Update brief description of function purpose (Reference-LAPACK 831)
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Martin Kroeker
GitHub
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*> \brief <b> CGEGS computes the eigenvalues, Schur form, and, optionally, the left and or/right Schur vectors of a complex matrix pair (A,B)</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 CGEGS + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgegs.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/cgegs.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/cgegs.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 CGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA,
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* VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK,
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* INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER JOBVSL, JOBVSR
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* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
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* ..
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* .. Array Arguments ..
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* REAL RWORK( * )
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* COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
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* $ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
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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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*> This routine is deprecated and has been replaced by routine CGGES.
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*>
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*> CGEGS computes the eigenvalues, Schur form, and, optionally, the
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*> left and or/right Schur vectors of a complex matrix pair (A,B).
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*> Given two square matrices A and B, the generalized Schur
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*> factorization has the form
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*>
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*> A = Q*S*Z**H, B = Q*T*Z**H
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*>
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*> where Q and Z are unitary matrices and S and T are upper triangular.
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*> The columns of Q are the left Schur vectors
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*> and the columns of Z are the right Schur vectors.
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*>
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*> If only the eigenvalues of (A,B) are needed, the driver routine
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*> CGEGV should be used instead. See CGEGV for a description of the
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*> eigenvalues of the generalized nonsymmetric eigenvalue problem
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*> (GNEP).
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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 (returned in VSL).
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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 (returned in VSR).
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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 COMPLEX array, dimension (LDA, N)
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*> On entry, the matrix A.
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*> On exit, the upper triangular matrix S from the generalized
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*> Schur factorization.
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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.
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*> On exit, the upper triangular matrix T from the generalized
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*> Schur factorization.
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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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*> The complex scalars alpha that define the eigenvalues of
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*> GNEP. ALPHA(j) = S(j,j), the diagonal element of the Schur
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*> form of A.
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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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*> The non-negative real scalars beta that define the
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*> eigenvalues of GNEP. BETA(j) = T(j,j), the diagonal element
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*> of the triangular factor T.
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*>
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*> Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
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*> represent the j-th eigenvalue of the matrix pair (A,B), in
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*> one of the forms lambda = alpha/beta or mu = beta/alpha.
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*> Since either lambda or mu may overflow, they should not,
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*> in general, be computed.
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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 COMPLEX array, dimension (LDVSL,N)
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*> If JOBVSL = 'V', the matrix of left Schur vectors Q.
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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 COMPLEX array, dimension (LDVSR,N)
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*> If JOBVSR = 'V', the matrix of right Schur vectors Z.
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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] 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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*> For good performance, LWORK must generally be larger.
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*> To compute the optimal value of LWORK, call ILAENV to get
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*> blocksizes (for CGEQRF, CUNMQR, and CUNGQR.) Then compute:
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*> NB -- MAX of the blocksizes for CGEQRF, CUNMQR, and CUNGQR;
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*> the optimal LWORK is N*(NB+1).
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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 (3*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 ALPHA(j) and BETA(j) should be correct for
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*> j=INFO+1,...,N.
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*> > N: errors that usually indicate LAPACK problems:
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*> =N+1: error return from CGGBAL
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*> =N+2: error return from CGEQRF
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*> =N+3: error return from CUNMQR
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*> =N+4: error return from CUNGQR
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*> =N+5: error return from CGGHRD
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*> =N+6: error return from CHGEQZ (other than failed
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*> iteration)
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*> =N+7: error return from CGGBAK (computing VSL)
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*> =N+8: error return from CGGBAK (computing VSR)
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*> =N+9: error return from CLASCL (various places)
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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 complexGEeigen
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*
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* =====================================================================
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SUBROUTINE CGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA,
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$ VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK,
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$ INFO )
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*
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* -- LAPACK driver routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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*
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* .. Scalar Arguments ..
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CHARACTER JOBVSL, JOBVSR
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INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
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* ..
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* .. Array Arguments ..
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REAL RWORK( * )
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COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
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$ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
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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 = 0.0E0, ONE = 1.0E0 )
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COMPLEX CZERO, CONE
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PARAMETER ( CZERO = ( 0.0E0, 0.0E0 ),
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$ CONE = ( 1.0E0, 0.0E0 ) )
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* ..
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* .. Local Scalars ..
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LOGICAL ILASCL, ILBSCL, ILVSL, ILVSR, LQUERY
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INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT,
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$ ILO, IRIGHT, IROWS, IRWORK, ITAU, IWORK,
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$ LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3
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REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
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$ SAFMIN, SMLNUM
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* ..
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* .. External Subroutines ..
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EXTERNAL CGEQRF, CGGBAK, CGGBAL, CGGHRD, CHGEQZ, CLACPY,
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$ CLASCL, CLASET, CUNGQR, CUNMQR, 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
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EXTERNAL ILAENV, LSAME, CLANGE, SLAMCH
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC INT, MAX
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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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* Test the input arguments
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*
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LWKMIN = MAX( 2*N, 1 )
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LWKOPT = LWKMIN
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WORK( 1 ) = LWKOPT
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LQUERY = ( LWORK.EQ.-1 )
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INFO = 0
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IF( IJOBVL.LE.0 ) THEN
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INFO = -1
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ELSE IF( IJOBVR.LE.0 ) THEN
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INFO = -2
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ELSE IF( N.LT.0 ) THEN
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INFO = -3
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -5
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ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -7
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ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
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INFO = -11
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ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
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INFO = -13
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ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
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INFO = -15
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END IF
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*
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IF( INFO.EQ.0 ) THEN
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NB1 = ILAENV( 1, 'CGEQRF', ' ', N, N, -1, -1 )
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NB2 = ILAENV( 1, 'CUNMQR', ' ', N, N, N, -1 )
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NB3 = ILAENV( 1, 'CUNGQR', ' ', N, N, N, -1 )
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NB = MAX( NB1, NB2, NB3 )
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LOPT = N*(NB+1)
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WORK( 1 ) = LOPT
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END IF
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*
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'CGEGS ', -INFO )
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RETURN
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ELSE IF( LQUERY ) THEN
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RETURN
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END IF
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*
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* Quick return if possible
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*
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IF( N.EQ.0 )
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$ RETURN
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*
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* Get machine constants
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*
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EPS = SLAMCH( 'E' )*SLAMCH( 'B' )
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SAFMIN = SLAMCH( 'S' )
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SMLNUM = N*SAFMIN / EPS
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BIGNUM = ONE / SMLNUM
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*
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* Scale A if max element outside range [SMLNUM,BIGNUM]
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*
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ANRM = CLANGE( 'M', N, N, A, LDA, RWORK )
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ILASCL = .FALSE.
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IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
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ANRMTO = SMLNUM
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ILASCL = .TRUE.
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ELSE IF( ANRM.GT.BIGNUM ) THEN
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ANRMTO = BIGNUM
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ILASCL = .TRUE.
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END IF
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*
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IF( ILASCL ) THEN
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CALL CLASCL( 'G', -1, -1, ANRM, ANRMTO, N, N, A, LDA, IINFO )
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IF( IINFO.NE.0 ) THEN
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INFO = N + 9
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RETURN
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END IF
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END IF
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*
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* Scale B if max element outside range [SMLNUM,BIGNUM]
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*
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BNRM = CLANGE( 'M', N, N, B, LDB, RWORK )
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ILBSCL = .FALSE.
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IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
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BNRMTO = SMLNUM
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ILBSCL = .TRUE.
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ELSE IF( BNRM.GT.BIGNUM ) THEN
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BNRMTO = BIGNUM
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ILBSCL = .TRUE.
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END IF
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*
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IF( ILBSCL ) THEN
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CALL CLASCL( 'G', -1, -1, BNRM, BNRMTO, N, N, B, LDB, IINFO )
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IF( IINFO.NE.0 ) THEN
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INFO = N + 9
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RETURN
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END IF
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END IF
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*
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* Permute the matrix to make it more nearly triangular
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*
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ILEFT = 1
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IRIGHT = N + 1
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IRWORK = IRIGHT + N
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IWORK = 1
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CALL CGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ),
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$ RWORK( IRIGHT ), RWORK( IRWORK ), IINFO )
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IF( IINFO.NE.0 ) THEN
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INFO = N + 1
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GO TO 10
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END IF
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*
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* Reduce B to triangular form, and initialize VSL and/or VSR
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*
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IROWS = IHI + 1 - ILO
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ICOLS = N + 1 - ILO
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ITAU = IWORK
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IWORK = ITAU + IROWS
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CALL CGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
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$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
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IF( IINFO.GE.0 )
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$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
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IF( IINFO.NE.0 ) THEN
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INFO = N + 2
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GO TO 10
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END IF
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*
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CALL CUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
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$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
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$ LWORK+1-IWORK, IINFO )
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IF( IINFO.GE.0 )
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$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
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IF( IINFO.NE.0 ) THEN
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INFO = N + 3
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GO TO 10
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END IF
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*
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IF( ILVSL ) THEN
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CALL CLASET( 'Full', N, N, CZERO, CONE, VSL, LDVSL )
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CALL CLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
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$ VSL( ILO+1, ILO ), LDVSL )
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CALL CUNGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
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$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
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$ IINFO )
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IF( IINFO.GE.0 )
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$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
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IF( IINFO.NE.0 ) THEN
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INFO = N + 4
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GO TO 10
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( ILVSR )
|
||||
$ CALL CLASET( 'Full', N, N, CZERO, CONE, VSR, LDVSR )
|
||||
*
|
||||
* Reduce to generalized Hessenberg form
|
||||
*
|
||||
CALL CGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
|
||||
$ LDVSL, VSR, LDVSR, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 5
|
||||
GO TO 10
|
||||
END IF
|
||||
*
|
||||
* Perform QZ algorithm, computing Schur vectors if desired
|
||||
*
|
||||
IWORK = ITAU
|
||||
CALL CHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
|
||||
$ ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK( IWORK ),
|
||||
$ LWORK+1-IWORK, RWORK( IRWORK ), IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
|
||||
INFO = IINFO
|
||||
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
|
||||
INFO = IINFO - N
|
||||
ELSE
|
||||
INFO = N + 6
|
||||
END IF
|
||||
GO TO 10
|
||||
END IF
|
||||
*
|
||||
* Apply permutation to VSL and VSR
|
||||
*
|
||||
IF( ILVSL ) THEN
|
||||
CALL CGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ),
|
||||
$ RWORK( IRIGHT ), N, VSL, LDVSL, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 7
|
||||
GO TO 10
|
||||
END IF
|
||||
END IF
|
||||
IF( ILVSR ) THEN
|
||||
CALL CGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ),
|
||||
$ RWORK( IRIGHT ), N, VSR, LDVSR, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 8
|
||||
GO TO 10
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Undo scaling
|
||||
*
|
||||
IF( ILASCL ) THEN
|
||||
CALL CLASCL( 'U', -1, -1, ANRMTO, ANRM, N, N, A, LDA, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
CALL CLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHA, N, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( ILBSCL ) THEN
|
||||
CALL CLASCL( 'U', -1, -1, BNRMTO, BNRM, N, N, B, LDB, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
CALL CLASCL( 'G', -1, -1, BNRMTO, BNRM, N, 1, BETA, N, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
10 CONTINUE
|
||||
WORK( 1 ) = LWKOPT
|
||||
*
|
||||
RETURN
|
||||
*
|
||||
* End of CGEGS
|
||||
*
|
||||
END
|
||||
|
|
@ -0,0 +1,703 @@
|
|||
*> \brief <b> CGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a complex matrix pair (A,B).</b>
|
||||
*
|
||||
* =========== DOCUMENTATION ===========
|
||||
*
|
||||
* Online html documentation available at
|
||||
* http://www.netlib.org/lapack/explore-html/
|
||||
*
|
||||
*> \htmlonly
|
||||
*> Download CGEGV + dependencies
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgegv.f">
|
||||
*> [TGZ]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgegv.f">
|
||||
*> [ZIP]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgegv.f">
|
||||
*> [TXT]</a>
|
||||
*> \endhtmlonly
|
||||
*
|
||||
* Definition:
|
||||
* ===========
|
||||
*
|
||||
* SUBROUTINE CGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
|
||||
* VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
* CHARACTER JOBVL, JOBVR
|
||||
* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
* REAL RWORK( * )
|
||||
* COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
|
||||
* $ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
|
||||
* $ WORK( * )
|
||||
* ..
|
||||
*
|
||||
*
|
||||
*> \par Purpose:
|
||||
* =============
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> This routine is deprecated and has been replaced by routine CGGEV.
|
||||
*>
|
||||
*> CGEGV computes the eigenvalues and, optionally, the left and/or right
|
||||
*> eigenvectors of a complex matrix pair (A,B).
|
||||
*> Given two square matrices A and B,
|
||||
*> the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
|
||||
*> eigenvalues lambda and corresponding (non-zero) eigenvectors x such
|
||||
*> that
|
||||
*> A*x = lambda*B*x.
|
||||
*>
|
||||
*> An alternate form is to find the eigenvalues mu and corresponding
|
||||
*> eigenvectors y such that
|
||||
*> mu*A*y = B*y.
|
||||
*>
|
||||
*> These two forms are equivalent with mu = 1/lambda and x = y if
|
||||
*> neither lambda nor mu is zero. In order to deal with the case that
|
||||
*> lambda or mu is zero or small, two values alpha and beta are returned
|
||||
*> for each eigenvalue, such that lambda = alpha/beta and
|
||||
*> mu = beta/alpha.
|
||||
*>
|
||||
*> The vectors x and y in the above equations are right eigenvectors of
|
||||
*> the matrix pair (A,B). Vectors u and v satisfying
|
||||
*> u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
|
||||
*> are left eigenvectors of (A,B).
|
||||
*>
|
||||
*> Note: this routine performs "full balancing" on A and B
|
||||
*> \endverbatim
|
||||
*
|
||||
* Arguments:
|
||||
* ==========
|
||||
*
|
||||
*> \param[in] JOBVL
|
||||
*> \verbatim
|
||||
*> JOBVL is CHARACTER*1
|
||||
*> = 'N': do not compute the left generalized eigenvectors;
|
||||
*> = 'V': compute the left generalized eigenvectors (returned
|
||||
*> in VL).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] JOBVR
|
||||
*> \verbatim
|
||||
*> JOBVR is CHARACTER*1
|
||||
*> = 'N': do not compute the right generalized eigenvectors;
|
||||
*> = 'V': compute the right generalized eigenvectors (returned
|
||||
*> in VR).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] N
|
||||
*> \verbatim
|
||||
*> N is INTEGER
|
||||
*> The order of the matrices A, B, VL, and VR. N >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] A
|
||||
*> \verbatim
|
||||
*> A is COMPLEX array, dimension (LDA, N)
|
||||
*> On entry, the matrix A.
|
||||
*> If JOBVL = 'V' or JOBVR = 'V', then on exit A
|
||||
*> contains the Schur form of A from the generalized Schur
|
||||
*> factorization of the pair (A,B) after balancing. If no
|
||||
*> eigenvectors were computed, then only the diagonal elements
|
||||
*> of the Schur form will be correct. See CGGHRD and CHGEQZ
|
||||
*> for details.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDA
|
||||
*> \verbatim
|
||||
*> LDA is INTEGER
|
||||
*> The leading dimension of A. LDA >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] B
|
||||
*> \verbatim
|
||||
*> B is COMPLEX array, dimension (LDB, N)
|
||||
*> On entry, the matrix B.
|
||||
*> If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
|
||||
*> upper triangular matrix obtained from B in the generalized
|
||||
*> Schur factorization of the pair (A,B) after balancing.
|
||||
*> If no eigenvectors were computed, then only the diagonal
|
||||
*> elements of B will be correct. See CGGHRD and CHGEQZ for
|
||||
*> details.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDB
|
||||
*> \verbatim
|
||||
*> LDB is INTEGER
|
||||
*> The leading dimension of B. LDB >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] ALPHA
|
||||
*> \verbatim
|
||||
*> ALPHA is COMPLEX array, dimension (N)
|
||||
*> The complex scalars alpha that define the eigenvalues of
|
||||
*> GNEP.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] BETA
|
||||
*> \verbatim
|
||||
*> BETA is COMPLEX array, dimension (N)
|
||||
*> The complex scalars beta that define the eigenvalues of GNEP.
|
||||
*>
|
||||
*> Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
|
||||
*> represent the j-th eigenvalue of the matrix pair (A,B), in
|
||||
*> one of the forms lambda = alpha/beta or mu = beta/alpha.
|
||||
*> Since either lambda or mu may overflow, they should not,
|
||||
*> in general, be computed.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] VL
|
||||
*> \verbatim
|
||||
*> VL is COMPLEX array, dimension (LDVL,N)
|
||||
*> If JOBVL = 'V', the left eigenvectors u(j) are stored
|
||||
*> in the columns of VL, in the same order as their eigenvalues.
|
||||
*> Each eigenvector is scaled so that its largest component has
|
||||
*> abs(real part) + abs(imag. part) = 1, except for eigenvectors
|
||||
*> corresponding to an eigenvalue with alpha = beta = 0, which
|
||||
*> are set to zero.
|
||||
*> Not referenced if JOBVL = 'N'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDVL
|
||||
*> \verbatim
|
||||
*> LDVL is INTEGER
|
||||
*> The leading dimension of the matrix VL. LDVL >= 1, and
|
||||
*> if JOBVL = 'V', LDVL >= N.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] VR
|
||||
*> \verbatim
|
||||
*> VR is COMPLEX array, dimension (LDVR,N)
|
||||
*> If JOBVR = 'V', the right eigenvectors x(j) are stored
|
||||
*> in the columns of VR, in the same order as their eigenvalues.
|
||||
*> Each eigenvector is scaled so that its largest component has
|
||||
*> abs(real part) + abs(imag. part) = 1, except for eigenvectors
|
||||
*> corresponding to an eigenvalue with alpha = beta = 0, which
|
||||
*> are set to zero.
