821 lines
27 KiB
Fortran
821 lines
27 KiB
Fortran
*> \brief <b> SGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors 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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*> \htmlonly
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*> Download SGGESX + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sggesx.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/sggesx.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/sggesx.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 SGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA,
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* B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL,
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* VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK,
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* LIWORK, BWORK, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER JOBVSL, JOBVSR, SENSE, SORT
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* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N,
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* $ SDIM
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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 A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
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* $ B( LDB, * ), BETA( * ), RCONDE( 2 ),
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* $ RCONDV( 2 ), VSL( LDVSL, * ), VSR( LDVSR, * ),
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* $ WORK( * )
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* ..
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* .. Function Arguments ..
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* LOGICAL SELCTG
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* EXTERNAL SELCTG
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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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*> SGGESX computes for a pair of N-by-N real nonsymmetric matrices
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*> (A,B), the generalized eigenvalues, the real Schur form (S,T), and,
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*> optionally, the left and/or right matrices of Schur vectors (VSL and
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*> VSR). This gives the generalized Schur factorization
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*>
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*> (A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T )
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*>
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*> Optionally, it also orders the eigenvalues so that a selected cluster
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*> of eigenvalues appears in the leading diagonal blocks of the upper
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*> quasi-triangular matrix S and the upper triangular matrix T; computes
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*> a reciprocal condition number for the average of the selected
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*> eigenvalues (RCONDE); and computes a reciprocal condition number for
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*> the right and left deflating subspaces corresponding to the selected
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*> eigenvalues (RCONDV). The leading columns of VSL and VSR then form
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*> an orthonormal basis for the corresponding left and right eigenspaces
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*> (deflating subspaces).
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*>
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*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
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*> or a ratio alpha/beta = w, such that A - w*B is singular. It is
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*> usually represented as the pair (alpha,beta), as there is a
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*> reasonable interpretation for beta=0 or for both being zero.
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*>
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*> A pair of matrices (S,T) is in generalized real Schur form if T is
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*> upper triangular with non-negative diagonal and S is block upper
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*> triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond
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*> to real generalized eigenvalues, while 2-by-2 blocks of S will be
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*> "standardized" by making the corresponding elements of T have the
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*> form:
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*> [ a 0 ]
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*> [ 0 b ]
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*>
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*> and the pair of corresponding 2-by-2 blocks in S and T will have a
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*> complex conjugate pair of generalized eigenvalues.
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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] JOBVSL
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*> \verbatim
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*> JOBVSL is CHARACTER*1
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*> = 'N': do not compute the left Schur vectors;
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*> = 'V': compute the left Schur vectors.
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*> \endverbatim
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*>
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*> \param[in] JOBVSR
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*> \verbatim
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*> JOBVSR is CHARACTER*1
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*> = 'N': do not compute the right Schur vectors;
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*> = 'V': compute the right Schur vectors.
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*> \endverbatim
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*>
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*> \param[in] SORT
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*> \verbatim
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*> SORT is CHARACTER*1
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*> Specifies whether or not to order the eigenvalues on the
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*> diagonal of the generalized Schur form.
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*> = 'N': Eigenvalues are not ordered;
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*> = 'S': Eigenvalues are ordered (see SELCTG).
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*> \endverbatim
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*>
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*> \param[in] SELCTG
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*> \verbatim
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*> SELCTG is procedure) LOGICAL FUNCTION of three REAL arguments
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*> SELCTG must be declared EXTERNAL in the calling subroutine.
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*> If SORT = 'N', SELCTG is not referenced.
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*> If SORT = 'S', SELCTG is used to select eigenvalues to sort
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*> to the top left of the Schur form.
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*> An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
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*> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
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*> one of a complex conjugate pair of eigenvalues is selected,
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*> then both complex eigenvalues are selected.
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*> Note that a selected complex eigenvalue may no longer satisfy
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*> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering,
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*> since ordering may change the value of complex eigenvalues
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*> (especially if the eigenvalue is ill-conditioned), in this
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*> case INFO is set to N+3.
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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 average of selected eigenvalues only;
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*> = 'V' : Computed for selected deflating subspaces only;
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*> = 'B' : Computed for both.
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*> If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.
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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, VSL, and VSR. 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 REAL array, dimension (LDA, N)
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*> On entry, the first of the pair of matrices.
