LSCALE, RSCALE, N,
|
|
$ VL,
|
|
$ LDVL, IERR )
|
|
*
|
|
DO 50 JC = 1, N
|
|
TEMP = ZERO
|
|
DO 30 JR = 1, N
|
|
TEMP = MAX( TEMP, CABS1( VL( JR, JC ) ) )
|
|
30 CONTINUE
|
|
IF( TEMP.LT.SMLNUM )
|
|
$ GO TO 50
|
|
TEMP = ONE / TEMP
|
|
DO 40 JR = 1, N
|
|
VL( JR, JC ) = VL( JR, JC )*TEMP
|
|
40 CONTINUE
|
|
50 CONTINUE
|
|
END IF
|
|
*
|
|
IF( ILVR ) THEN
|
|
CALL CGGBAK( BALANC, 'R', N, ILO, IHI, LSCALE, RSCALE, N,
|
|
$ VR,
|
|
$ LDVR, IERR )
|
|
DO 80 JC = 1, N
|
|
TEMP = ZERO
|
|
DO 60 JR = 1, N
|
|
TEMP = MAX( TEMP, CABS1( VR( JR, JC ) ) )
|
|
60 CONTINUE
|
|
IF( TEMP.LT.SMLNUM )
|
|
$ GO TO 80
|
|
TEMP = ONE / TEMP
|
|
DO 70 JR = 1, N
|
|
VR( JR, JC ) = VR( JR, JC )*TEMP
|
|
70 CONTINUE
|
|
80 CONTINUE
|
|
END IF
|
|
*
|
|
* Undo scaling if necessary
|
|
*
|
|
90 CONTINUE
|
|
*
|
|
IF( ILASCL )
|
|
$ CALL CLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHA, N, IERR )
|
|
*
|
|
IF( ILBSCL )
|
|
$ CALL CLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
|
|
*
|
|
WORK( 1 ) = SROUNDUP_LWORK(MAXWRK)
|
|
RETURN
|
|
*
|
|
* End of CGGEVX
|
|
*
|
|
END
|