|
||||
*> Not referenced if JOBVR = 'N'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDVR
|
||||
*> \verbatim
|
||||
*> LDVR is INTEGER
|
||||
*> The leading dimension of the matrix VR. LDVR >= 1, and
|
||||
*> if JOBVR = 'V', LDVR >= N.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] WORK
|
||||
*> \verbatim
|
||||
*> WORK is COMPLEX array, dimension (MAX(1,LWORK))
|
||||
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LWORK
|
||||
*> \verbatim
|
||||
*> LWORK is INTEGER
|
||||
*> The dimension of the array WORK. LWORK >= max(1,2*N).
|
||||
*> For good performance, LWORK must generally be larger.
|
||||
*> To compute the optimal value of LWORK, call ILAENV to get
|
||||
*> blocksizes (for CGEQRF, CUNMQR, and CUNGQR.) Then compute:
|
||||
*> NB -- MAX of the blocksizes for CGEQRF, CUNMQR, and CUNGQR;
|
||||
*> The optimal LWORK is MAX( 2*N, N*(NB+1) ).
|
||||
*>
|
||||
*> If LWORK = -1, then a workspace query is assumed; the routine
|
||||
*> only calculates the optimal size of the WORK array, returns
|
||||
*> this value as the first entry of the WORK array, and no error
|
||||
*> message related to LWORK is issued by XERBLA.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] RWORK
|
||||
*> \verbatim
|
||||
*> RWORK is REAL array, dimension (8*N)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] INFO
|
||||
*> \verbatim
|
||||
*> INFO is INTEGER
|
||||
*> = 0: successful exit
|
||||
*> < 0: if INFO = -i, the i-th argument had an illegal value.
|
||||
*> =1,...,N:
|
||||
*> The QZ iteration failed. No eigenvectors have been
|
||||
*> calculated, but ALPHA(j) and BETA(j) should be
|
||||
*> correct for j=INFO+1,...,N.
|
||||
*> > N: errors that usually indicate LAPACK problems:
|
||||
*> =N+1: error return from CGGBAL
|
||||
*> =N+2: error return from CGEQRF
|
||||
*> =N+3: error return from CUNMQR
|
||||
*> =N+4: error return from CUNGQR
|
||||
*> =N+5: error return from CGGHRD
|
||||
*> =N+6: error return from CHGEQZ (other than failed
|
||||
*> iteration)
|
||||
*> =N+7: error return from CTGEVC
|
||||
*> =N+8: error return from CGGBAK (computing VL)
|
||||
*> =N+9: error return from CGGBAK (computing VR)
|
||||
*> =N+10: error return from CLASCL (various calls)
|
||||
*> \endverbatim
|
||||
*
|
||||
* Authors:
|
||||
* ========
|
||||
*
|
||||
*> \author Univ. of Tennessee
|
||||
*> \author Univ. of California Berkeley
|
||||
*> \author Univ. of Colorado Denver
|
||||
*> \author NAG Ltd.
|
||||
*
|
||||
*> \ingroup complexGEeigen
|
||||
*
|
||||
*> \par Further Details:
|
||||
* =====================
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> Balancing
|
||||
*> ---------
|
||||
*>
|
||||
*> This driver calls CGGBAL to both permute and scale rows and columns
|
||||
*> of A and B. The permutations PL and PR are chosen so that PL*A*PR
|
||||
*> and PL*B*R will be upper triangular except for the diagonal blocks
|
||||
*> A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
|
||||
*> possible. The diagonal scaling matrices DL and DR are chosen so
|
||||
*> that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
|
||||
*> one (except for the elements that start out zero.)
|
||||
*>
|
||||
*> After the eigenvalues and eigenvectors of the balanced matrices
|
||||
*> have been computed, CGGBAK transforms the eigenvectors back to what
|
||||
*> they would have been (in perfect arithmetic) if they had not been
|
||||
*> balanced.
|
||||
*>
|
||||
*> Contents of A and B on Exit
|
||||
*> -------- -- - --- - -- ----
|
||||
*>
|
||||
*> If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
|
||||
*> both), then on exit the arrays A and B will contain the complex Schur
|
||||
*> form[*] of the "balanced" versions of A and B. If no eigenvectors
|
||||
*> are computed, then only the diagonal blocks will be correct.
|
||||
*>
|
||||
*> [*] In other words, upper triangular form.
|
||||
*> \endverbatim
|
||||
*>
|
||||
* =====================================================================
|
||||
SUBROUTINE CGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
|
||||
$ VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER JOBVL, JOBVR
|
||||
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
REAL RWORK( * )
|
||||
COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
|
||||
$ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
|
||||
$ WORK( * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
REAL ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
|
||||
COMPLEX CZERO, CONE
|
||||
PARAMETER ( CZERO = ( 0.0E0, 0.0E0 ),
|
||||
$ CONE = ( 1.0E0, 0.0E0 ) )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL ILIMIT, ILV, ILVL, ILVR, LQUERY
|
||||
CHARACTER CHTEMP
|
||||
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
|
||||
$ IN, IRIGHT, IROWS, IRWORK, ITAU, IWORK, JC, JR,
|
||||
$ LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3
|
||||
REAL ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM,
|
||||
$ BNRM1, BNRM2, EPS, SAFMAX, SAFMIN, SALFAI,
|
||||
$ SALFAR, SBETA, SCALE, TEMP
|
||||
COMPLEX X
|
||||
* ..
|
||||
* .. Local Arrays ..
|
||||
LOGICAL LDUMMA( 1 )
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL CGEQRF, CGGBAK, CGGBAL, CGGHRD, CHGEQZ, CLACPY,
|
||||
$ CLASCL, CLASET, CTGEVC, CUNGQR, CUNMQR, XERBLA
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
INTEGER ILAENV
|
||||
REAL CLANGE, SLAMCH
|
||||
EXTERNAL ILAENV, LSAME, CLANGE, SLAMCH
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC ABS, AIMAG, CMPLX, INT, MAX, REAL
|
||||
* ..
|
||||
* .. Statement Functions ..
|
||||
REAL ABS1
|
||||
* ..
|
||||
* .. Statement Function definitions ..
|
||||
ABS1( X ) = ABS( REAL( X ) ) + ABS( AIMAG( X ) )
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
* Decode the input arguments
|
||||
*
|
||||
IF( LSAME( JOBVL, 'N' ) ) THEN
|
||||
IJOBVL = 1
|
||||
ILVL = .FALSE.
|
||||
ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
|
||||
IJOBVL = 2
|
||||
ILVL = .TRUE.
|
||||
ELSE
|
||||
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
|
||||
*
|
||||
* Test the input arguments
|
||||
*
|
||||
LWKMIN = MAX( 2*N, 1 )
|
||||
LWKOPT = LWKMIN
|
||||
WORK( 1 ) = LWKOPT
|
||||
LQUERY = ( LWORK.EQ.-1 )
|
||||
INFO = 0
|
||||
IF( IJOBVL.LE.0 ) THEN
|
||||
INFO = -1
|
||||
ELSE IF( IJOBVR.LE.0 ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -5
|
||||
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -7
|
||||
ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
|
||||
INFO = -11
|
||||
ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
|
||||
INFO = -13
|
||||
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
|
||||
INFO = -15
|
||||
END IF
|
||||
*
|
||||
IF( INFO.EQ.0 ) THEN
|
||||
NB1 = ILAENV( 1, 'CGEQRF', ' ', N, N, -1, -1 )
|
||||
NB2 = ILAENV( 1, 'CUNMQR', ' ', N, N, N, -1 )
|
||||
NB3 = ILAENV( 1, 'CUNGQR', ' ', N, N, N, -1 )
|
||||
NB = MAX( NB1, NB2, NB3 )
|
||||
LOPT = MAX( 2*N, N*(NB+1) )
|
||||
WORK( 1 ) = LOPT
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'CGEGV ', -INFO )
|
||||
RETURN
|
||||
ELSE IF( LQUERY ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Quick return if possible
|
||||
*
|
||||
IF( N.EQ.0 )
|
||||
$ RETURN
|
||||
*
|
||||
* Get machine constants
|
||||
*
|
||||
EPS = SLAMCH( 'E' )*SLAMCH( 'B' )
|
||||
SAFMIN = SLAMCH( 'S' )
|
||||
SAFMIN = SAFMIN + SAFMIN
|
||||
SAFMAX = ONE / SAFMIN
|
||||
*
|
||||
* Scale A
|
||||
*
|
||||
ANRM = CLANGE( 'M', N, N, A, LDA, RWORK )
|
||||
ANRM1 = ANRM
|
||||
ANRM2 = ONE
|
||||
IF( ANRM.LT.ONE ) THEN
|
||||
IF( SAFMAX*ANRM.LT.ONE ) THEN
|
||||
ANRM1 = SAFMIN
|
||||
ANRM2 = SAFMAX*ANRM
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( ANRM.GT.ZERO ) THEN
|
||||
CALL CLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 10
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Scale B
|
||||
*
|
||||
BNRM = CLANGE( 'M', N, N, B, LDB, RWORK )
|
||||
BNRM1 = BNRM
|
||||
BNRM2 = ONE
|
||||
IF( BNRM.LT.ONE ) THEN
|
||||
IF( SAFMAX*BNRM.LT.ONE ) THEN
|
||||
BNRM1 = SAFMIN
|
||||
BNRM2 = SAFMAX*BNRM
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( BNRM.GT.ZERO ) THEN
|
||||
CALL CLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 10
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Permute the matrix to make it more nearly triangular
|
||||
* Also "balance" the matrix.
|
||||
*
|
||||
ILEFT = 1
|
||||
IRIGHT = N + 1
|
||||
IRWORK = IRIGHT + N
|
||||
CALL CGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ),
|
||||
$ RWORK( IRIGHT ), RWORK( IRWORK ), IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 1
|
||||
GO TO 80
|
||||
END IF
|
||||
*
|
||||
* Reduce B to triangular form, and initialize VL and/or VR
|
||||
*
|
||||
IROWS = IHI + 1 - ILO
|
||||
IF( ILV ) THEN
|
||||
ICOLS = N + 1 - ILO
|
||||
ELSE
|
||||
ICOLS = IROWS
|
||||
END IF
|
||||
ITAU = 1
|
||||
IWORK = ITAU + IROWS
|
||||
CALL CGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
|
||||
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 2
|
||||
GO TO 80
|
||||
END IF
|
||||
*
|
||||
CALL CUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
|
||||
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
|
||||
$ LWORK+1-IWORK, IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 3
|
||||
GO TO 80
|
||||
END IF
|
||||
*
|
||||
IF( ILVL ) THEN
|
||||
CALL CLASET( 'Full', N, N, CZERO, CONE, VL, LDVL )
|
||||
CALL CLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
|
||||
$ VL( ILO+1, ILO ), LDVL )
|
||||
CALL CUNGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
|
||||
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
|
||||
$ IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 4
|
||||
GO TO 80
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( ILVR )
|
||||
$ CALL CLASET( 'Full', N, N, CZERO, CONE, VR, LDVR )
|
||||
*
|
||||
* Reduce to generalized Hessenberg form
|
||||
*
|
||||
IF( ILV ) THEN
|
||||
*
|
||||
* Eigenvectors requested -- work on whole matrix.
|
||||
*
|
||||
CALL CGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
|
||||
$ LDVL, VR, LDVR, IINFO )
|
||||
ELSE
|
||||
CALL CGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
|
||||
$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO )
|
||||
END IF
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 5
|
||||
GO TO 80
|
||||
END IF
|
||||
*
|
||||
* Perform QZ algorithm
|
||||
*
|
||||
IWORK = ITAU
|
||||
IF( ILV ) 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( IWORK ),
|
||||
$ LWORK+1-IWORK, RWORK( IRWORK ), IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
|
||||
INFO = IINFO
|
||||
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
|
||||
INFO = IINFO - N
|
||||
ELSE
|
||||
INFO = N + 6
|
||||
END IF
|
||||
GO TO 80
|
||||
END IF
|
||||
*
|
||||
IF( ILV ) THEN
|
||||
*
|
||||
* Compute Eigenvectors
|
||||
*
|
||||
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( IWORK ), RWORK( IRWORK ),
|
||||
$ IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 7
|
||||
GO TO 80
|
||||
END IF
|
||||
*
|
||||
* Undo balancing on VL and VR, rescale
|
||||
*
|
||||
IF( ILVL ) THEN
|
||||
CALL CGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ),
|
||||
$ RWORK( IRIGHT ), N, VL, LDVL, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 8
|
||||
GO TO 80
|
||||
END IF
|
||||
DO 30 JC = 1, N
|
||||
TEMP = ZERO
|
||||
DO 10 JR = 1, N
|
||||
TEMP = MAX( TEMP, ABS1( VL( JR, JC ) ) )
|
||||
10 CONTINUE
|
||||
IF( TEMP.LT.SAFMIN )
|
||||
$ GO TO 30
|
||||
TEMP = ONE / TEMP
|
||||
DO 20 JR = 1, N
|
||||
VL( JR, JC ) = VL( JR, JC )*TEMP
|
||||
20 CONTINUE
|
||||
30 CONTINUE
|
||||
END IF
|
||||
IF( ILVR ) THEN
|
||||
CALL CGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ),
|
||||
$ RWORK( IRIGHT ), N, VR, LDVR, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
GO TO 80
|
||||
END IF
|
||||
DO 60 JC = 1, N
|
||||
TEMP = ZERO
|
||||
DO 40 JR = 1, N
|
||||
TEMP = MAX( TEMP, ABS1( VR( JR, JC ) ) )
|
||||
40 CONTINUE
|
||||
IF( TEMP.LT.SAFMIN )
|
||||
$ GO TO 60
|
||||
TEMP = ONE / TEMP
|
||||
DO 50 JR = 1, N
|
||||
VR( JR, JC ) = VR( JR, JC )*TEMP
|
||||
50 CONTINUE
|
||||
60 CONTINUE
|
||||
END IF
|
||||
*
|
||||
* End of eigenvector calculation
|
||||
*
|
||||
END IF
|
||||
*
|
||||
* Undo scaling in alpha, beta
|
||||
*
|
||||
* Note: this does not give the alpha and beta for the unscaled
|
||||
* problem.
|
||||
*
|
||||
* Un-scaling is limited to avoid underflow in alpha and beta
|
||||
* if they are significant.
|
||||
*
|
||||
DO 70 JC = 1, N
|
||||
ABSAR = ABS( REAL( ALPHA( JC ) ) )
|
||||
ABSAI = ABS( AIMAG( ALPHA( JC ) ) )
|
||||
ABSB = ABS( REAL( BETA( JC ) ) )
|
||||
SALFAR = ANRM*REAL( ALPHA( JC ) )
|
||||
SALFAI = ANRM*AIMAG( ALPHA( JC ) )
|
||||
SBETA = BNRM*REAL( BETA( JC ) )
|
||||
ILIMIT = .FALSE.
|
||||
SCALE = ONE
|
||||
*
|
||||
* Check for significant underflow in imaginary part of ALPHA
|
||||
*
|
||||
IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE.
|
||||
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN
|
||||
ILIMIT = .TRUE.
|
||||
SCALE = ( SAFMIN / ANRM1 ) / MAX( SAFMIN, ANRM2*ABSAI )
|
||||
END IF
|
||||
*
|
||||
* Check for significant underflow in real part of ALPHA
|
||||
*
|
||||
IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE.
|
||||
$ MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN
|
||||
ILIMIT = .TRUE.
|
||||
SCALE = MAX( SCALE, ( SAFMIN / ANRM1 ) /
|
||||
$ MAX( SAFMIN, ANRM2*ABSAR ) )
|
||||
END IF
|
||||
*
|
||||
* Check for significant underflow in BETA
|
||||
*
|
||||
IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE.
|
||||
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN
|
||||
ILIMIT = .TRUE.
|
||||
SCALE = MAX( SCALE, ( SAFMIN / BNRM1 ) /
|
||||
$ MAX( SAFMIN, BNRM2*ABSB ) )
|
||||
END IF
|
||||
*
|
||||
* Check for possible overflow when limiting scaling
|
||||
*
|
||||
IF( ILIMIT ) THEN
|
||||
TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ),
|
||||
$ ABS( SBETA ) )
|
||||
IF( TEMP.GT.ONE )
|
||||
$ SCALE = SCALE / TEMP
|
||||
IF( SCALE.LT.ONE )
|
||||
$ ILIMIT = .FALSE.
|
||||
END IF
|
||||
*
|
||||
* Recompute un-scaled ALPHA, BETA if necessary.
|
||||
*
|
||||
IF( ILIMIT ) THEN
|
||||
SALFAR = ( SCALE*REAL( ALPHA( JC ) ) )*ANRM
|
||||
SALFAI = ( SCALE*AIMAG( ALPHA( JC ) ) )*ANRM
|
||||
SBETA = ( SCALE*BETA( JC ) )*BNRM
|
||||
END IF
|
||||
ALPHA( JC ) = CMPLX( SALFAR, SALFAI )
|
||||
BETA( JC ) = SBETA
|
||||
70 CONTINUE
|
||||
*
|
||||
80 CONTINUE
|
||||
WORK( 1 ) = LWKOPT
|
||||
*
|
||||
RETURN
|
||||
*
|
||||
* End of CGEGV
|
||||
*
|
||||
END
|
||||
|
|
@ -0,0 +1,538 @@
|
|||
*> \brief <b> DGEGS computes the eigenvalues, real Schur form, and, optionally, the left and/or right Schur vectors of a real matrix pair (A,B)</b>
|
||||
*
|
||||
* =========== DOCUMENTATION ===========
|
||||
*
|
||||
* Online html documentation available at
|
||||
* http://www.netlib.org/lapack/explore-html/
|
||||
*
|
||||
*> \htmlonly
|
||||
*> Download DGEGS + dependencies
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgegs.f">
|
||||
*> [TGZ]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgegs.f">
|
||||
*> [ZIP]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgegs.f">
|
||||
*> [TXT]</a>
|
||||
*> \endhtmlonly
|
||||
*
|
||||
* Definition:
|
||||
* ===========
|
||||
*
|
||||
* SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
|
||||
* ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
|
||||
* LWORK, INFO )
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
* CHARACTER JOBVSL, JOBVSR
|
||||
* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
|
||||
* $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
|
||||
* $ VSR( LDVSR, * ), WORK( * )
|
||||
* ..