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*> On exit, A has been overwritten by its generalized Schur
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*> form S.
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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 REAL array, dimension (LDB, N)
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*> On entry, the second of the pair of matrices.
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*> On exit, B has been overwritten by its generalized Schur
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*> form T.
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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] SDIM
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*> \verbatim
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*> SDIM is INTEGER
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*> If SORT = 'N', SDIM = 0.
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*> If SORT = 'S', SDIM = number of eigenvalues (after sorting)
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*> for which SELCTG is true. (Complex conjugate pairs for which
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*> SELCTG is true for either eigenvalue count as 2.)
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*> \endverbatim
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*>
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*> \param[out] ALPHAR
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*> \verbatim
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*> ALPHAR is REAL array, dimension (N)
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*> \endverbatim
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*>
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*> \param[out] ALPHAI
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*> \verbatim
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*> ALPHAI is REAL 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 REAL array, dimension (N)
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*> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
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*> be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i
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*> and BETA(j),j=1,...,N are the diagonals of the complex Schur
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*> form (S,T) that would result if the 2-by-2 diagonal blocks of
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*> the real Schur form of (A,B) were further reduced to
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*> triangular form using 2-by-2 complex unitary transformations.
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*> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
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*> positive, then the j-th and (j+1)-st eigenvalues are a
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*> complex conjugate pair, with ALPHAI(j+1) negative.
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*>
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*> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
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*> may easily over- or underflow, and BETA(j) may even be zero.
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*> Thus, the user should avoid naively computing the ratio.
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*> However, ALPHAR and ALPHAI will be always less than and
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*> usually comparable with norm(A) in magnitude, and BETA always
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*> less than and usually comparable with norm(B).
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*> \endverbatim
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*>
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*> \param[out] VSL
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*> \verbatim
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*> VSL is REAL array, dimension (LDVSL,N)
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*> If JOBVSL = 'V', VSL will contain the left Schur vectors.
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*> Not referenced if JOBVSL = 'N'.
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*> \endverbatim
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*>
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*> \param[in] LDVSL
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*> \verbatim
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*> LDVSL is INTEGER
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*> The leading dimension of the matrix VSL. LDVSL >=1, and
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*> if JOBVSL = 'V', LDVSL >= N.
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*> \endverbatim
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*>
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*> \param[out] VSR
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*> \verbatim
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*> VSR is REAL array, dimension (LDVSR,N)
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*> If JOBVSR = 'V', VSR will contain the right Schur vectors.
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*> Not referenced if JOBVSR = 'N'.
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*> \endverbatim
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*>
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*> \param[in] LDVSR
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*> \verbatim
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*> LDVSR is INTEGER
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*> The leading dimension of the matrix VSR. LDVSR >= 1, and
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*> if JOBVSR = 'V', LDVSR >= N.
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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 ( 2 )
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*> If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
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*> reciprocal condition numbers for the average of the selected
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*> eigenvalues.
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*> Not referenced if SENSE = 'N' or 'V'.
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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 ( 2 )
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*> If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
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*> reciprocal condition numbers for the selected deflating
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*> subspaces.
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*> Not referenced if SENSE = 'N' or 'E'.
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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 REAL 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.
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*> If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
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*> LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else
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*> LWORK >= max( 8*N, 6*N+16 ).
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*> Note that 2*SDIM*(N-SDIM) <= N*N/2.
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*> Note also that an error is only returned if
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*> LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B'
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*> this may not be large enough.
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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 bound on the optimal size of the WORK
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*> array and the minimum size of the IWORK array, returns these
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*> values as the first entries of the WORK and IWORK arrays, and
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*> no error message related to LWORK or LIWORK is issued by
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*> XERBLA.
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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 (MAX(1,LIWORK))
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*> On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
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*> \endverbatim
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*>
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*> \param[in] LIWORK
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*> \verbatim
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*> LIWORK is INTEGER
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*> The dimension of the array IWORK.
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*> If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
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*> LIWORK >= N+6.
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*>
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*> If LIWORK = -1, then a workspace query is assumed; the
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*> routine only calculates the bound on the optimal size of the
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*> WORK array and the minimum size of the IWORK array, returns
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*> these values as the first entries of the WORK and IWORK
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*> arrays, and no error message related to LWORK or LIWORK is
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*> issued by XERBLA.