|
||||
*
|
||||
*
|
||||
*> \par Purpose:
|
||||
* =============
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> This routine is deprecated and has been replaced by routine DGGES.
|
||||
*>
|
||||
*> DGEGS computes the eigenvalues, real Schur form, and, optionally,
|
||||
*> left and or/right Schur vectors of a real matrix pair (A,B).
|
||||
*> Given two square matrices A and B, the generalized real Schur
|
||||
*> factorization has the form
|
||||
*>
|
||||
*> A = Q*S*Z**T, B = Q*T*Z**T
|
||||
*>
|
||||
*> where Q and Z are orthogonal matrices, T is upper triangular, and S
|
||||
*> is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal
|
||||
*> blocks, the 2-by-2 blocks corresponding to complex conjugate pairs
|
||||
*> of eigenvalues of (A,B). The columns of Q are the left Schur vectors
|
||||
*> and the columns of Z are the right Schur vectors.
|
||||
*>
|
||||
*> If only the eigenvalues of (A,B) are needed, the driver routine
|
||||
*> DGEGV should be used instead. See DGEGV for a description of the
|
||||
*> eigenvalues of the generalized nonsymmetric eigenvalue problem
|
||||
*> (GNEP).
|
||||
*> \endverbatim
|
||||
*
|
||||
* Arguments:
|
||||
* ==========
|
||||
*
|
||||
*> \param[in] JOBVSL
|
||||
*> \verbatim
|
||||
*> JOBVSL is CHARACTER*1
|
||||
*> = 'N': do not compute the left Schur vectors;
|
||||
*> = 'V': compute the left Schur vectors (returned in VSL).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] JOBVSR
|
||||
*> \verbatim
|
||||
*> JOBVSR is CHARACTER*1
|
||||
*> = 'N': do not compute the right Schur vectors;
|
||||
*> = 'V': compute the right Schur vectors (returned in VSR).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] N
|
||||
*> \verbatim
|
||||
*> N is INTEGER
|
||||
*> The order of the matrices A, B, VSL, and VSR. N >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] A
|
||||
*> \verbatim
|
||||
*> A is DOUBLE PRECISION array, dimension (LDA, N)
|
||||
*> On entry, the matrix A.
|
||||
*> On exit, the upper quasi-triangular matrix S from the
|
||||
*> generalized real Schur factorization.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDA
|
||||
*> \verbatim
|
||||
*> LDA is INTEGER
|
||||
*> The leading dimension of A. LDA >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] B
|
||||
*> \verbatim
|
||||
*> B is DOUBLE PRECISION array, dimension (LDB, N)
|
||||
*> On entry, the matrix B.
|
||||
*> On exit, the upper triangular matrix T from the generalized
|
||||
*> real Schur factorization.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDB
|
||||
*> \verbatim
|
||||
*> LDB is INTEGER
|
||||
*> The leading dimension of B. LDB >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] ALPHAR
|
||||
*> \verbatim
|
||||
*> ALPHAR is DOUBLE PRECISION array, dimension (N)
|
||||
*> The real parts of each scalar alpha defining an eigenvalue
|
||||
*> of GNEP.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] ALPHAI
|
||||
*> \verbatim
|
||||
*> ALPHAI is DOUBLE PRECISION array, dimension (N)
|
||||
*> The imaginary parts of each scalar alpha defining an
|
||||
*> eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th
|
||||
*> eigenvalue is real; if positive, then the j-th and (j+1)-st
|
||||
*> eigenvalues are a complex conjugate pair, with
|
||||
*> ALPHAI(j+1) = -ALPHAI(j).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] BETA
|
||||
*> \verbatim
|
||||
*> BETA is DOUBLE PRECISION array, dimension (N)
|
||||
*> The scalars beta that define the eigenvalues of GNEP.
|
||||
*> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
|
||||
*> beta = BETA(j) represent the j-th eigenvalue of the matrix
|
||||
*> pair (A,B), in one of the forms lambda = alpha/beta or
|
||||
*> mu = beta/alpha. Since either lambda or mu may overflow,
|
||||
*> they should not, in general, be computed.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] VSL
|
||||
*> \verbatim
|
||||
*> VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
|
||||
*> If JOBVSL = 'V', the matrix of left Schur vectors Q.
|
||||
*> Not referenced if JOBVSL = 'N'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDVSL
|
||||
*> \verbatim
|
||||
*> LDVSL is INTEGER
|
||||
*> The leading dimension of the matrix VSL. LDVSL >=1, and
|
||||
*> if JOBVSL = 'V', LDVSL >= N.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] VSR
|
||||
*> \verbatim
|
||||
*> VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
|
||||
*> If JOBVSR = 'V', the matrix of right Schur vectors Z.
|
||||
*> Not referenced if JOBVSR = 'N'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDVSR
|
||||
*> \verbatim
|
||||
*> LDVSR is INTEGER
|
||||
*> The leading dimension of the matrix VSR. LDVSR >= 1, and
|
||||
*> if JOBVSR = 'V', LDVSR >= N.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] WORK
|
||||
*> \verbatim
|
||||
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
|
||||
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LWORK
|
||||
*> \verbatim
|
||||
*> LWORK is INTEGER
|
||||
*> The dimension of the array WORK. LWORK >= max(1,4*N).
|
||||
*> For good performance, LWORK must generally be larger.
|
||||
*> To compute the optimal value of LWORK, call ILAENV to get
|
||||
*> blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute:
|
||||
*> NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR
|
||||
*> The optimal LWORK is 2*N + N*(NB+1).
|
||||
*>
|
||||
*> If LWORK = -1, then a workspace query is assumed; the routine
|
||||
*> only calculates the optimal size of the WORK array, returns
|
||||
*> this value as the first entry of the WORK array, and no error
|
||||
*> message related to LWORK is issued by XERBLA.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] INFO
|
||||
*> \verbatim
|
||||
*> INFO is INTEGER
|
||||
*> = 0: successful exit
|
||||
*> < 0: if INFO = -i, the i-th argument had an illegal value.
|
||||
*> = 1,...,N:
|
||||
*> The QZ iteration failed. (A,B) are not in Schur
|
||||
*> form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
|
||||
*> be correct for j=INFO+1,...,N.
|
||||
*> > N: errors that usually indicate LAPACK problems:
|
||||
*> =N+1: error return from DGGBAL
|
||||
*> =N+2: error return from DGEQRF
|
||||
*> =N+3: error return from DORMQR
|
||||
*> =N+4: error return from DORGQR
|
||||
*> =N+5: error return from DGGHRD
|
||||
*> =N+6: error return from DHGEQZ (other than failed
|
||||
*> iteration)
|
||||
*> =N+7: error return from DGGBAK (computing VSL)
|
||||
*> =N+8: error return from DGGBAK (computing VSR)
|
||||
*> =N+9: error return from DLASCL (various places)
|
||||
*> \endverbatim
|
||||
*
|
||||
* Authors:
|
||||
* ========
|
||||
*
|
||||
*> \author Univ. of Tennessee
|
||||
*> \author Univ. of California Berkeley
|
||||
*> \author Univ. of Colorado Denver
|
||||
*> \author NAG Ltd.
|
||||
*
|
||||
*> \ingroup doubleGEeigen
|
||||
*
|
||||
* =====================================================================
|
||||
SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
|
||||
$ ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
|
||||
$ LWORK, INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER JOBVSL, JOBVSR
|
||||
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
|
||||
$ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
|
||||
$ VSR( LDVSR, * ), WORK( * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
DOUBLE PRECISION ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL ILASCL, ILBSCL, ILVSL, ILVSR, LQUERY
|
||||
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
|
||||
$ IRIGHT, IROWS, ITAU, IWORK, LOPT, LWKMIN,
|
||||
$ LWKOPT, NB, NB1, NB2, NB3
|
||||
DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
|
||||
$ SAFMIN, SMLNUM
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY,
|
||||
$ DLASCL, DLASET, DORGQR, DORMQR, XERBLA
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
INTEGER ILAENV
|
||||
DOUBLE PRECISION DLAMCH, DLANGE
|
||||
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC INT, MAX
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
* Decode the input arguments
|
||||
*
|
||||
IF( LSAME( JOBVSL, 'N' ) ) THEN
|
||||
IJOBVL = 1
|
||||
ILVSL = .FALSE.
|
||||
ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
|
||||
IJOBVL = 2
|
||||
ILVSL = .TRUE.
|
||||
ELSE
|
||||
IJOBVL = -1
|
||||
ILVSL = .FALSE.
|
||||
END IF
|
||||
*
|
||||
IF( LSAME( JOBVSR, 'N' ) ) THEN
|
||||
IJOBVR = 1
|
||||
ILVSR = .FALSE.
|
||||
ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
|
||||
IJOBVR = 2
|
||||
ILVSR = .TRUE.
|
||||
ELSE
|
||||
IJOBVR = -1
|
||||
ILVSR = .FALSE.
|
||||
END IF
|
||||
*
|
||||
* Test the input arguments
|
||||
*
|
||||
LWKMIN = MAX( 4*N, 1 )
|
||||
LWKOPT = LWKMIN
|
||||
WORK( 1 ) = LWKOPT
|
||||
LQUERY = ( LWORK.EQ.-1 )
|
||||
INFO = 0
|
||||
IF( IJOBVL.LE.0 ) THEN
|
||||
INFO = -1
|
||||
ELSE IF( IJOBVR.LE.0 ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -5
|
||||
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -7
|
||||
ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
|
||||
INFO = -12
|
||||
ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
|
||||
INFO = -14
|
||||
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
|
||||
INFO = -16
|
||||
END IF
|
||||
*
|
||||
IF( INFO.EQ.0 ) THEN
|
||||
NB1 = ILAENV( 1, 'DGEQRF', ' ', N, N, -1, -1 )
|
||||
NB2 = ILAENV( 1, 'DORMQR', ' ', N, N, N, -1 )
|
||||
NB3 = ILAENV( 1, 'DORGQR', ' ', N, N, N, -1 )
|
||||
NB = MAX( NB1, NB2, NB3 )
|
||||
LOPT = 2*N + N*( NB+1 )
|
||||
WORK( 1 ) = LOPT
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'DGEGS ', -INFO )
|
||||
RETURN
|
||||
ELSE IF( LQUERY ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Quick return if possible
|
||||
*
|
||||
IF( N.EQ.0 )
|
||||
$ RETURN
|
||||
*
|
||||
* Get machine constants
|
||||
*
|
||||
EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
|
||||
SAFMIN = DLAMCH( 'S' )
|
||||
SMLNUM = N*SAFMIN / EPS
|
||||
BIGNUM = ONE / SMLNUM
|
||||
*
|
||||
* Scale A if max element outside range [SMLNUM,BIGNUM]
|
||||
*
|
||||
ANRM = DLANGE( '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 ) THEN
|
||||
CALL DLASCL( 'G', -1, -1, ANRM, ANRMTO, N, N, A, LDA, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Scale B if max element outside range [SMLNUM,BIGNUM]
|
||||
*
|
||||
BNRM = DLANGE( '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 ) THEN
|
||||
CALL DLASCL( 'G', -1, -1, BNRM, BNRMTO, N, N, B, LDB, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Permute the matrix to make it more nearly triangular
|
||||
* Workspace layout: (2*N words -- "work..." not actually used)
|
||||
* left_permutation, right_permutation, work...
|
||||
*
|
||||
ILEFT = 1
|
||||
IRIGHT = N + 1
|
||||
IWORK = IRIGHT + N
|
||||
CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
|
||||
$ WORK( IRIGHT ), WORK( IWORK ), IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 1
|
||||
GO TO 10
|
||||
END IF
|
||||
*
|
||||
* Reduce B to triangular form, and initialize VSL and/or VSR
|
||||
* Workspace layout: ("work..." must have at least N words)
|
||||
* left_permutation, right_permutation, tau, work...
|
||||
*
|
||||
IROWS = IHI + 1 - ILO
|
||||
ICOLS = N + 1 - ILO
|
||||
ITAU = IWORK
|
||||
IWORK = ITAU + IROWS
|
||||
CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
|
||||
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 2
|
||||
GO TO 10
|
||||
END IF
|
||||
*
|
||||
CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
|
||||
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
|
||||
$ LWORK+1-IWORK, IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 3
|
||||
GO TO 10
|
||||
END IF
|
||||
*
|
||||
IF( ILVSL ) THEN
|
||||
CALL DLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL )
|
||||
CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
|
||||
$ VSL( ILO+1, ILO ), LDVSL )
|
||||
CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
|
||||
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
|
||||
$ IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 4
|
||||
GO TO 10
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( ILVSR )
|
||||
$ CALL DLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR )
|
||||
*
|
||||
* Reduce to generalized Hessenberg form
|
||||
*
|
||||
CALL DGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
|
||||
$ LDVSL, VSR, LDVSR, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 5
|
||||
GO TO 10
|
||||
END IF
|
||||
*
|
||||
* Perform QZ algorithm, computing Schur vectors if desired
|
||||
* Workspace layout: ("work..." must have at least 1 word)
|
||||
* left_permutation, right_permutation, work...
|
||||
*
|
||||
IWORK = ITAU
|
||||
CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
|
||||
$ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
|
||||
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
|
||||
INFO = IINFO
|
||||
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
|
||||
INFO = IINFO - N
|
||||
ELSE
|
||||
INFO = N + 6
|
||||
END IF
|
||||
GO TO 10
|
||||
END IF
|
||||
*
|
||||
* Apply permutation to VSL and VSR
|
||||
*
|
||||
IF( ILVSL ) THEN
|
||||
CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
|
||||
$ WORK( IRIGHT ), N, VSL, LDVSL, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 7
|
||||
GO TO 10
|
||||
END IF
|
||||
END IF
|
||||
IF( ILVSR ) THEN
|
||||
CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
|
||||
$ WORK( IRIGHT ), N, VSR, LDVSR, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 8
|
||||
GO TO 10
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Undo scaling
|
||||
*
|
||||
IF( ILASCL ) THEN
|
||||
CALL DLASCL( 'H', -1, -1, ANRMTO, ANRM, N, N, A, LDA, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
CALL DLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAR, N,
|
||||
$ IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
CALL DLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAI, N,
|
||||
$ IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( ILBSCL ) THEN
|
||||
CALL DLASCL( 'U', -1, -1, BNRMTO, BNRM, N, N, B, LDB, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
CALL DLASCL( 'G', -1, -1, BNRMTO, BNRM, N, 1, BETA, N, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
10 CONTINUE
|
||||
WORK( 1 ) = LWKOPT
|
||||
*
|
||||
RETURN
|
||||
*
|
||||
* End of DGEGS
|
||||
*
|
||||
END
|
||||
|
|
@ -0,0 +1,766 @@
|
|||
*> \brief <b> DGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a real matrix pair (A,B).</b>
|
||||
*
|
||||
* =========== DOCUMENTATION ===========
|
||||
*
|
||||
* Online html documentation available at
|
||||
* http://www.netlib.org/lapack/explore-html/
|
||||
*
|
||||
*> \htmlonly
|
||||
*> Download DGEGV + dependencies
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgegv.f">
|
||||
*> [TGZ]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgegv.f">
|
||||
*> [ZIP]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgegv.f">
|
||||
*> [TXT]</a>
|
||||
*> \endhtmlonly
|
||||
*
|
||||
* Definition:
|
||||
* ===========
|
||||
*
|
||||
* SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
|
||||
* BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
* CHARACTER JOBVL, JOBVR
|
||||
* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
|
||||
* $ B( LDB, * ), BETA( * ), VL( LDVL, * ),
|
||||
* $ VR( LDVR, * ), WORK( * )
|
||||
* ..
|
||||
*
|
||||
*
|
||||
*> \par Purpose:
|
||||
* =============
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> This routine is deprecated and has been replaced by routine DGGEV.
|
||||
*>
|
||||
*> DGEGV computes the eigenvalues and, optionally, the left and/or right
|
||||
*> eigenvectors of a real matrix pair (A,B).
|
||||
*> Given two square matrices A and B,
|
||||
*> the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
|
||||
*> eigenvalues lambda and corresponding (non-zero) eigenvectors x such
|
||||
*> that
|
||||
*>
|
||||
*> A*x = lambda*B*x.
|
||||
*>
|
||||
*> An alternate form is to find the eigenvalues mu and corresponding
|
||||
*> eigenvectors y such that
|
||||
*>
|
||||
*> mu*A*y = B*y.
|
||||
*>
|
||||
*> These two forms are equivalent with mu = 1/lambda and x = y if
|
||||
*> neither lambda nor mu is zero. In order to deal with the case that
|
||||
*> lambda or mu is zero or small, two values alpha and beta are returned
|
||||
*> for each eigenvalue, such that lambda = alpha/beta and
|
||||
*> mu = beta/alpha.
|
||||
*>
|
||||
*> The vectors x and y in the above equations are right eigenvectors of
|
||||
*> the matrix pair (A,B). Vectors u and v satisfying
|
||||
*>
|
||||
*> u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
|
||||
*>
|
||||
*> are left eigenvectors of (A,B).
|
||||
*>
|
||||
*> Note: this routine performs "full balancing" on A and B
|
||||
*> \endverbatim
|
||||
*
|
||||
* Arguments:
|
||||
* ==========
|
||||
*
|
||||
*> \param[in] JOBVL
|
||||
*> \verbatim
|
||||
*> JOBVL is CHARACTER*1
|
||||
*> = 'N': do not compute the left generalized eigenvectors;
|
||||
*> = 'V': compute the left generalized eigenvectors (returned
|
||||
*> in VL).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] JOBVR
|
||||
*> \verbatim
|
||||
*> JOBVR is CHARACTER*1
|
||||
*> = 'N': do not compute the right generalized eigenvectors;
|
||||
*> = 'V': compute the right generalized eigenvectors (returned
|
||||
*> in VR).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] N
|
||||
*> \verbatim
|
||||
*> N is INTEGER
|
||||
*> The order of the matrices A, B, VL, and VR. N >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] A
|
||||
*> \verbatim
|
||||
*> A is DOUBLE PRECISION array, dimension (LDA, N)
|
||||
*> On entry, the matrix A.