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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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*> Not referenced if SORT = 'N'.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit
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*> < 0: if INFO = -i, the i-th argument had an illegal value.
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*> = 1,...,N:
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*> The QZ iteration failed. (A,B) are not in Schur
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*> form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
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*> be correct for j=INFO+1,...,N.
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*> > N: =N+1: other than QZ iteration failed in SHGEQZ
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*> =N+2: after reordering, roundoff changed values of
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*> some complex eigenvalues so that leading
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*> eigenvalues in the Generalized Schur form no
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*> longer satisfy SELCTG=.TRUE. This could also
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*> be caused due to scaling.
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*> =N+3: reordering failed in STGSEN.
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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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*> \date November 2011
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*
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*> \ingroup realGEeigen
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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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*> An approximate (asymptotic) bound on the average absolute error of
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*> the selected eigenvalues is
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*>
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*> EPS * norm((A, B)) / RCONDE( 1 ).
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*>
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*> An approximate (asymptotic) bound on the maximum angular error in
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*> the computed deflating subspaces is
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*>
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*> EPS * norm((A, B)) / RCONDV( 2 ).
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*>
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*> See LAPACK User's Guide, section 4.11 for more information.
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*> \endverbatim
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*>
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* =====================================================================
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SUBROUTINE SGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA,
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$ B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL,
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$ VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK,
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$ LIWORK, BWORK, INFO )
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*
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* -- LAPACK driver routine (version 3.4.0) --
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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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* November 2011
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*
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* .. Scalar Arguments ..
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CHARACTER JOBVSL, JOBVSR, SENSE, SORT
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INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N,
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$ SDIM
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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 A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
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$ B( LDB, * ), BETA( * ), RCONDE( 2 ),
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$ RCONDV( 2 ), VSL( LDVSL, * ), VSR( LDVSR, * ),
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$ WORK( * )
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* ..
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* .. Function Arguments ..
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LOGICAL SELCTG
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EXTERNAL SELCTG
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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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* ..
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* .. Local Scalars ..
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LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
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$ LQUERY, LST2SL, WANTSB, WANTSE, WANTSN, WANTST,
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$ WANTSV
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INTEGER I, ICOLS, IERR, IHI, IJOB, IJOBVL, IJOBVR,
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$ ILEFT, ILO, IP, IRIGHT, IROWS, ITAU, IWRK,
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$ LIWMIN, LWRK, MAXWRK, MINWRK
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REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PL,
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$ PR, SAFMAX, SAFMIN, SMLNUM
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* ..
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* .. Local Arrays ..
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REAL DIF( 2 )
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* ..
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* .. External Subroutines ..
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EXTERNAL SGEQRF, SGGBAK, SGGBAL, SGGHRD, SHGEQZ, SLABAD,
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$ SLACPY, SLASCL, SLASET, SORGQR, SORMQR, STGSEN,
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$ 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 SLAMCH, SLANGE
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EXTERNAL LSAME, ILAENV, SLAMCH, SLANGE
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX, SQRT
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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( JOBVSL, 'N' ) ) THEN
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IJOBVL = 1
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ILVSL = .FALSE.
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ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
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IJOBVL = 2
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ILVSL = .TRUE.
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ELSE
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IJOBVL = -1
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ILVSL = .FALSE.
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END IF
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*
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IF( LSAME( JOBVSR, 'N' ) ) THEN
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IJOBVR = 1
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ILVSR = .FALSE.
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ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
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IJOBVR = 2
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ILVSR = .TRUE.
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ELSE
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IJOBVR = -1
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ILVSR = .FALSE.
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END IF
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*
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WANTST = LSAME( SORT, 'S' )
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WANTSN = LSAME( SENSE, 'N' )
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WANTSE = LSAME( SENSE, 'E' )
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WANTSV = LSAME( SENSE, 'V' )
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WANTSB = LSAME( SENSE, 'B' )
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LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
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IF( WANTSN ) THEN
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IJOB = 0
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ELSE IF( WANTSE ) THEN
|
|
IJOB = 1
|
|
ELSE IF( WANTSV ) THEN
|
|
IJOB = 2
|
|
ELSE IF( WANTSB ) THEN
|
|
IJOB = 4
|
|
END IF
|
|
*
|
|
* Test the input arguments
|
|
*
|
|
INFO = 0
|
|
IF( IJOBVL.LE.0 ) THEN
|
|
INFO = -1
|
|
ELSE IF( IJOBVR.LE.0 ) THEN
|
|
INFO = -2
|
|
ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
|
|
INFO = -3
|
|
ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSV .OR. WANTSB ) .OR.