|
||||
*> If JOBVL = 'V' or JOBVR = 'V', then on exit A
|
||||
*> contains the real Schur form of A from the generalized Schur
|
||||
*> factorization of the pair (A,B) after balancing.
|
||||
*> If no eigenvectors were computed, then only the diagonal
|
||||
*> blocks from the Schur form will be correct. See DGGHRD and
|
||||
*> DHGEQZ for details.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDA
|
||||
*> \verbatim
|
||||
*> LDA is INTEGER
|
||||
*> The leading dimension of A. LDA >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] B
|
||||
*> \verbatim
|
||||
*> B is DOUBLE PRECISION array, dimension (LDB, N)
|
||||
*> On entry, the matrix B.
|
||||
*> If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
|
||||
*> upper triangular matrix obtained from B in the generalized
|
||||
*> Schur factorization of the pair (A,B) after balancing.
|
||||
*> If no eigenvectors were computed, then only those elements of
|
||||
*> B corresponding to the diagonal blocks from the Schur form of
|
||||
*> A will be correct. See DGGHRD and DHGEQZ for details.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDB
|
||||
*> \verbatim
|
||||
*> LDB is INTEGER
|
||||
*> The leading dimension of B. LDB >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] ALPHAR
|
||||
*> \verbatim
|
||||
*> ALPHAR is DOUBLE PRECISION array, dimension (N)
|
||||
*> The real parts of each scalar alpha defining an eigenvalue of
|
||||
*> GNEP.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] ALPHAI
|
||||
*> \verbatim
|
||||
*> ALPHAI is DOUBLE PRECISION array, dimension (N)
|
||||
*> The imaginary parts of each scalar alpha defining an
|
||||
*> eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th
|
||||
*> eigenvalue is real; if positive, then the j-th and
|
||||
*> (j+1)-st eigenvalues are a complex conjugate pair, with
|
||||
*> ALPHAI(j+1) = -ALPHAI(j).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] BETA
|
||||
*> \verbatim
|
||||
*> BETA is DOUBLE PRECISION array, dimension (N)
|
||||
*> The scalars beta that define the eigenvalues of GNEP.
|
||||
*>
|
||||
*> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
|
||||
*> beta = BETA(j) represent the j-th eigenvalue of the matrix
|
||||
*> pair (A,B), in one of the forms lambda = alpha/beta or
|
||||
*> mu = beta/alpha. Since either lambda or mu may overflow,
|
||||
*> they should not, in general, be computed.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] VL
|
||||
*> \verbatim
|
||||
*> VL is DOUBLE PRECISION array, dimension (LDVL,N)
|
||||
*> If JOBVL = 'V', the left eigenvectors u(j) are stored
|
||||
*> in the columns of VL, in the same order as their eigenvalues.
|
||||
*> If the j-th eigenvalue is real, then u(j) = VL(:,j).
|
||||
*> If the j-th and (j+1)-st eigenvalues form a complex conjugate
|
||||
*> pair, then
|
||||
*> u(j) = VL(:,j) + i*VL(:,j+1)
|
||||
*> and
|
||||
*> u(j+1) = VL(:,j) - i*VL(:,j+1).
|
||||
*>
|
||||
*> Each eigenvector is scaled so that its largest component has
|
||||
*> abs(real part) + abs(imag. part) = 1, except for eigenvectors
|
||||
*> corresponding to an eigenvalue with alpha = beta = 0, which
|
||||
*> are set to zero.
|
||||
*> Not referenced if JOBVL = 'N'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDVL
|
||||
*> \verbatim
|
||||
*> LDVL is INTEGER
|
||||
*> The leading dimension of the matrix VL. LDVL >= 1, and
|
||||
*> if JOBVL = 'V', LDVL >= N.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] VR
|
||||
*> \verbatim
|
||||
*> VR is DOUBLE PRECISION array, dimension (LDVR,N)
|
||||
*> If JOBVR = 'V', the right eigenvectors x(j) are stored
|
||||
*> in the columns of VR, in the same order as their eigenvalues.
|
||||
*> If the j-th eigenvalue is real, then x(j) = VR(:,j).
|
||||
*> If the j-th and (j+1)-st eigenvalues form a complex conjugate
|
||||
*> pair, then
|
||||
*> x(j) = VR(:,j) + i*VR(:,j+1)
|
||||
*> and
|
||||
*> x(j+1) = VR(:,j) - i*VR(:,j+1).
|
||||
*>
|
||||
*> Each eigenvector is scaled so that its largest component has
|
||||
*> abs(real part) + abs(imag. part) = 1, except for eigenvalues
|
||||
*> corresponding to an eigenvalue with alpha = beta = 0, which
|
||||
*> are set to zero.
|
||||
*> Not referenced if JOBVR = 'N'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDVR
|
||||
*> \verbatim
|
||||
*> LDVR is INTEGER
|
||||
*> The leading dimension of the matrix VR. LDVR >= 1, and
|
||||
*> if JOBVR = 'V', LDVR >= N.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] WORK
|
||||
*> \verbatim
|
||||
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
|
||||
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LWORK
|
||||
*> \verbatim
|
||||
*> LWORK is INTEGER
|
||||
*> The dimension of the array WORK. LWORK >= max(1,8*N).
|
||||
*> For good performance, LWORK must generally be larger.
|
||||
*> To compute the optimal value of LWORK, call ILAENV to get
|
||||
*> blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute:
|
||||
*> NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR;
|
||||
*> The optimal LWORK is:
|
||||
*> 2*N + MAX( 6*N, N*(NB+1) ).
|
||||
*>
|
||||
*> If LWORK = -1, then a workspace query is assumed; the routine
|
||||
*> only calculates the optimal size of the WORK array, returns
|
||||
*> this value as the first entry of the WORK array, and no error
|
||||
*> message related to LWORK is issued by XERBLA.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] INFO
|
||||
*> \verbatim
|
||||
*> INFO is INTEGER
|
||||
*> = 0: successful exit
|
||||
*> < 0: if INFO = -i, the i-th argument had an illegal value.
|
||||
*> = 1,...,N:
|
||||
*> The QZ iteration failed. No eigenvectors have been
|
||||
*> calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
|
||||
*> should be correct for j=INFO+1,...,N.
|
||||
*> > N: errors that usually indicate LAPACK problems:
|
||||
*> =N+1: error return from DGGBAL
|
||||
*> =N+2: error return from DGEQRF
|
||||
*> =N+3: error return from DORMQR
|
||||
*> =N+4: error return from DORGQR
|
||||
*> =N+5: error return from DGGHRD
|
||||
*> =N+6: error return from DHGEQZ (other than failed
|
||||
*> iteration)
|
||||
*> =N+7: error return from DTGEVC
|
||||
*> =N+8: error return from DGGBAK (computing VL)
|
||||
*> =N+9: error return from DGGBAK (computing VR)
|
||||
*> =N+10: error return from DLASCL (various calls)
|
||||
*> \endverbatim
|
||||
*
|
||||
* Authors:
|
||||
* ========
|
||||
*
|
||||
*> \author Univ. of Tennessee
|
||||
*> \author Univ. of California Berkeley
|
||||
*> \author Univ. of Colorado Denver
|
||||
*> \author NAG Ltd.
|
||||
*
|
||||
*> \ingroup doubleGEeigen
|
||||
*
|
||||
*> \par Further Details:
|
||||
* =====================
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> Balancing
|
||||
*> ---------
|
||||
*>
|
||||
*> This driver calls DGGBAL to both permute and scale rows and columns
|
||||
*> of A and B. The permutations PL and PR are chosen so that PL*A*PR
|
||||
*> and PL*B*R will be upper triangular except for the diagonal blocks
|
||||
*> A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
|
||||
*> possible. The diagonal scaling matrices DL and DR are chosen so
|
||||
*> that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
|
||||
*> one (except for the elements that start out zero.)
|
||||
*>
|
||||
*> After the eigenvalues and eigenvectors of the balanced matrices
|
||||
*> have been computed, DGGBAK transforms the eigenvectors back to what
|
||||
*> they would have been (in perfect arithmetic) if they had not been
|
||||
*> balanced.
|
||||
*>
|
||||
*> Contents of A and B on Exit
|
||||
*> -------- -- - --- - -- ----
|
||||
*>
|
||||
*> If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
|
||||
*> both), then on exit the arrays A and B will contain the real Schur
|
||||
*> form[*] of the "balanced" versions of A and B. If no eigenvectors
|
||||
*> are computed, then only the diagonal blocks will be correct.
|
||||
*>
|
||||
*> [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations",
|
||||
*> by Golub & van Loan, pub. by Johns Hopkins U. Press.
|
||||
*> \endverbatim
|
||||
*>
|
||||
* =====================================================================
|
||||
SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
|
||||
$ BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER JOBVL, JOBVR
|
||||
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
|
||||
$ B( LDB, * ), BETA( * ), VL( LDVL, * ),
|
||||
$ VR( LDVR, * ), WORK( * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
DOUBLE PRECISION ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL ILIMIT, ILV, ILVL, ILVR, LQUERY
|
||||
CHARACTER CHTEMP
|
||||
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
|
||||
$ IN, IRIGHT, IROWS, ITAU, IWORK, JC, JR, LOPT,
|
||||
$ LWKMIN, LWKOPT, NB, NB1, NB2, NB3
|
||||
DOUBLE PRECISION ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM,
|
||||
$ BNRM1, BNRM2, EPS, ONEPLS, SAFMAX, SAFMIN,
|
||||
$ SALFAI, SALFAR, SBETA, SCALE, TEMP
|
||||
* ..
|
||||
* .. Local Arrays ..
|
||||
LOGICAL LDUMMA( 1 )
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY,
|
||||
$ DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, XERBLA
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
INTEGER ILAENV
|
||||
DOUBLE PRECISION DLAMCH, DLANGE
|
||||
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC ABS, INT, MAX
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
* Decode the input arguments
|
||||
*
|
||||
IF( LSAME( JOBVL, 'N' ) ) THEN
|
||||
IJOBVL = 1
|
||||
ILVL = .FALSE.
|
||||
ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
|
||||
IJOBVL = 2
|
||||
ILVL = .TRUE.
|
||||
ELSE
|
||||
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
|
||||
*
|
||||
* Test the input arguments
|
||||
*
|
||||
LWKMIN = MAX( 8*N, 1 )
|
||||
LWKOPT = LWKMIN
|
||||
WORK( 1 ) = LWKOPT
|
||||
LQUERY = ( LWORK.EQ.-1 )
|
||||
INFO = 0
|
||||
IF( IJOBVL.LE.0 ) THEN
|
||||
INFO = -1
|
||||
ELSE IF( IJOBVR.LE.0 ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -5
|
||||
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -7
|
||||
ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
|
||||
INFO = -12
|
||||
ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
|
||||
INFO = -14
|
||||
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
|
||||
INFO = -16
|
||||
END IF
|
||||
*
|
||||
IF( INFO.EQ.0 ) THEN
|
||||
NB1 = ILAENV( 1, 'DGEQRF', ' ', N, N, -1, -1 )
|
||||
NB2 = ILAENV( 1, 'DORMQR', ' ', N, N, N, -1 )
|
||||
NB3 = ILAENV( 1, 'DORGQR', ' ', N, N, N, -1 )
|
||||
NB = MAX( NB1, NB2, NB3 )
|
||||
LOPT = 2*N + MAX( 6*N, N*( NB+1 ) )
|
||||
WORK( 1 ) = LOPT
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'DGEGV ', -INFO )
|
||||
RETURN
|
||||
ELSE IF( LQUERY ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Quick return if possible
|
||||
*
|
||||
IF( N.EQ.0 )
|
||||
$ RETURN
|
||||
*
|
||||
* Get machine constants
|
||||
*
|
||||
EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
|
||||
SAFMIN = DLAMCH( 'S' )
|
||||
SAFMIN = SAFMIN + SAFMIN
|
||||
SAFMAX = ONE / SAFMIN
|
||||
ONEPLS = ONE + ( 4*EPS )
|
||||
*
|
||||
* Scale A
|
||||
*
|
||||
ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
|
||||
ANRM1 = ANRM
|
||||
ANRM2 = ONE
|
||||
IF( ANRM.LT.ONE ) THEN
|
||||
IF( SAFMAX*ANRM.LT.ONE ) THEN
|
||||
ANRM1 = SAFMIN
|
||||
ANRM2 = SAFMAX*ANRM
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( ANRM.GT.ZERO ) THEN
|
||||
CALL DLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 10
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Scale B
|
||||
*
|
||||
BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
|
||||
BNRM1 = BNRM
|
||||
BNRM2 = ONE
|
||||
IF( BNRM.LT.ONE ) THEN
|
||||
IF( SAFMAX*BNRM.LT.ONE ) THEN
|
||||
BNRM1 = SAFMIN
|
||||
BNRM2 = SAFMAX*BNRM
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( BNRM.GT.ZERO ) THEN
|
||||
CALL DLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 10
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Permute the matrix to make it more nearly triangular
|
||||
* Workspace layout: (8*N words -- "work" requires 6*N words)
|
||||
* left_permutation, right_permutation, work...
|
||||
*
|
||||
ILEFT = 1
|
||||
IRIGHT = N + 1
|
||||
IWORK = IRIGHT + N
|
||||
CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
|
||||
$ WORK( IRIGHT ), WORK( IWORK ), IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 1
|
||||
GO TO 120
|
||||
END IF
|
||||
*
|
||||
* Reduce B to triangular form, and initialize VL and/or VR
|
||||
* Workspace layout: ("work..." must have at least N words)
|
||||
* left_permutation, right_permutation, tau, work...
|
||||
*
|
||||
IROWS = IHI + 1 - ILO
|
||||
IF( ILV ) THEN
|
||||
ICOLS = N + 1 - ILO
|
||||
ELSE
|
||||
ICOLS = IROWS
|
||||
END IF
|
||||
ITAU = IWORK
|
||||
IWORK = ITAU + IROWS
|
||||
CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
|
||||
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 2
|
||||
GO TO 120
|
||||
END IF
|
||||
*
|
||||
CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
|
||||
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
|
||||
$ LWORK+1-IWORK, IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 3
|
||||
GO TO 120
|
||||
END IF
|
||||
*
|
||||
IF( ILVL ) THEN
|
||||
CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
|
||||
CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
|
||||
$ VL( ILO+1, ILO ), LDVL )
|
||||
CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
|
||||
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
|
||||
$ IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 4
|
||||
GO TO 120
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( ILVR )
|
||||
$ CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
|
||||
*
|
||||
* Reduce to generalized Hessenberg form
|
||||
*
|
||||
IF( ILV ) THEN
|
||||
*
|
||||
* Eigenvectors requested -- work on whole matrix.
|
||||
*
|
||||
CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
|
||||
$ LDVL, VR, LDVR, IINFO )
|
||||
ELSE
|
||||
CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
|
||||
$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO )
|
||||
END IF
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 5
|
||||
GO TO 120
|
||||
END IF
|
||||
*
|
||||
* Perform QZ algorithm
|
||||
* Workspace layout: ("work..." must have at least 1 word)
|
||||
* left_permutation, right_permutation, work...
|
||||
*
|
||||
IWORK = ITAU
|
||||
IF( ILV ) THEN
|
||||
CHTEMP = 'S'
|
||||
ELSE
|
||||
CHTEMP = 'E'
|
||||
END IF
|
||||
CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
|
||||
$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR,
|
||||
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
|
||||
INFO = IINFO
|
||||
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
|
||||
INFO = IINFO - N
|
||||
ELSE
|
||||
INFO = N + 6
|
||||
END IF
|
||||
GO TO 120
|
||||
END IF
|
||||
*
|
||||
IF( ILV ) THEN
|
||||
*
|
||||
* Compute Eigenvectors (DTGEVC requires 6*N words of workspace)
|
||||
*
|
||||
IF( ILVL ) THEN
|
||||
IF( ILVR ) THEN
|
||||
CHTEMP = 'B'
|
||||
ELSE
|
||||
CHTEMP = 'L'
|
||||
END IF
|
||||
ELSE
|
||||
CHTEMP = 'R'
|
||||
END IF
|
||||
*
|
||||
CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
|
||||
$ VR, LDVR, N, IN, WORK( IWORK ), IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 7
|
||||
GO TO 120
|
||||
END IF
|
||||
*
|
||||
* Undo balancing on VL and VR, rescale
|
||||
*
|
||||
IF( ILVL ) THEN
|
||||
CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
|
||||
$ WORK( IRIGHT ), N, VL, LDVL, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 8
|
||||
GO TO 120
|
||||
END IF
|
||||
DO 50 JC = 1, N
|
||||
IF( ALPHAI( JC ).LT.ZERO )
|
||||
$ GO TO 50
|
||||
TEMP = ZERO
|
||||
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
||||
DO 10 JR = 1, N
|
||||
TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
|
||||
10 CONTINUE
|
||||
ELSE
|
||||
DO 20 JR = 1, N
|
||||
TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
|
||||
$ ABS( VL( JR, JC+1 ) ) )
|
||||
20 CONTINUE
|
||||
END IF
|
||||
IF( TEMP.LT.SAFMIN )
|
||||
$ GO TO 50
|
||||
TEMP = ONE / TEMP
|
||||
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
||||
DO 30 JR = 1, N
|
||||
VL( JR, JC ) = VL( JR, JC )*TEMP
|
||||
30 CONTINUE
|
||||
ELSE
|
||||
DO 40 JR = 1, N
|
||||
VL( JR, JC ) = VL( JR, JC )*TEMP
|
||||
VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
|
||||
40 CONTINUE
|
||||
END IF
|
||||
50 CONTINUE
|
||||
END IF
|
||||
IF( ILVR ) THEN
|
||||
CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
|
||||
$ WORK( IRIGHT ), N, VR, LDVR, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
GO TO 120
|
||||
END IF
|
||||
DO 100 JC = 1, N
|
||||
IF( ALPHAI( JC ).LT.ZERO )
|
||||
$ GO TO 100
|
||||
TEMP = ZERO
|
||||
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
||||
DO 60 JR = 1, N
|
||||
TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
|
||||
60 CONTINUE
|
||||
ELSE
|
||||
DO 70 JR = 1, N
|
||||
TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
|
||||
$ ABS( VR( JR, JC+1 ) ) )
|
||||
70 CONTINUE
|
||||
END IF
|
||||
IF( TEMP.LT.SAFMIN )
|
||||
$ GO TO 100
|
||||
TEMP = ONE / TEMP
|
||||
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
||||
DO 80 JR = 1, N
|
||||
VR( JR, JC ) = VR( JR, JC )*TEMP
|
||||
80 CONTINUE
|
||||
ELSE
|
||||
DO 90 JR = 1, N
|
||||
VR( JR, JC ) = VR( JR, JC )*TEMP
|
||||
VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
|
||||
90 CONTINUE
|
||||
END IF
|
||||
100 CONTINUE
|
||||
END IF
|
||||
*
|
||||
* End of eigenvector calculation
|
||||
*
|
||||
END IF
|
||||
*
|
||||
* Undo scaling in alpha, beta
|
||||
*
|
||||
* Note: this does not give the alpha and beta for the unscaled
|
||||
* problem.
|
||||
*
|
||||
* Un-scaling is limited to avoid underflow in alpha and beta
|
||||
* if they are significant.
|
||||
*
|
||||
DO 110 JC = 1, N
|
||||
ABSAR = ABS( ALPHAR( JC ) )
|
||||
ABSAI = ABS( ALPHAI( JC ) )
|
||||
ABSB = ABS( BETA( JC ) )
|
||||
SALFAR = ANRM*ALPHAR( JC )
|
||||
SALFAI = ANRM*ALPHAI( JC )
|
||||
SBETA = BNRM*BETA( JC )
|
||||
ILIMIT = .FALSE.
|
||||
SCALE = ONE
|
||||
*
|
||||
* Check for significant underflow in ALPHAI
|
||||
*
|
||||
IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE.
|
||||
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN
|
||||
ILIMIT = .TRUE.
|
||||
SCALE = ( ONEPLS*SAFMIN / ANRM1 ) /
|
||||
$ MAX( ONEPLS*SAFMIN, ANRM2*ABSAI )
|
||||
*
|
||||
ELSE IF( SALFAI.EQ.ZERO ) THEN
|
||||
*
|
||||
* If insignificant underflow in ALPHAI, then make the
|
||||
* conjugate eigenvalue real.
|
||||
*
|
||||
IF( ALPHAI( JC ).LT.ZERO .AND. JC.GT.1 ) THEN
|
||||
ALPHAI( JC-1 ) = ZERO
|
||||
ELSE IF( ALPHAI( JC ).GT.ZERO .AND. JC.LT.N ) THEN
|
||||
ALPHAI( JC+1 ) = ZERO
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Check for significant underflow in ALPHAR
|
||||
*
|
||||
IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE.
|
||||
$ MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN
|
||||
ILIMIT = .TRUE.
|
||||
SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / ANRM1 ) /
|
||||
$ MAX( ONEPLS*SAFMIN, ANRM2*ABSAR ) )
|
||||
END IF
|
||||
*
|
||||
* Check for significant underflow in BETA
|
||||
*
|
||||
IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE.
|
||||
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN
|
||||
ILIMIT = .TRUE.
|
||||
SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / BNRM1 ) /
|
||||
$ MAX( ONEPLS*SAFMIN, BNRM2*ABSB ) )
|
||||
END IF
|
||||
*
|
||||
* Check for possible overflow when limiting scaling
|
||||
*
|
||||
IF( ILIMIT ) THEN
|
||||
TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ),
|
||||
$ ABS( SBETA ) )
|
||||
IF( TEMP.GT.ONE )
|
||||
$ SCALE = SCALE / TEMP
|
||||
IF( SCALE.LT.ONE )
|
||||
$ ILIMIT = .FALSE.
|
||||
END IF
|
||||
*
|
||||
* Recompute un-scaled ALPHAR, ALPHAI, BETA if necessary.
|
||||
*
|
||||
IF( ILIMIT ) THEN
|
||||
SALFAR = ( SCALE*ALPHAR( JC ) )*ANRM
|
||||
SALFAI = ( SCALE*ALPHAI( JC ) )*ANRM
|
||||
SBETA = ( SCALE*BETA( JC ) )*BNRM
|
||||
END IF
|
||||
ALPHAR( JC ) = SALFAR
|
||||
ALPHAI( JC ) = SALFAI
|
||||
BETA( JC ) = SBETA
|
||||
110 CONTINUE
|
||||
*
|
||||
120 CONTINUE
|
||||
WORK( 1 ) = LWKOPT
|
||||
*
|
||||
RETURN
|
||||
*
|
||||
* End of DGEGV
|
||||
*
|
||||
END
|
||||
|
|
@ -0,0 +1,538 @@
|
|||
*> \brief <b> SGEGS computes the eigenvalues, real Schur form, and, optionally, the left and/or right Schur vectors of a real matrix pair (A,B)</b>
|
||||
*
|
||||
* =========== DOCUMENTATION ===========
|
||||
*
|
||||
* Online html documentation available at
|
||||
* http://www.netlib.org/lapack/explore-html/
|
||||
*
|
||||
*> \htmlonly
|
||||
*> Download SGEGS + dependencies
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgegs.f">
|
||||
*> [TGZ]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgegs.f">
|
||||
*> [ZIP]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgegs.f">
|
||||
*> [TXT]</a>
|
||||
*> \endhtmlonly
|
||||
*
|
||||
* Definition:
|
||||
* ===========
|
||||
*
|
||||
* SUBROUTINE SGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
|
||||
* ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
|
||||
* LWORK, INFO )
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
* CHARACTER JOBVSL, JOBVSR
|
||||
* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
* REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
|
||||
* $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
|
||||
* $ VSR( LDVSR, * ), WORK( * )
|
||||
* ..
|
||||
*
|
||||
*
|
||||
*> \par Purpose:
|
||||
* =============
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> This routine is deprecated and has been replaced by routine SGGES.
|
||||
*>
|
||||
*> SGEGS computes the eigenvalues, real Schur form, and, optionally,
|
||||
*> left and or/right Schur vectors of a real matrix pair (A,B).
|
||||
*> Given two square matrices A and B, the generalized real Schur
|
||||
*> factorization has the form
|
||||
*>
|
||||
*> A = Q*S*Z**T, B = Q*T*Z**T
|
||||
*>
|
||||
*> where Q and Z are orthogonal matrices, T is upper triangular, and S
|
||||
*> is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal
|
||||
*> blocks, the 2-by-2 blocks corresponding to complex conjugate pairs
|
||||
*> of eigenvalues of (A,B). The columns of Q are the left Schur vectors
|
||||
*> and the columns of Z are the right Schur vectors.
|
||||
*>
|
||||
*> If only the eigenvalues of (A,B) are needed, the driver routine
|
||||
*> SGEGV should be used instead. See SGEGV for a description of the
|
||||
*> eigenvalues of the generalized nonsymmetric eigenvalue problem
|
||||
*> (GNEP).
|
||||
*> \endverbatim
|
||||
*
|
||||
* Arguments:
|
||||
* ==========
|
||||
*
|
||||
*> \param[in] JOBVSL
|
||||
*> \verbatim
|
||||
*> JOBVSL is CHARACTER*1
|
||||
*> = 'N': do not compute the left Schur vectors;
|
||||
*> = 'V': compute the left Schur vectors (returned in VSL).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] JOBVSR
|
||||
*> \verbatim
|
||||
*> JOBVSR is CHARACTER*1
|
||||
*> = 'N': do not compute the right Schur vectors;
|
||||
*> = 'V': compute the right Schur vectors (returned in VSR).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] N
|
||||
*> \verbatim
|
||||
*> N is INTEGER
|
||||
*> The order of the matrices A, B, VSL, and VSR. N >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] A
|
||||
*> \verbatim
|
||||
*> A is REAL array, dimension (LDA, N)
|
||||
*> On entry, the matrix A.
|
||||
*> On exit, the upper quasi-triangular matrix S from the
|
||||
*> generalized real Schur factorization.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDA
|
||||
*> \verbatim
|
||||
*> LDA is INTEGER
|
||||
*> The leading dimension of A. LDA >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] B
|
||||
*> \verbatim
|
||||
*> B is REAL array, dimension (LDB, N)
|
||||
*> On entry, the matrix B.
|
||||
*> On exit, the upper triangular matrix T from the generalized
|
||||
*> real Schur factorization.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDB
|
||||
*> \verbatim
|
||||
*> LDB is INTEGER
|
||||
*> The leading dimension of B. LDB >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] ALPHAR
|
||||
*> \verbatim
|
||||
*> ALPHAR is REAL array, dimension (N)
|
||||
*> The real parts of each scalar alpha defining an eigenvalue
|
||||
*> of GNEP.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] ALPHAI
|
||||
*> \verbatim
|
||||
*> ALPHAI is REAL array, dimension (N)
|
||||
*> The imaginary parts of each scalar alpha defining an
|
||||
*> eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th
|
||||
*> eigenvalue is real; if positive, then the j-th and (j+1)-st
|
||||
*> eigenvalues are a complex conjugate pair, with
|
||||
*> ALPHAI(j+1) = -ALPHAI(j).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] BETA
|
||||
*> \verbatim
|
||||
*> BETA is REAL array, dimension (N)
|
||||
*> The scalars beta that define the eigenvalues of GNEP.
|
||||
*> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
|
||||
*> beta = BETA(j) represent the j-th eigenvalue of the matrix
|
||||
*> pair (A,B), in one of the forms lambda = alpha/beta or
|
||||
*> mu = beta/alpha. Since either lambda or mu may overflow,
|
||||
*> they should not, in general, be computed.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] VSL
|
||||
*> \verbatim
|
||||
*> VSL is REAL array, dimension (LDVSL,N)
|
||||
*> If JOBVSL = 'V', the matrix of left Schur vectors Q.
|
||||
*> Not referenced if JOBVSL = 'N'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDVSL
|
||||
*> \verbatim
|
||||
*> LDVSL is INTEGER
|
||||
*> The leading dimension of the matrix VSL. LDVSL >=1, and
|
||||
*> if JOBVSL = 'V', LDVSL >= N.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] VSR
|
||||
*> \verbatim
|
||||
*> VSR is REAL array, dimension (LDVSR,N)
|
||||
*> If JOBVSR = 'V', the matrix of right Schur vectors Z.
|
||||
*> Not referenced if JOBVSR = 'N'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDVSR
|
||||
*> \verbatim
|
||||
*> LDVSR is INTEGER
|
||||
*> The leading dimension of the matrix VSR. LDVSR >= 1, and
|
||||
*> if JOBVSR = 'V', LDVSR >= N.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] WORK
|
||||
*> \verbatim
|
||||
*> WORK is REAL array, dimension (MAX(1,LWORK))
|
||||
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LWORK
|
||||
*> \verbatim
|
||||
*> LWORK is INTEGER
|
||||
*> The dimension of the array WORK. LWORK >= max(1,4*N).
|
||||
*> For good performance, LWORK must generally be larger.
|
||||
*> To compute the optimal value of LWORK, call ILAENV to get
|
||||
*> blocksizes (for SGEQRF, SORMQR, and SORGQR.) Then compute:
|
||||
*> NB -- MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR
|
||||
*> The optimal LWORK is 2*N + N*(NB+1).
|
||||
*>
|
||||
*> If LWORK = -1, then a workspace query is assumed; the routine
|
||||
*> only calculates the optimal size of the WORK array, returns
|
||||
*> this value as the first entry of the WORK array, and no error
|
||||
*> message related to LWORK is issued by XERBLA.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] INFO
|
||||
*> \verbatim
|
||||
*> INFO is INTEGER
|
||||
*> = 0: successful exit
|
||||
*> < 0: if INFO = -i, the i-th argument had an illegal value.
|
||||
*> = 1,...,N:
|
||||
*> The QZ iteration failed. (A,B) are not in Schur
|
||||
*> form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
|
||||
*> be correct for j=INFO+1,...,N.
|
||||
*> > N: errors that usually indicate LAPACK problems:
|
||||
*> =N+1: error return from SGGBAL
|
||||
*> =N+2: error return from SGEQRF
|
||||
*> =N+3: error return from SORMQR
|
||||
*> =N+4: error return from SORGQR
|
||||
*> =N+5: error return from SGGHRD
|
||||
*> =N+6: error return from SHGEQZ (other than failed
|
||||
*> iteration)
|
||||
*> =N+7: error return from SGGBAK (computing VSL)
|
||||
*> =N+8: error return from SGGBAK (computing VSR)
|
||||
*> =N+9: error return from SLASCL (various places)
|
||||
*> \endverbatim
|
||||
*
|
||||
* Authors:
|
||||
* ========
|
||||
*
|
||||
*> \author Univ. of Tennessee
|
||||
*> \author Univ. of California Berkeley
|
||||
*> \author Univ. of Colorado Denver
|
||||
*> \author NAG Ltd.
|
||||
*
|
||||
*> \ingroup realGEeigen
|
||||
*
|
||||
* =====================================================================
|
||||
SUBROUTINE SGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
|
||||
$ ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
|
||||
$ LWORK, INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER JOBVSL, JOBVSR
|
||||
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
|
||||
$ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
|
||||
$ VSR( LDVSR, * ), WORK( * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
REAL ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL ILASCL, ILBSCL, ILVSL, ILVSR, LQUERY
|
||||
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT,
|
||||
$ ILO, IRIGHT, IROWS, ITAU, IWORK, LOPT, LWKMIN,
|
||||
$ LWKOPT, NB, NB1, NB2, NB3
|
||||
REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
|
||||
$ SAFMIN, SMLNUM
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL SGEQRF, SGGBAK, SGGBAL, SGGHRD, SHGEQZ, SLACPY,
|
||||
$ SLASCL, SLASET, SORGQR, SORMQR, XERBLA
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
INTEGER ILAENV
|
||||
REAL SLAMCH, SLANGE
|
||||
EXTERNAL ILAENV, LSAME, SLAMCH, SLANGE
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC INT, MAX
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
* Decode the input arguments
|
||||
*
|
||||
IF( LSAME( JOBVSL, 'N' ) ) THEN
|
||||
IJOBVL = 1
|
||||
ILVSL = .FALSE.
|
||||
ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
|
||||
IJOBVL = 2
|
||||
ILVSL = .TRUE.
|
||||
ELSE
|
||||
IJOBVL = -1
|
||||
ILVSL = .FALSE.
|
||||
END IF
|
||||
*
|
||||
IF( LSAME( JOBVSR, 'N' ) ) THEN
|
||||
IJOBVR = 1
|
||||
ILVSR = .FALSE.
|
||||
ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
|
||||
IJOBVR = 2
|
||||
ILVSR = .TRUE.
|
||||
ELSE
|
||||
IJOBVR = -1
|
||||
ILVSR = .FALSE.
|
||||
END IF
|
||||
*
|
||||
* Test the input arguments
|
||||
*
|
||||
LWKMIN = MAX( 4*N, 1 )
|
||||
LWKOPT = LWKMIN
|
||||
WORK( 1 ) = LWKOPT
|
||||
LQUERY = ( LWORK.EQ.-1 )
|
||||
INFO = 0
|
||||
IF( IJOBVL.LE.0 ) THEN
|
||||
INFO = -1
|
||||
ELSE IF( IJOBVR.LE.0 ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -5
|
||||
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -7
|
||||
ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
|
||||
INFO = -12
|
||||
ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
|
||||
INFO = -14
|
||||
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
|
||||
INFO = -16
|
||||
END IF
|
||||
*
|
||||
IF( INFO.EQ.0 ) THEN
|
||||
NB1 = ILAENV( 1, 'SGEQRF', ' ', N, N, -1, -1 )
|
||||
NB2 = ILAENV( 1, 'SORMQR', ' ', N, N, N, -1 )
|
||||
NB3 = ILAENV( 1, 'SORGQR', ' ', N, N, N, -1 )
|
||||
NB = MAX( NB1, NB2, NB3 )
|
||||
LOPT = 2*N+N*(NB+1)
|
||||
WORK( 1 ) = LOPT
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'SGEGS ', -INFO )
|
||||
RETURN
|
||||
ELSE IF( LQUERY ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Quick return if possible
|
||||
*
|
||||
IF( N.EQ.0 )
|
||||
$ RETURN
|
||||
*
|
||||
* Get machine constants
|
||||
*
|
||||
EPS = SLAMCH( 'E' )*SLAMCH( 'B' )
|
||||
SAFMIN = SLAMCH( 'S' )
|
||||
SMLNUM = N*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 ) THEN
|
||||
CALL SLASCL( 'G', -1, -1, ANRM, ANRMTO, N, N, A, LDA, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* 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 ) THEN
|
||||
CALL SLASCL( 'G', -1, -1, BNRM, BNRMTO, N, N, B, LDB, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Permute the matrix to make it more nearly triangular
|
||||
* Workspace layout: (2*N words -- "work..." not actually used)
|
||||
* left_permutation, right_permutation, work...
|
||||
*
|
||||
ILEFT = 1
|
||||
IRIGHT = N + 1
|
||||
IWORK = IRIGHT + N
|
||||
CALL SGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
|
||||
$ WORK( IRIGHT ), WORK( IWORK ), IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 1
|
||||
GO TO 10
|
||||
END IF
|
||||
*
|
||||
* Reduce B to triangular form, and initialize VSL and/or VSR
|
||||
* Workspace layout: ("work..." must have at least N words)
|
||||
* left_permutation, right_permutation, tau, work...