|
|
$ ( .NOT.WANTST .AND. .NOT.WANTSN ) ) THEN
|
|
INFO = -5
|
|
ELSE IF( N.LT.0 ) THEN
|
|
INFO = -6
|
|
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
|
INFO = -8
|
|
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
|
|
INFO = -10
|
|
ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
|
|
INFO = -16
|
|
ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
|
|
INFO = -18
|
|
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.)
|
|
*
|
|
IF( INFO.EQ.0 ) THEN
|
|
IF( N.GT.0) THEN
|
|
MINWRK = MAX( 8*N, 6*N + 16 )
|
|
MAXWRK = MINWRK - N +
|
|
$ N*ILAENV( 1, 'SGEQRF', ' ', N, 1, N, 0 )
|
|
MAXWRK = MAX( MAXWRK, MINWRK - N +
|
|
$ N*ILAENV( 1, 'SORMQR', ' ', N, 1, N, -1 ) )
|
|
IF( ILVSL ) THEN
|
|
MAXWRK = MAX( MAXWRK, MINWRK - N +
|
|
$ N*ILAENV( 1, 'SORGQR', ' ', N, 1, N, -1 ) )
|
|
END IF
|
|
LWRK = MAXWRK
|
|
IF( IJOB.GE.1 )
|
|
$ LWRK = MAX( LWRK, N*N/2 )
|
|
ELSE
|
|
MINWRK = 1
|
|
MAXWRK = 1
|
|
LWRK = 1
|
|
END IF
|
|
WORK( 1 ) = LWRK
|
|
IF( WANTSN .OR. N.EQ.0 ) THEN
|
|
LIWMIN = 1
|
|
ELSE
|
|
LIWMIN = N + 6
|
|
END IF
|
|
IWORK( 1 ) = LIWMIN
|
|
*
|
|
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
|
|
INFO = -22
|
|
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
|
|
INFO = -24
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'SGGESX', -INFO )
|
|
RETURN
|
|
ELSE IF (LQUERY) THEN
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Quick return if possible
|
|
*
|
|
IF( N.EQ.0 ) THEN
|
|
SDIM = 0
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Get machine constants
|
|
*
|
|
EPS = SLAMCH( 'P' )
|
|
SAFMIN = SLAMCH( 'S' )
|
|
SAFMAX = ONE / SAFMIN
|
|
CALL SLABAD( SAFMIN, SAFMAX )
|
|
SMLNUM = SQRT( SAFMIN ) / EPS
|
|
BIGNUM = ONE / SMLNUM
|
|
*
|
|
* Scale A if max element outside range [SMLNUM,BIGNUM]
|
|
*
|
|
ANRM = SLANGE( 'M', N, N, A, LDA, WORK )
|
|
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 SLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
|
|
*
|
|
* Scale B if max element outside range [SMLNUM,BIGNUM]
|
|
*
|
|
BNRM = SLANGE( 'M', N, N, B, LDB, WORK )
|
|
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 SLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
|
|
*
|
|
* Permute the matrix to make it more nearly triangular
|
|
* (Workspace: need 6*N + 2*N for permutation parameters)
|
|
*
|
|
ILEFT = 1
|
|
IRIGHT = N + 1
|
|
IWRK = IRIGHT + N
|
|
CALL SGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
|
|
$ WORK( IRIGHT ), WORK( IWRK ), IERR )
|
|
*
|
|
* Reduce B to triangular form (QR decomposition of B)
|
|
* (Workspace: need N, prefer N*NB)
|
|
*
|
|
IROWS = IHI + 1 - ILO
|
|
ICOLS = N + 1 - ILO
|
|
ITAU = IWRK
|
|
IWRK = ITAU + IROWS
|
|
CALL SGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
|
|
$ WORK( IWRK ), LWORK+1-IWRK, IERR )
|
|
*
|
|
* Apply the orthogonal transformation to matrix A
|
|
* (Workspace: need N, prefer N*NB)
|
|
*
|
|
CALL SORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
|
|
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
|
|
$ LWORK+1-IWRK, IERR )
|
|
*
|
|
* Initialize VSL
|
|
* (Workspace: need N, prefer N*NB)
|
|
*
|
|
IF( ILVSL ) THEN
|
|