|
||||
*
|
||||
IROWS = IHI + 1 - ILO
|
||||
ICOLS = N + 1 - ILO
|
||||
ITAU = IWORK
|
||||
IWORK = ITAU + IROWS
|
||||
CALL SGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
|
||||
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 2
|
||||
GO TO 10
|
||||
END IF
|
||||
*
|
||||
CALL SORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
|
||||
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
|
||||
$ LWORK+1-IWORK, IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 3
|
||||
GO TO 10
|
||||
END IF
|
||||
*
|
||||
IF( ILVSL ) THEN
|
||||
CALL SLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL )
|
||||
CALL SLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
|
||||
$ VSL( ILO+1, ILO ), LDVSL )
|
||||
CALL SORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
|
||||
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
|
||||
$ IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 4
|
||||
GO TO 10
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( ILVSR )
|
||||
$ CALL SLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR )
|
||||
*
|
||||
* Reduce to generalized Hessenberg form
|
||||
*
|
||||
CALL SGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
|
||||
$ LDVSL, VSR, LDVSR, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 5
|
||||
GO TO 10
|
||||
END IF
|
||||
*
|
||||
* Perform QZ algorithm, computing Schur vectors if desired
|
||||
* Workspace layout: ("work..." must have at least 1 word)
|
||||
* left_permutation, right_permutation, work...
|
||||
*
|
||||
IWORK = ITAU
|
||||
CALL SHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
|
||||
$ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
|
||||
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
|
||||
INFO = IINFO
|
||||
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
|
||||
INFO = IINFO - N
|
||||
ELSE
|
||||
INFO = N + 6
|
||||
END IF
|
||||
GO TO 10
|
||||
END IF
|
||||
*
|
||||
* Apply permutation to VSL and VSR
|
||||
*
|
||||
IF( ILVSL ) THEN
|
||||
CALL SGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
|
||||
$ WORK( IRIGHT ), N, VSL, LDVSL, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 7
|
||||
GO TO 10
|
||||
END IF
|
||||
END IF
|
||||
IF( ILVSR ) THEN
|
||||
CALL SGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
|
||||
$ WORK( IRIGHT ), N, VSR, LDVSR, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 8
|
||||
GO TO 10
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Undo scaling
|
||||
*
|
||||
IF( ILASCL ) THEN
|
||||
CALL SLASCL( 'H', -1, -1, ANRMTO, ANRM, N, N, A, LDA, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
CALL SLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAR, N,
|
||||
$ IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
CALL SLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAI, N,
|
||||
$ IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( ILBSCL ) THEN
|
||||
CALL SLASCL( 'U', -1, -1, BNRMTO, BNRM, N, N, B, LDB, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
CALL SLASCL( 'G', -1, -1, BNRMTO, BNRM, N, 1, BETA, N, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
10 CONTINUE
|
||||
WORK( 1 ) = LWKOPT
|
||||
*
|
||||
RETURN
|
||||
*
|
||||
* End of SGEGS
|
||||
*
|
||||
END
|
||||
|
|
@ -0,0 +1,766 @@
|
|||
*> \brief <b> SGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a real matrix pair (A,B).</b>
|
||||
*
|
||||
* =========== DOCUMENTATION ===========
|
||||
*
|
||||
* Online html documentation available at
|
||||
* http://www.netlib.org/lapack/explore-html/
|
||||
*
|
||||
*> \htmlonly
|
||||
*> Download SGEGV + dependencies
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgegv.f">
|
||||
*> [TGZ]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgegv.f">
|
||||
*> [ZIP]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgegv.f">
|
||||
*> [TXT]</a>
|
||||
*> \endhtmlonly
|
||||
*
|
||||
* Definition:
|
||||
* ===========
|
||||
*
|
||||
* SUBROUTINE SGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
|
||||
* BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
* CHARACTER JOBVL, JOBVR
|
||||
* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
* REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
|
||||
* $ B( LDB, * ), BETA( * ), VL( LDVL, * ),
|
||||
* $ VR( LDVR, * ), WORK( * )
|
||||
* ..
|
||||
*
|
||||
*
|
||||
*> \par Purpose:
|
||||
* =============
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> This routine is deprecated and has been replaced by routine SGGEV.
|
||||
*>
|
||||
*> SGEGV computes the eigenvalues and, optionally, the left and/or right
|
||||
*> eigenvectors of a real matrix pair (A,B).
|
||||
*> Given two square matrices A and B,
|
||||
*> the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
|
||||
*> eigenvalues lambda and corresponding (non-zero) eigenvectors x such
|
||||
*> that
|
||||
*>
|
||||
*> A*x = lambda*B*x.
|
||||
*>
|
||||
*> An alternate form is to find the eigenvalues mu and corresponding
|
||||
*> eigenvectors y such that
|
||||
*>
|
||||
*> mu*A*y = B*y.
|
||||
*>
|
||||
*> These two forms are equivalent with mu = 1/lambda and x = y if
|
||||
*> neither lambda nor mu is zero. In order to deal with the case that
|
||||
*> lambda or mu is zero or small, two values alpha and beta are returned
|
||||
*> for each eigenvalue, such that lambda = alpha/beta and
|
||||
*> mu = beta/alpha.
|
||||
*>
|
||||
*> The vectors x and y in the above equations are right eigenvectors of
|
||||
*> the matrix pair (A,B). Vectors u and v satisfying
|
||||
*>
|
||||
*> u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
|
||||
*>
|
||||
*> are left eigenvectors of (A,B).
|
||||
*>
|
||||
*> Note: this routine performs "full balancing" on A and B
|
||||
*> \endverbatim
|
||||
*
|
||||
* Arguments:
|
||||
* ==========
|
||||
*
|
||||
*> \param[in] JOBVL
|
||||
*> \verbatim
|
||||
*> JOBVL is CHARACTER*1
|
||||
*> = 'N': do not compute the left generalized eigenvectors;
|
||||
*> = 'V': compute the left generalized eigenvectors (returned
|
||||
*> in VL).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] JOBVR
|
||||
*> \verbatim
|
||||
*> JOBVR is CHARACTER*1
|
||||
*> = 'N': do not compute the right generalized eigenvectors;
|
||||
*> = 'V': compute the right generalized eigenvectors (returned
|
||||
*> in VR).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] N
|
||||
*> \verbatim
|
||||
*> N is INTEGER
|
||||
*> The order of the matrices A, B, VL, and VR. N >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] A
|
||||
*> \verbatim
|
||||
*> A is REAL array, dimension (LDA, N)
|
||||
*> On entry, the matrix A.
|
||||
*> If JOBVL = 'V' or JOBVR = 'V', then on exit A
|
||||
*> contains the real Schur form of A from the generalized Schur
|
||||
*> factorization of the pair (A,B) after balancing.
|
||||
*> If no eigenvectors were computed, then only the diagonal
|
||||
*> blocks from the Schur form will be correct. See SGGHRD and
|
||||
*> SHGEQZ for details.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDA
|
||||
*> \verbatim
|
||||
*> LDA is INTEGER
|
||||
*> The leading dimension of A. LDA >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] B
|
||||
*> \verbatim
|
||||
*> B is REAL array, dimension (LDB, N)
|
||||
*> On entry, the matrix B.
|
||||
*> If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
|
||||
*> upper triangular matrix obtained from B in the generalized
|
||||
*> Schur factorization of the pair (A,B) after balancing.
|
||||
*> If no eigenvectors were computed, then only those elements of
|
||||
*> B corresponding to the diagonal blocks from the Schur form of
|
||||
*> A will be correct. See SGGHRD and SHGEQZ for details.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDB
|
||||
*> \verbatim
|
||||
*> LDB is INTEGER
|
||||
*> The leading dimension of B. LDB >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] ALPHAR
|
||||
*> \verbatim
|
||||
*> ALPHAR is REAL array, dimension (N)
|
||||
*> The real parts of each scalar alpha defining an eigenvalue of
|
||||
*> GNEP.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] ALPHAI
|
||||
*> \verbatim
|
||||
*> ALPHAI is REAL array, dimension (N)
|
||||
*> The imaginary parts of each scalar alpha defining an
|
||||
*> eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th
|
||||
*> eigenvalue is real; if positive, then the j-th and
|
||||
*> (j+1)-st eigenvalues are a complex conjugate pair, with
|
||||
*> ALPHAI(j+1) = -ALPHAI(j).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] BETA
|
||||
*> \verbatim
|
||||
*> BETA is REAL array, dimension (N)
|
||||
*> The scalars beta that define the eigenvalues of GNEP.
|
||||
*>
|
||||
*> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
|
||||
*> beta = BETA(j) represent the j-th eigenvalue of the matrix
|
||||
*> pair (A,B), in one of the forms lambda = alpha/beta or
|
||||
*> mu = beta/alpha. Since either lambda or mu may overflow,
|
||||
*> they should not, in general, be computed.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] VL
|
||||
*> \verbatim
|
||||
*> VL is REAL array, dimension (LDVL,N)
|
||||
*> If JOBVL = 'V', the left eigenvectors u(j) are stored
|
||||
*> in the columns of VL, in the same order as their eigenvalues.
|
||||
*> If the j-th eigenvalue is real, then u(j) = VL(:,j).
|
||||
*> If the j-th and (j+1)-st eigenvalues form a complex conjugate
|
||||
*> pair, then
|
||||
*> u(j) = VL(:,j) + i*VL(:,j+1)
|
||||
*> and
|
||||
*> u(j+1) = VL(:,j) - i*VL(:,j+1).
|
||||
*>
|
||||
*> Each eigenvector is scaled so that its largest component has
|
||||
*> abs(real part) + abs(imag. part) = 1, except for eigenvectors
|
||||
*> corresponding to an eigenvalue with alpha = beta = 0, which
|
||||
*> are set to zero.
|
||||
*> Not referenced if JOBVL = 'N'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDVL
|
||||
*> \verbatim
|
||||
*> LDVL is INTEGER
|
||||
*> The leading dimension of the matrix VL. LDVL >= 1, and
|
||||
*> if JOBVL = 'V', LDVL >= N.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] VR
|
||||
*> \verbatim
|
||||
*> VR is REAL array, dimension (LDVR,N)
|
||||
*> If JOBVR = 'V', the right eigenvectors x(j) are stored
|
||||
*> in the columns of VR, in the same order as their eigenvalues.
|
||||
*> If the j-th eigenvalue is real, then x(j) = VR(:,j).
|
||||
*> If the j-th and (j+1)-st eigenvalues form a complex conjugate
|
||||
*> pair, then
|
||||
*> x(j) = VR(:,j) + i*VR(:,j+1)
|
||||
*> and
|
||||
*> x(j+1) = VR(:,j) - i*VR(:,j+1).
|
||||
*>
|
||||
*> Each eigenvector is scaled so that its largest component has
|
||||
*> abs(real part) + abs(imag. part) = 1, except for eigenvalues
|
||||
*> corresponding to an eigenvalue with alpha = beta = 0, which
|
||||
*> are set to zero.
|
||||
*> Not referenced if JOBVR = 'N'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDVR
|
||||
*> \verbatim
|
||||
*> LDVR is INTEGER
|
||||
*> The leading dimension of the matrix VR. LDVR >= 1, and
|
||||
*> if JOBVR = 'V', LDVR >= N.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] WORK
|
||||
*> \verbatim
|
||||
*> WORK is REAL array, dimension (MAX(1,LWORK))
|
||||
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LWORK
|
||||
*> \verbatim
|
||||
*> LWORK is INTEGER
|
||||
*> The dimension of the array WORK. LWORK >= max(1,8*N).
|
||||
*> For good performance, LWORK must generally be larger.
|
||||
*> To compute the optimal value of LWORK, call ILAENV to get
|
||||
*> blocksizes (for SGEQRF, SORMQR, and SORGQR.) Then compute:
|
||||
*> NB -- MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR;
|
||||
*> The optimal LWORK is:
|
||||
*> 2*N + MAX( 6*N, N*(NB+1) ).
|
||||
*>
|
||||
*> If LWORK = -1, then a workspace query is assumed; the routine
|
||||
*> only calculates the optimal size of the WORK array, returns
|
||||
*> this value as the first entry of the WORK array, and no error
|
||||
*> message related to LWORK is issued by XERBLA.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] INFO
|
||||
*> \verbatim
|
||||
*> INFO is INTEGER
|
||||
*> = 0: successful exit
|
||||
*> < 0: if INFO = -i, the i-th argument had an illegal value.
|
||||
*> = 1,...,N:
|
||||
*> The QZ iteration failed. No eigenvectors have been
|
||||
*> calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
|
||||
*> should be correct for j=INFO+1,...,N.
|
||||
*> > N: errors that usually indicate LAPACK problems:
|
||||
*> =N+1: error return from SGGBAL
|
||||
*> =N+2: error return from SGEQRF
|
||||
*> =N+3: error return from SORMQR
|
||||
*> =N+4: error return from SORGQR
|
||||
*> =N+5: error return from SGGHRD
|
||||
*> =N+6: error return from SHGEQZ (other than failed
|
||||
*> iteration)
|
||||
*> =N+7: error return from STGEVC
|
||||
*> =N+8: error return from SGGBAK (computing VL)
|
||||
*> =N+9: error return from SGGBAK (computing VR)
|
||||
*> =N+10: error return from SLASCL (various calls)
|
||||
*> \endverbatim
|
||||
*
|
||||
* Authors:
|
||||
* ========
|
||||
*
|
||||
*> \author Univ. of Tennessee
|
||||
*> \author Univ. of California Berkeley
|
||||
*> \author Univ. of Colorado Denver
|
||||
*> \author NAG Ltd.
|
||||
*
|
||||
*> \ingroup realGEeigen
|
||||
*
|
||||
*> \par Further Details:
|
||||
* =====================
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> Balancing
|
||||
*> ---------
|
||||
*>
|
||||
*> This driver calls SGGBAL to both permute and scale rows and columns
|
||||
*> of A and B. The permutations PL and PR are chosen so that PL*A*PR
|
||||
*> and PL*B*R will be upper triangular except for the diagonal blocks
|
||||
*> A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
|
||||
*> possible. The diagonal scaling matrices DL and DR are chosen so
|
||||
*> that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
|
||||
*> one (except for the elements that start out zero.)
|
||||
*>
|
||||
*> After the eigenvalues and eigenvectors of the balanced matrices
|
||||
*> have been computed, SGGBAK transforms the eigenvectors back to what
|
||||
*> they would have been (in perfect arithmetic) if they had not been
|
||||
*> balanced.
|
||||
*>
|
||||
*> Contents of A and B on Exit
|
||||
*> -------- -- - --- - -- ----
|
||||
*>
|
||||
*> If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
|
||||
*> both), then on exit the arrays A and B will contain the real Schur
|
||||
*> form[*] of the "balanced" versions of A and B. If no eigenvectors
|
||||
*> are computed, then only the diagonal blocks will be correct.
|
||||
*>
|
||||
*> [*] See SHGEQZ, SGEGS, or read the book "Matrix Computations",
|
||||
*> by Golub & van Loan, pub. by Johns Hopkins U. Press.
|
||||
*> \endverbatim
|
||||
*>
|
||||
* =====================================================================
|
||||
SUBROUTINE SGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
|
||||
$ BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER JOBVL, JOBVR
|
||||
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
|
||||
$ B( LDB, * ), BETA( * ), VL( LDVL, * ),
|
||||
$ VR( LDVR, * ), WORK( * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
REAL ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL ILIMIT, ILV, ILVL, ILVR, LQUERY
|
||||
CHARACTER CHTEMP
|
||||
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
|
||||
$ IN, IRIGHT, IROWS, ITAU, IWORK, JC, JR, LOPT,
|
||||
$ LWKMIN, LWKOPT, NB, NB1, NB2, NB3
|
||||
REAL ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM,
|
||||
$ BNRM1, BNRM2, EPS, ONEPLS, SAFMAX, SAFMIN,
|
||||
$ SALFAI, SALFAR, SBETA, SCALE, TEMP
|
||||
* ..
|
||||
* .. Local Arrays ..
|
||||
LOGICAL LDUMMA( 1 )
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL SGEQRF, SGGBAK, SGGBAL, SGGHRD, SHGEQZ, SLACPY,
|
||||
$ SLASCL, SLASET, SORGQR, SORMQR, STGEVC, XERBLA
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
INTEGER ILAENV
|
||||
REAL SLAMCH, SLANGE
|
||||
EXTERNAL ILAENV, LSAME, SLAMCH, SLANGE
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC ABS, INT, MAX
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
* Decode the input arguments
|
||||
*
|
||||
IF( LSAME( JOBVL, 'N' ) ) THEN
|
||||
IJOBVL = 1
|
||||
ILVL = .FALSE.
|
||||
ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
|
||||
IJOBVL = 2
|
||||
ILVL = .TRUE.
|
||||
ELSE
|
||||
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
|
||||
*
|
||||
* Test the input arguments
|
||||
*
|
||||
LWKMIN = MAX( 8*N, 1 )
|
||||
LWKOPT = LWKMIN
|
||||
WORK( 1 ) = LWKOPT
|
||||
LQUERY = ( LWORK.EQ.-1 )
|
||||
INFO = 0
|
||||
IF( IJOBVL.LE.0 ) THEN
|
||||
INFO = -1
|
||||
ELSE IF( IJOBVR.LE.0 ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -5
|
||||
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -7
|
||||
ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
|
||||
INFO = -12
|
||||
ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
|
||||
INFO = -14
|
||||
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
|
||||
INFO = -16
|
||||
END IF
|
||||
*
|
||||
IF( INFO.EQ.0 ) THEN
|
||||
NB1 = ILAENV( 1, 'SGEQRF', ' ', N, N, -1, -1 )
|
||||
NB2 = ILAENV( 1, 'SORMQR', ' ', N, N, N, -1 )
|
||||
NB3 = ILAENV( 1, 'SORGQR', ' ', N, N, N, -1 )
|
||||
NB = MAX( NB1, NB2, NB3 )
|
||||
LOPT = 2*N + MAX( 6*N, N*(NB+1) )
|
||||
WORK( 1 ) = LOPT
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'SGEGV ', -INFO )
|
||||
RETURN
|
||||
ELSE IF( LQUERY ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Quick return if possible
|
||||
*
|
||||
IF( N.EQ.0 )
|
||||
$ RETURN
|
||||
*
|
||||
* Get machine constants
|
||||
*
|
||||
EPS = SLAMCH( 'E' )*SLAMCH( 'B' )
|
||||
SAFMIN = SLAMCH( 'S' )
|
||||
SAFMIN = SAFMIN + SAFMIN
|
||||
SAFMAX = ONE / SAFMIN
|
||||
ONEPLS = ONE + ( 4*EPS )
|
||||
*
|
||||
* Scale A
|
||||
*
|
||||
ANRM = SLANGE( 'M', N, N, A, LDA, WORK )
|
||||
ANRM1 = ANRM
|
||||
ANRM2 = ONE
|
||||
IF( ANRM.LT.ONE ) THEN
|
||||
IF( SAFMAX*ANRM.LT.ONE ) THEN
|
||||
ANRM1 = SAFMIN
|
||||
ANRM2 = SAFMAX*ANRM
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( ANRM.GT.ZERO ) THEN
|
||||
CALL SLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 10
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Scale B
|
||||
*
|
||||
BNRM = SLANGE( 'M', N, N, B, LDB, WORK )
|
||||
BNRM1 = BNRM
|
||||
BNRM2 = ONE
|
||||
IF( BNRM.LT.ONE ) THEN
|
||||
IF( SAFMAX*BNRM.LT.ONE ) THEN
|
||||
BNRM1 = SAFMIN
|
||||
BNRM2 = SAFMAX*BNRM
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( BNRM.GT.ZERO ) THEN
|
||||
CALL SLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 10
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Permute the matrix to make it more nearly triangular
|
||||
* Workspace layout: (8*N words -- "work" requires 6*N words)
|
||||
* left_permutation, right_permutation, work...