CALL SLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL )
|
|
IF( IROWS.GT.1 ) THEN
|
|
CALL SLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
|
|
$ VSL( ILO+1, ILO ), LDVSL )
|
|
END IF
|
|
CALL SORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
|
|
$ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
|
|
END IF
|
|
*
|
|
* Initialize VSR
|
|
*
|
|
IF( ILVSR )
|
|
$ CALL SLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR )
|
|
*
|
|
* Reduce to generalized Hessenberg form
|
|
* (Workspace: none needed)
|
|
*
|
|
CALL SGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
|
|
$ LDVSL, VSR, LDVSR, IERR )
|
|
*
|
|
SDIM = 0
|
|
*
|
|
* Perform QZ algorithm, computing Schur vectors if desired
|
|
* (Workspace: need N)
|
|
*
|
|
IWRK = ITAU
|
|
CALL SHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
|
|
$ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
|
|
$ WORK( IWRK ), LWORK+1-IWRK, 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 50
|
|
END IF
|
|
*
|
|
* Sort eigenvalues ALPHA/BETA and compute the reciprocal of
|
|
* condition number(s)
|
|
* (Workspace: If IJOB >= 1, need MAX( 8*(N+1), 2*SDIM*(N-SDIM) )
|
|
* otherwise, need 8*(N+1) )
|
|
*
|
|
IF( WANTST ) THEN
|
|
*
|
|
* Undo scaling on eigenvalues before SELCTGing
|
|
*
|
|
IF( ILASCL ) THEN
|
|
CALL SLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N,
|
|
$ IERR )
|
|
CALL SLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N,
|
|
$ IERR )
|
|
END IF
|
|
IF( ILBSCL )
|
|
$ CALL SLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
|
|
*
|
|
* Select eigenvalues
|
|
*
|
|
DO 10 I = 1, N
|
|
BWORK( I ) = SELCTG( ALPHAR( I ), ALPHAI( I ), BETA( I ) )
|
|
10 CONTINUE
|
|
*
|
|
* Reorder eigenvalues, transform Generalized Schur vectors, and
|
|
* compute reciprocal condition numbers
|
|
*
|
|
CALL STGSEN( IJOB, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB,
|
|
$ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
|
|
$ SDIM, PL, PR, DIF, WORK( IWRK ), LWORK-IWRK+1,
|
|
$ IWORK, LIWORK, IERR )
|
|
*
|
|
IF( IJOB.GE.1 )
|
|
$ MAXWRK = MAX( MAXWRK, 2*SDIM*( N-SDIM ) )
|
|
IF( IERR.EQ.-22 ) THEN
|
|
*
|
|
* not enough real workspace
|
|
*
|
|
INFO = -22
|
|
ELSE
|
|
IF( IJOB.EQ.1 .OR. IJOB.EQ.4 ) THEN
|
|
RCONDE( 1 ) = PL
|
|
RCONDE( 2 ) = PR
|
|
END IF
|
|
IF( IJOB.EQ.2 .OR. IJOB.EQ.4 ) THEN
|
|
RCONDV( 1 ) = DIF( 1 )
|
|
RCONDV( 2 ) = DIF( 2 )
|
|
END IF
|
|
IF( IERR.EQ.1 )
|
|
$ INFO = N + 3
|
|
END IF
|
|
*
|
|
END IF
|
|
*
|
|
* Apply permutation to VSL and VSR
|
|
* (Workspace: none needed)
|
|
*
|
|
IF( ILVSL )
|
|
$ CALL SGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
|
|
$ WORK( IRIGHT ), N, VSL, LDVSL, IERR )
|
|
*
|
|
IF( ILVSR )
|
|
$ CALL SGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
|
|
$ WORK( IRIGHT ), N, VSR, LDVSR, IERR )
|
|
*
|
|
* Check if unscaling would cause over/underflow, if so, rescale
|
|
* (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
|
|
* B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
|
|
*
|
|
IF( ILASCL ) THEN
|
|
DO 20 I = 1, N
|
|
IF( ALPHAI( I ).NE.ZERO ) THEN
|
|
IF( ( ALPHAR( I ) / SAFMAX ).GT.( ANRMTO / ANRM ) .OR.