|
||||
*
|
||||
ILEFT = 1
|
||||
IRIGHT = N + 1
|
||||
IWORK = IRIGHT + N
|
||||
CALL SGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
|
||||
$ WORK( IRIGHT ), WORK( IWORK ), IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 1
|
||||
GO TO 120
|
||||
END IF
|
||||
*
|
||||
* Reduce B to triangular form, and initialize VL and/or VR
|
||||
* Workspace layout: ("work..." must have at least N words)
|
||||
* left_permutation, right_permutation, tau, work...
|
||||
*
|
||||
IROWS = IHI + 1 - ILO
|
||||
IF( ILV ) THEN
|
||||
ICOLS = N + 1 - ILO
|
||||
ELSE
|
||||
ICOLS = IROWS
|
||||
END IF
|
||||
ITAU = IWORK
|
||||
IWORK = ITAU + IROWS
|
||||
CALL SGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
|
||||
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 2
|
||||
GO TO 120
|
||||
END IF
|
||||
*
|
||||
CALL SORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
|
||||
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
|
||||
$ LWORK+1-IWORK, IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 3
|
||||
GO TO 120
|
||||
END IF
|
||||
*
|
||||
IF( ILVL ) THEN
|
||||
CALL SLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
|
||||
CALL SLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
|
||||
$ VL( ILO+1, ILO ), LDVL )
|
||||
CALL SORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
|
||||
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
|
||||
$ IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 4
|
||||
GO TO 120
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( ILVR )
|
||||
$ CALL SLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
|
||||
*
|
||||
* Reduce to generalized Hessenberg form
|
||||
*
|
||||
IF( ILV ) THEN
|
||||
*
|
||||
* Eigenvectors requested -- work on whole matrix.
|
||||
*
|
||||
CALL SGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
|
||||
$ LDVL, VR, LDVR, IINFO )
|
||||
ELSE
|
||||
CALL SGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
|
||||
$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO )
|
||||
END IF
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 5
|
||||
GO TO 120
|
||||
END IF
|
||||
*
|
||||
* Perform QZ algorithm
|
||||
* Workspace layout: ("work..." must have at least 1 word)
|
||||
* left_permutation, right_permutation, work...
|
||||
*
|
||||
IWORK = ITAU
|
||||
IF( ILV ) THEN
|
||||
CHTEMP = 'S'
|
||||
ELSE
|
||||
CHTEMP = 'E'
|
||||
END IF
|
||||
CALL SHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
|
||||
$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR,
|
||||
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
|
||||
INFO = IINFO
|
||||
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
|
||||
INFO = IINFO - N
|
||||
ELSE
|
||||
INFO = N + 6
|
||||
END IF
|
||||
GO TO 120
|
||||
END IF
|
||||
*
|
||||
IF( ILV ) THEN
|
||||
*
|
||||
* Compute Eigenvectors (STGEVC requires 6*N words of workspace)
|
||||
*
|
||||
IF( ILVL ) THEN
|
||||
IF( ILVR ) THEN
|
||||
CHTEMP = 'B'
|
||||
ELSE
|
||||
CHTEMP = 'L'
|
||||
END IF
|
||||
ELSE
|
||||
CHTEMP = 'R'
|
||||
END IF
|
||||
*
|
||||
CALL STGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
|
||||
$ VR, LDVR, N, IN, WORK( IWORK ), IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 7
|
||||
GO TO 120
|
||||
END IF
|
||||
*
|
||||
* Undo balancing on VL and VR, rescale
|
||||
*
|
||||
IF( ILVL ) THEN
|
||||
CALL SGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
|
||||
$ WORK( IRIGHT ), N, VL, LDVL, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 8
|
||||
GO TO 120
|
||||
END IF
|
||||
DO 50 JC = 1, N
|
||||
IF( ALPHAI( JC ).LT.ZERO )
|
||||
$ GO TO 50
|
||||
TEMP = ZERO
|
||||
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
||||
DO 10 JR = 1, N
|
||||
TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
|
||||
10 CONTINUE
|
||||
ELSE
|
||||
DO 20 JR = 1, N
|
||||
TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
|
||||
$ ABS( VL( JR, JC+1 ) ) )
|
||||
20 CONTINUE
|
||||
END IF
|
||||
IF( TEMP.LT.SAFMIN )
|
||||
$ GO TO 50
|
||||
TEMP = ONE / TEMP
|
||||
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
||||
DO 30 JR = 1, N
|
||||
VL( JR, JC ) = VL( JR, JC )*TEMP
|
||||
30 CONTINUE
|
||||
ELSE
|
||||
DO 40 JR = 1, N
|
||||
VL( JR, JC ) = VL( JR, JC )*TEMP
|
||||
VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
|
||||
40 CONTINUE
|
||||
END IF
|
||||
50 CONTINUE
|
||||
END IF
|
||||
IF( ILVR ) THEN
|
||||
CALL SGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
|
||||
$ WORK( IRIGHT ), N, VR, LDVR, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
GO TO 120
|
||||
END IF
|
||||
DO 100 JC = 1, N
|
||||
IF( ALPHAI( JC ).LT.ZERO )
|
||||
$ GO TO 100
|
||||
TEMP = ZERO
|
||||
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
||||
DO 60 JR = 1, N
|
||||
TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
|
||||
60 CONTINUE
|
||||
ELSE
|
||||
DO 70 JR = 1, N
|
||||
TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
|
||||
$ ABS( VR( JR, JC+1 ) ) )
|
||||
70 CONTINUE
|
||||
END IF
|
||||
IF( TEMP.LT.SAFMIN )
|
||||
$ GO TO 100
|
||||
TEMP = ONE / TEMP
|
||||
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
||||
DO 80 JR = 1, N
|
||||
VR( JR, JC ) = VR( JR, JC )*TEMP
|
||||
80 CONTINUE
|
||||
ELSE
|
||||
DO 90 JR = 1, N
|
||||
VR( JR, JC ) = VR( JR, JC )*TEMP
|
||||
VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
|
||||
90 CONTINUE
|
||||
END IF
|
||||
100 CONTINUE
|
||||
END IF
|
||||
*
|
||||
* End of eigenvector calculation
|
||||
*
|
||||
END IF
|
||||
*
|
||||
* Undo scaling in alpha, beta
|
||||
*
|
||||
* Note: this does not give the alpha and beta for the unscaled
|
||||
* problem.
|
||||
*
|
||||
* Un-scaling is limited to avoid underflow in alpha and beta
|
||||
* if they are significant.
|
||||
*
|
||||
DO 110 JC = 1, N
|
||||
ABSAR = ABS( ALPHAR( JC ) )
|
||||
ABSAI = ABS( ALPHAI( JC ) )
|
||||
ABSB = ABS( BETA( JC ) )
|
||||
SALFAR = ANRM*ALPHAR( JC )
|
||||
SALFAI = ANRM*ALPHAI( JC )
|
||||
SBETA = BNRM*BETA( JC )
|
||||
ILIMIT = .FALSE.
|
||||
SCALE = ONE
|
||||
*
|
||||
* Check for significant underflow in ALPHAI
|
||||
*
|
||||
IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE.
|
||||
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN
|
||||
ILIMIT = .TRUE.
|
||||
SCALE = ( ONEPLS*SAFMIN / ANRM1 ) /
|
||||
$ MAX( ONEPLS*SAFMIN, ANRM2*ABSAI )
|
||||
*
|
||||
ELSE IF( SALFAI.EQ.ZERO ) THEN
|
||||
*
|
||||
* If insignificant underflow in ALPHAI, then make the
|
||||
* conjugate eigenvalue real.
|
||||
*
|
||||
IF( ALPHAI( JC ).LT.ZERO .AND. JC.GT.1 ) THEN
|
||||
ALPHAI( JC-1 ) = ZERO
|
||||
ELSE IF( ALPHAI( JC ).GT.ZERO .AND. JC.LT.N ) THEN
|
||||
ALPHAI( JC+1 ) = ZERO
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Check for significant underflow in ALPHAR
|
||||
*
|
||||
IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE.
|
||||
$ MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN
|
||||
ILIMIT = .TRUE.
|
||||
SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / ANRM1 ) /
|
||||
$ MAX( ONEPLS*SAFMIN, ANRM2*ABSAR ) )
|
||||
END IF
|
||||
*
|
||||
* Check for significant underflow in BETA
|
||||
*
|
||||
IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE.
|
||||
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN
|
||||
ILIMIT = .TRUE.
|
||||
SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / BNRM1 ) /
|
||||
$ MAX( ONEPLS*SAFMIN, BNRM2*ABSB ) )
|
||||
END IF
|
||||
*
|
||||
* Check for possible overflow when limiting scaling
|
||||
*
|
||||
IF( ILIMIT ) THEN
|
||||
TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ),
|
||||
$ ABS( SBETA ) )
|
||||
IF( TEMP.GT.ONE )
|
||||
$ SCALE = SCALE / TEMP
|
||||
IF( SCALE.LT.ONE )
|
||||
$ ILIMIT = .FALSE.
|
||||
END IF
|
||||
*
|
||||
* Recompute un-scaled ALPHAR, ALPHAI, BETA if necessary.
|
||||
*
|
||||
IF( ILIMIT ) THEN
|
||||
SALFAR = ( SCALE*ALPHAR( JC ) )*ANRM
|
||||
SALFAI = ( SCALE*ALPHAI( JC ) )*ANRM
|
||||
SBETA = ( SCALE*BETA( JC ) )*BNRM
|
||||
END IF
|
||||
ALPHAR( JC ) = SALFAR
|
||||
ALPHAI( JC ) = SALFAI
|
||||
BETA( JC ) = SBETA
|
||||
110 CONTINUE
|
||||
*
|
||||
120 CONTINUE
|
||||
WORK( 1 ) = LWKOPT
|
||||
*
|
||||
RETURN
|
||||
*
|
||||
* End of SGEGV
|
||||
*
|
||||
END
|
||||
|
|
@ -0,0 +1,528 @@
|
|||
*> \brief <b> ZGEGS computes the eigenvalues, Schur form, and, optionally, the left and or/right Schur vectors of a complex matrix pair (A,B)</b>
|
||||
*
|
||||
* =========== DOCUMENTATION ===========
|
||||
*
|
||||
* Online html documentation available at
|
||||
* http://www.netlib.org/lapack/explore-html/
|
||||
*
|
||||
*> \htmlonly
|
||||
*> Download ZGEGS + dependencies
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgegs.f">
|
||||
*> [TGZ]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgegs.f">
|
||||
*> [ZIP]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgegs.f">
|
||||
*> [TXT]</a>
|
||||
*> \endhtmlonly
|
||||
*
|
||||
* Definition:
|
||||
* ===========
|
||||
*
|
||||
* SUBROUTINE ZGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA,
|
||||
* VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK,
|
||||
* INFO )
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
* CHARACTER JOBVSL, JOBVSR
|
||||
* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
* DOUBLE PRECISION RWORK( * )
|
||||
* COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
|
||||
* $ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
|
||||
* $ WORK( * )
|
||||
* ..
|
||||
*
|
||||
*
|
||||
*> \par Purpose:
|
||||
* =============
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> This routine is deprecated and has been replaced by routine ZGGES.
|
||||
*>
|
||||
*> ZGEGS computes the eigenvalues, Schur form, and, optionally, the
|
||||
*> left and or/right Schur vectors of a complex matrix pair (A,B).
|
||||
*> Given two square matrices A and B, the generalized Schur
|
||||
*> factorization has the form
|
||||
*>
|
||||
*> A = Q*S*Z**H, B = Q*T*Z**H
|
||||
*>
|
||||
*> where Q and Z are unitary matrices and S and T are upper triangular.
|
||||
*> The columns of Q are the left Schur vectors
|
||||
*> and the columns of Z are the right Schur vectors.
|
||||
*>
|
||||
*> If only the eigenvalues of (A,B) are needed, the driver routine
|
||||
*> ZGEGV should be used instead. See ZGEGV for a description of the
|
||||
*> eigenvalues of the generalized nonsymmetric eigenvalue problem
|
||||
*> (GNEP).
|
||||
*> \endverbatim
|
||||
*
|
||||
* Arguments:
|
||||
* ==========
|
||||
*
|
||||
*> \param[in] JOBVSL
|
||||
*> \verbatim
|
||||
*> JOBVSL is CHARACTER*1
|
||||
*> = 'N': do not compute the left Schur vectors;
|
||||
*> = 'V': compute the left Schur vectors (returned in VSL).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] JOBVSR
|
||||
*> \verbatim
|
||||
*> JOBVSR is CHARACTER*1
|
||||
*> = 'N': do not compute the right Schur vectors;
|
||||
*> = 'V': compute the right Schur vectors (returned in VSR).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] N
|
||||
*> \verbatim
|
||||
*> N is INTEGER
|
||||
*> The order of the matrices A, B, VSL, and VSR. N >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] A
|
||||
*> \verbatim
|
||||
*> A is COMPLEX*16 array, dimension (LDA, N)
|
||||
*> On entry, the matrix A.
|
||||
*> On exit, the upper triangular matrix S from the generalized
|
||||
*> Schur factorization.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDA
|
||||
*> \verbatim
|
||||
*> LDA is INTEGER
|
||||
*> The leading dimension of A. LDA >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] B
|
||||
*> \verbatim
|
||||
*> B is COMPLEX*16 array, dimension (LDB, N)
|
||||
*> On entry, the matrix B.
|
||||
*> On exit, the upper triangular matrix T from the generalized
|
||||
*> Schur factorization.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDB
|
||||
*> \verbatim
|
||||
*> LDB is INTEGER
|
||||
*> The leading dimension of B. LDB >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] ALPHA
|
||||
*> \verbatim
|
||||
*> ALPHA is COMPLEX*16 array, dimension (N)
|
||||
*> The complex scalars alpha that define the eigenvalues of
|
||||
*> GNEP. ALPHA(j) = S(j,j), the diagonal element of the Schur
|
||||
*> form of A.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] BETA
|
||||
*> \verbatim
|
||||
*> BETA is COMPLEX*16 array, dimension (N)
|
||||
*> The non-negative real scalars beta that define the
|
||||
*> eigenvalues of GNEP. BETA(j) = T(j,j), the diagonal element
|
||||
*> of the triangular factor T.
|
||||
*>
|
||||
*> Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
|
||||
*> represent the j-th eigenvalue of the matrix pair (A,B), in
|
||||
*> one of the forms lambda = alpha/beta or mu = beta/alpha.
|
||||
*> Since either lambda or mu may overflow, they should not,
|
||||
*> in general, be computed.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] VSL
|
||||
*> \verbatim
|
||||
*> VSL is COMPLEX*16 array, dimension (LDVSL,N)
|
||||
*> If JOBVSL = 'V', the matrix of left Schur vectors Q.
|
||||
*> Not referenced if JOBVSL = 'N'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDVSL
|
||||
*> \verbatim
|
||||
*> LDVSL is INTEGER
|
||||
*> The leading dimension of the matrix VSL. LDVSL >= 1, and
|
||||
*> if JOBVSL = 'V', LDVSL >= N.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] VSR
|
||||
*> \verbatim
|
||||
*> VSR is COMPLEX*16 array, dimension (LDVSR,N)
|
||||
*> If JOBVSR = 'V', the matrix of right Schur vectors Z.
|
||||
*> Not referenced if JOBVSR = 'N'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDVSR
|
||||
*> \verbatim
|
||||
*> LDVSR is INTEGER
|
||||
*> The leading dimension of the matrix VSR. LDVSR >= 1, and
|
||||
*> if JOBVSR = 'V', LDVSR >= N.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] WORK
|
||||
*> \verbatim
|
||||
*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
|
||||
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LWORK
|
||||
*> \verbatim
|
||||
*> LWORK is INTEGER
|
||||
*> The dimension of the array WORK. LWORK >= max(1,2*N).
|
||||
*> For good performance, LWORK must generally be larger.
|
||||
*> To compute the optimal value of LWORK, call ILAENV to get
|
||||
*> blocksizes (for ZGEQRF, ZUNMQR, and CUNGQR.) Then compute:
|
||||
*> NB -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and CUNGQR;
|
||||
*> the optimal LWORK is N*(NB+1).
|
||||
*>
|
||||
*> If LWORK = -1, then a workspace query is assumed; the routine
|
||||
*> only calculates the optimal size of the WORK array, returns
|
||||
*> this value as the first entry of the WORK array, and no error
|
||||
*> message related to LWORK is issued by XERBLA.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] RWORK
|
||||
*> \verbatim
|
||||
*> RWORK is DOUBLE PRECISION array, dimension (3*N)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] INFO
|
||||
*> \verbatim
|
||||
*> INFO is INTEGER
|
||||
*> = 0: successful exit
|
||||
*> < 0: if INFO = -i, the i-th argument had an illegal value.