|
|
$ ( SAFMIN / ALPHAR( I ) ).GT.( ANRM / ANRMTO ) )
|
|
$ THEN
|
|
WORK( 1 ) = ABS( A( I, I ) / ALPHAR( I ) )
|
|
BETA( I ) = BETA( I )*WORK( 1 )
|
|
ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
|
|
ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
|
|
ELSE IF( ( ALPHAI( I ) / SAFMAX ).GT.( ANRMTO / ANRM )
|
|
$ .OR. ( SAFMIN / ALPHAI( I ) ).GT.( ANRM / ANRMTO ) )
|
|
$ THEN
|
|
WORK( 1 ) = ABS( A( I, I+1 ) / ALPHAI( I ) )
|
|
BETA( I ) = BETA( I )*WORK( 1 )
|
|
ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
|
|
ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
|
|
END IF
|
|
END IF
|
|
20 CONTINUE
|
|
END IF
|
|
*
|
|
IF( ILBSCL ) THEN
|
|
DO 25 I = 1, N
|
|
IF( ALPHAI( I ).NE.ZERO ) THEN
|
|
IF( ( BETA( I ) / SAFMAX ).GT.( BNRMTO / BNRM ) .OR.
|
|
$ ( SAFMIN / BETA( I ) ).GT.( BNRM / BNRMTO ) ) THEN
|
|
WORK( 1 ) = ABS( B( I, I ) / BETA( I ) )
|
|
BETA( I ) = BETA( I )*WORK( 1 )
|
|
ALPHAR( I ) = ALPHAR( I )*WORK( 1 )
|
|
ALPHAI( I ) = ALPHAI( I )*WORK( 1 )
|
|
END IF
|
|
END IF
|
|
25 CONTINUE
|
|
END IF
|
|
*
|
|
* Undo scaling
|
|
*
|
|
IF( ILASCL ) THEN
|
|
CALL SLASCL( 'H', 0, 0, ANRMTO, ANRM, N, N, A, LDA, IERR )
|
|
CALL SLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR )
|
|
CALL SLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR )
|
|
END IF
|
|
*
|
|
IF( ILBSCL ) THEN
|
|
CALL SLASCL( 'U', 0, 0, BNRMTO, BNRM, N, N, B, LDB, IERR )
|
|
CALL SLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
|
|
END IF
|
|
*
|
|
IF( WANTST ) THEN
|
|
*
|
|
* Check if reordering is correct
|
|
*
|
|
LASTSL = .TRUE.
|
|
LST2SL = .TRUE.
|
|
SDIM = 0
|
|
IP = 0
|
|
DO 40 I = 1, N
|
|
CURSL = SELCTG( ALPHAR( I ), ALPHAI( I ), BETA( I ) )
|
|
IF( ALPHAI( I ).EQ.ZERO ) THEN
|
|
IF( CURSL )
|
|
$ SDIM = SDIM + 1
|
|
IP = 0
|
|
IF( CURSL .AND. .NOT.LASTSL )
|
|
$ INFO = N + 2
|
|
ELSE
|
|
IF( IP.EQ.1 ) THEN
|
|
*
|
|
* Last eigenvalue of conjugate pair
|
|
*
|
|
CURSL = CURSL .OR. LASTSL
|
|
LASTSL = CURSL
|
|
IF( CURSL )
|
|
$ SDIM = SDIM + 2
|
|
IP = -1
|
|
IF( CURSL .AND. .NOT.LST2SL )
|
|
$ INFO = N + 2
|
|
ELSE
|
|
*
|
|
* First eigenvalue of conjugate pair
|
|
*
|
|
IP = 1
|
|
END IF
|
|
END IF
|
|
LST2SL = LASTSL
|
|
LASTSL = CURSL
|
|
40 CONTINUE
|
|
*
|
|
END IF
|
|
*
|
|
50 CONTINUE
|
|
*
|
|
WORK( 1 ) = MAXWRK
|
|
IWORK( 1 ) = LIWMIN
|
|
*
|
|
RETURN
|
|
*
|
|
* End of SGGESX
|
|
*
|
|
END
|