|
||||
*> =1,...,N:
|
||||
*> The QZ iteration failed. (A,B) are not in Schur
|
||||
*> form, but ALPHA(j) and BETA(j) should be correct for
|
||||
*> j=INFO+1,...,N.
|
||||
*> > N: errors that usually indicate LAPACK problems:
|
||||
*> =N+1: error return from ZGGBAL
|
||||
*> =N+2: error return from ZGEQRF
|
||||
*> =N+3: error return from ZUNMQR
|
||||
*> =N+4: error return from ZUNGQR
|
||||
*> =N+5: error return from ZGGHRD
|
||||
*> =N+6: error return from ZHGEQZ (other than failed
|
||||
*> iteration)
|
||||
*> =N+7: error return from ZGGBAK (computing VSL)
|
||||
*> =N+8: error return from ZGGBAK (computing VSR)
|
||||
*> =N+9: error return from ZLASCL (various places)
|
||||
*> \endverbatim
|
||||
*
|
||||
* Authors:
|
||||
* ========
|
||||
*
|
||||
*> \author Univ. of Tennessee
|
||||
*> \author Univ. of California Berkeley
|
||||
*> \author Univ. of Colorado Denver
|
||||
*> \author NAG Ltd.
|
||||
*
|
||||
*> \ingroup complex16GEeigen
|
||||
*
|
||||
* =====================================================================
|
||||
SUBROUTINE ZGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA,
|
||||
$ VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK,
|
||||
$ INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER JOBVSL, JOBVSR
|
||||
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
DOUBLE PRECISION RWORK( * )
|
||||
COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
|
||||
$ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
|
||||
$ WORK( * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
DOUBLE PRECISION ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
|
||||
COMPLEX*16 CZERO, CONE
|
||||
PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ),
|
||||
$ CONE = ( 1.0D0, 0.0D0 ) )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL ILASCL, ILBSCL, ILVSL, ILVSR, LQUERY
|
||||
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
|
||||
$ IRIGHT, IROWS, IRWORK, ITAU, IWORK, LOPT,
|
||||
$ LWKMIN, LWKOPT, NB, NB1, NB2, NB3
|
||||
DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
|
||||
$ SAFMIN, SMLNUM
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL XERBLA, ZGEQRF, ZGGBAK, ZGGBAL, ZGGHRD, ZHGEQZ,
|
||||
$ ZLACPY, ZLASCL, ZLASET, ZUNGQR, ZUNMQR
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
INTEGER ILAENV
|
||||
DOUBLE PRECISION DLAMCH, ZLANGE
|
||||
EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC INT, MAX
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
* Decode the input arguments
|
||||
*
|
||||
IF( LSAME( JOBVSL, 'N' ) ) THEN
|
||||
IJOBVL = 1
|
||||
ILVSL = .FALSE.
|
||||
ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
|
||||
IJOBVL = 2
|
||||
ILVSL = .TRUE.
|
||||
ELSE
|
||||
IJOBVL = -1
|
||||
ILVSL = .FALSE.
|
||||
END IF
|
||||
*
|
||||
IF( LSAME( JOBVSR, 'N' ) ) THEN
|
||||
IJOBVR = 1
|
||||
ILVSR = .FALSE.
|
||||
ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
|
||||
IJOBVR = 2
|
||||
ILVSR = .TRUE.
|
||||
ELSE
|
||||
IJOBVR = -1
|
||||
ILVSR = .FALSE.
|
||||
END IF
|
||||
*
|
||||
* Test the input arguments
|
||||
*
|
||||
LWKMIN = MAX( 2*N, 1 )
|
||||
LWKOPT = LWKMIN
|
||||
WORK( 1 ) = LWKOPT
|
||||
LQUERY = ( LWORK.EQ.-1 )
|
||||
INFO = 0
|
||||
IF( IJOBVL.LE.0 ) THEN
|
||||
INFO = -1
|
||||
ELSE IF( IJOBVR.LE.0 ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -5
|
||||
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -7
|
||||
ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
|
||||
INFO = -11
|
||||
ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
|
||||
INFO = -13
|
||||
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
|
||||
INFO = -15
|
||||
END IF
|
||||
*
|
||||
IF( INFO.EQ.0 ) THEN
|
||||
NB1 = ILAENV( 1, 'ZGEQRF', ' ', N, N, -1, -1 )
|
||||
NB2 = ILAENV( 1, 'ZUNMQR', ' ', N, N, N, -1 )
|
||||
NB3 = ILAENV( 1, 'ZUNGQR', ' ', N, N, N, -1 )
|
||||
NB = MAX( NB1, NB2, NB3 )
|
||||
LOPT = N*( NB+1 )
|
||||
WORK( 1 ) = LOPT
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'ZGEGS ', -INFO )
|
||||
RETURN
|
||||
ELSE IF( LQUERY ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Quick return if possible
|
||||
*
|
||||
IF( N.EQ.0 )
|
||||
$ RETURN
|
||||
*
|
||||
* Get machine constants
|
||||
*
|
||||
EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
|
||||
SAFMIN = DLAMCH( 'S' )
|
||||
SMLNUM = N*SAFMIN / EPS
|
||||
BIGNUM = ONE / SMLNUM
|
||||
*
|
||||
* Scale A if max element outside range [SMLNUM,BIGNUM]
|
||||
*
|
||||
ANRM = ZLANGE( '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 ) THEN
|
||||
CALL ZLASCL( 'G', -1, -1, ANRM, ANRMTO, N, N, A, LDA, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Scale B if max element outside range [SMLNUM,BIGNUM]
|
||||
*
|
||||
BNRM = ZLANGE( '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 ) THEN
|
||||
CALL ZLASCL( 'G', -1, -1, BNRM, BNRMTO, N, N, B, LDB, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Permute the matrix to make it more nearly triangular
|
||||
*
|
||||
ILEFT = 1
|
||||
IRIGHT = N + 1
|
||||
IRWORK = IRIGHT + N
|
||||
IWORK = 1
|
||||
CALL ZGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ),
|
||||
$ RWORK( IRIGHT ), RWORK( IRWORK ), IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 1
|
||||
GO TO 10
|
||||
END IF
|
||||
*
|
||||
* Reduce B to triangular form, and initialize VSL and/or VSR
|
||||
*
|
||||
IROWS = IHI + 1 - ILO
|
||||
ICOLS = N + 1 - ILO
|
||||
ITAU = IWORK
|
||||
IWORK = ITAU + IROWS
|
||||
CALL ZGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
|
||||
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 2
|
||||
GO TO 10
|
||||
END IF
|
||||
*
|
||||
CALL ZUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
|
||||
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
|
||||
$ LWORK+1-IWORK, IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 3
|
||||
GO TO 10
|
||||
END IF
|
||||
*
|
||||
IF( ILVSL ) THEN
|
||||
CALL ZLASET( 'Full', N, N, CZERO, CONE, VSL, LDVSL )
|
||||
CALL ZLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
|
||||
$ VSL( ILO+1, ILO ), LDVSL )
|
||||
CALL ZUNGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
|
||||
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
|
||||
$ IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 4
|
||||
GO TO 10
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( ILVSR )
|
||||
$ CALL ZLASET( 'Full', N, N, CZERO, CONE, VSR, LDVSR )
|
||||
*
|
||||
* Reduce to generalized Hessenberg form
|
||||
*
|
||||
CALL ZGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
|
||||
$ LDVSL, VSR, LDVSR, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 5
|
||||
GO TO 10
|
||||
END IF
|
||||
*
|
||||
* Perform QZ algorithm, computing Schur vectors if desired
|
||||
*
|
||||
IWORK = ITAU
|
||||
CALL ZHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
|
||||
$ ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK( IWORK ),
|
||||
$ LWORK+1-IWORK, RWORK( IRWORK ), IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
|
||||
INFO = IINFO
|
||||
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
|
||||
INFO = IINFO - N
|
||||
ELSE
|
||||
INFO = N + 6
|
||||
END IF
|
||||
GO TO 10
|
||||
END IF
|
||||
*
|
||||
* Apply permutation to VSL and VSR
|
||||
*
|
||||
IF( ILVSL ) THEN
|
||||
CALL ZGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ),
|
||||
$ RWORK( IRIGHT ), N, VSL, LDVSL, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 7
|
||||
GO TO 10
|
||||
END IF
|
||||
END IF
|
||||
IF( ILVSR ) THEN
|
||||
CALL ZGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ),
|
||||
$ RWORK( IRIGHT ), N, VSR, LDVSR, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 8
|
||||
GO TO 10
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Undo scaling
|
||||
*
|
||||
IF( ILASCL ) THEN
|
||||
CALL ZLASCL( 'U', -1, -1, ANRMTO, ANRM, N, N, A, LDA, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
CALL ZLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHA, N, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( ILBSCL ) THEN
|
||||
CALL ZLASCL( 'U', -1, -1, BNRMTO, BNRM, N, N, B, LDB, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
CALL ZLASCL( 'G', -1, -1, BNRMTO, BNRM, N, 1, BETA, N, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
10 CONTINUE
|
||||
WORK( 1 ) = LWKOPT
|
||||
*
|
||||
RETURN
|
||||
*
|
||||
* End of ZGEGS
|
||||
*
|
||||
END
|
||||
|
|
@ -0,0 +1,703 @@
|
|||
*> \brief <b> ZGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a complex matrix pair (A,B).</b>
|
||||
*
|
||||
* =========== DOCUMENTATION ===========
|
||||
*
|
||||
* Online html documentation available at
|
||||
* http://www.netlib.org/lapack/explore-html/
|
||||
*
|
||||
*> \htmlonly
|
||||
*> Download ZGEGV + dependencies
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgegv.f">
|
||||
*> [TGZ]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgegv.f">
|
||||
*> [ZIP]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgegv.f">
|
||||
*> [TXT]</a>
|
||||
*> \endhtmlonly
|
||||
*
|
||||
* Definition:
|
||||
* ===========
|
||||
*
|
||||
* SUBROUTINE ZGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
|
||||
* VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
* CHARACTER JOBVL, JOBVR
|
||||
* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
* DOUBLE PRECISION RWORK( * )
|
||||
* COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
|
||||
* $ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
|
||||
* $ WORK( * )
|
||||
* ..
|
||||
*
|
||||
*
|
||||
*> \par Purpose:
|
||||
* =============
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> This routine is deprecated and has been replaced by routine ZGGEV.
|
||||
*>
|
||||
*> ZGEGV computes the eigenvalues and, optionally, the left and/or right
|
||||
*> eigenvectors of a complex matrix pair (A,B).
|
||||
*> Given two square matrices A and B,
|
||||
*> the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
|
||||
*> eigenvalues lambda and corresponding (non-zero) eigenvectors x such
|
||||
*> that
|
||||
*> A*x = lambda*B*x.
|
||||
*>
|
||||
*> An alternate form is to find the eigenvalues mu and corresponding
|
||||
*> eigenvectors y such that
|
||||
*> mu*A*y = B*y.
|
||||
*>
|
||||
*> These two forms are equivalent with mu = 1/lambda and x = y if
|
||||
*> neither lambda nor mu is zero. In order to deal with the case that
|
||||
*> lambda or mu is zero or small, two values alpha and beta are returned
|
||||
*> for each eigenvalue, such that lambda = alpha/beta and
|
||||
*> mu = beta/alpha.
|
||||
*>
|
||||
*> The vectors x and y in the above equations are right eigenvectors of
|
||||
*> the matrix pair (A,B). Vectors u and v satisfying
|
||||
*> u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
|
||||
*> are left eigenvectors of (A,B).
|
||||
*>
|
||||
*> Note: this routine performs "full balancing" on A and B
|
||||
*> \endverbatim
|
||||
*
|
||||
* Arguments:
|
||||
* ==========
|
||||
*
|
||||
*> \param[in] JOBVL
|
||||
*> \verbatim
|
||||
*> JOBVL is CHARACTER*1
|
||||
*> = 'N': do not compute the left generalized eigenvectors;
|
||||
*> = 'V': compute the left generalized eigenvectors (returned
|
||||
*> in VL).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] JOBVR
|
||||
*> \verbatim
|
||||
*> JOBVR is CHARACTER*1
|
||||
*> = 'N': do not compute the right generalized eigenvectors;
|
||||
*> = 'V': compute the right generalized eigenvectors (returned
|
||||
*> in VR).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] N
|
||||
*> \verbatim
|
||||
*> N is INTEGER
|
||||
*> The order of the matrices A, B, VL, and VR. N >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] A
|
||||
*> \verbatim
|
||||
*> A is COMPLEX*16 array, dimension (LDA, N)
|
||||
*> On entry, the matrix A.
|
||||
*> If JOBVL = 'V' or JOBVR = 'V', then on exit A
|
||||
*> contains the Schur form of A from the generalized Schur
|
||||
*> factorization of the pair (A,B) after balancing. If no
|
||||
*> eigenvectors were computed, then only the diagonal elements
|
||||
*> of the Schur form will be correct. See ZGGHRD and ZHGEQZ
|
||||
*> for details.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDA
|
||||
*> \verbatim
|
||||
*> LDA is INTEGER
|
||||
*> The leading dimension of A. LDA >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] B
|
||||
*> \verbatim
|
||||
*> B is COMPLEX*16 array, dimension (LDB, N)
|
||||
*> On entry, the matrix B.
|
||||
*> If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
|
||||
*> upper triangular matrix obtained from B in the generalized
|
||||
*> Schur factorization of the pair (A,B) after balancing.
|
||||
*> If no eigenvectors were computed, then only the diagonal
|
||||
*> elements of B will be correct. See ZGGHRD and ZHGEQZ for
|
||||
*> details.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDB
|
||||
*> \verbatim
|
||||
*> LDB is INTEGER
|
||||
*> The leading dimension of B. LDB >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] ALPHA
|
||||
*> \verbatim
|
||||
*> ALPHA is COMPLEX*16 array, dimension (N)
|
||||
*> The complex scalars alpha that define the eigenvalues of
|
||||
*> GNEP.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] BETA
|
||||
*> \verbatim
|
||||
*> BETA is COMPLEX*16 array, dimension (N)
|
||||
*> The complex scalars beta that define the eigenvalues of GNEP.
|
||||
*>
|
||||
*> Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
|
||||
*> represent the j-th eigenvalue of the matrix pair (A,B), in
|
||||
*> one of the forms lambda = alpha/beta or mu = beta/alpha.
|
||||
*> Since either lambda or mu may overflow, they should not,
|
||||
*> in general, be computed.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] VL
|
||||
*> \verbatim
|
||||
*> VL is COMPLEX*16 array, dimension (LDVL,N)
|
||||
*> If JOBVL = 'V', the left eigenvectors u(j) are stored
|
||||
*> in the columns of VL, in the same order as their eigenvalues.
|
||||
*> Each eigenvector is scaled so that its largest component has
|
||||
*> abs(real part) + abs(imag. part) = 1, except for eigenvectors
|
||||
*> corresponding to an eigenvalue with alpha = beta = 0, which
|
||||
*> are set to zero.
|
||||
*> Not referenced if JOBVL = 'N'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDVL
|
||||
*> \verbatim
|
||||
*> LDVL is INTEGER
|
||||
*> The leading dimension of the matrix VL. LDVL >= 1, and
|
||||
*> if JOBVL = 'V', LDVL >= N.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] VR
|
||||
*> \verbatim
|
||||
*> VR is COMPLEX*16 array, dimension (LDVR,N)
|
||||
*> If JOBVR = 'V', the right eigenvectors x(j) are stored
|
||||
*> in the columns of VR, in the same order as their eigenvalues.
|
||||
*> Each eigenvector is scaled so that its largest component has
|
||||
*> abs(real part) + abs(imag. part) = 1, except for eigenvectors
|
||||
*> corresponding to an eigenvalue with alpha = beta = 0, which
|
||||
*> are set to zero.
|
||||
*> Not referenced if JOBVR = 'N'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDVR
|
||||
*> \verbatim
|
||||
*> LDVR is INTEGER
|
||||
*> The leading dimension of the matrix VR. LDVR >= 1, and
|
||||
*> if JOBVR = 'V', LDVR >= N.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] WORK
|
||||
*> \verbatim
|
||||
*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
|
||||
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LWORK
|
||||
*> \verbatim
|
||||
*> LWORK is INTEGER
|
||||
*> The dimension of the array WORK. LWORK >= max(1,2*N).
|
||||
*> For good performance, LWORK must generally be larger.
|
||||
*> To compute the optimal value of LWORK, call ILAENV to get
|
||||
*> blocksizes (for ZGEQRF, ZUNMQR, and ZUNGQR.) Then compute:
|
||||
*> NB -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and ZUNGQR;
|
||||
*> The optimal LWORK is MAX( 2*N, N*(NB+1) ).
|
||||
*>
|
||||
*> If LWORK = -1, then a workspace query is assumed; the routine
|
||||
*> only calculates the optimal size of the WORK array, returns
|
||||
*> this value as the first entry of the WORK array, and no error
|
||||
*> message related to LWORK is issued by XERBLA.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] RWORK
|
||||
*> \verbatim
|
||||
*> RWORK is DOUBLE PRECISION array, dimension (8*N)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] INFO
|
||||
*> \verbatim
|
||||
*> INFO is INTEGER
|
||||
*> = 0: successful exit
|
||||
*> < 0: if INFO = -i, the i-th argument had an illegal value.
|
||||
*> =1,...,N:
|
||||
*> The QZ iteration failed. No eigenvectors have been
|
||||
*> calculated, but ALPHA(j) and BETA(j) should be
|
||||
*> correct for j=INFO+1,...,N.
|
||||
*> > N: errors that usually indicate LAPACK problems:
|
||||
*> =N+1: error return from ZGGBAL
|
||||
*> =N+2: error return from ZGEQRF
|
||||
*> =N+3: error return from ZUNMQR
|
||||
*> =N+4: error return from ZUNGQR
|
||||
*> =N+5: error return from ZGGHRD
|
||||
*> =N+6: error return from ZHGEQZ (other than failed
|
||||
*> iteration)
|
||||
*> =N+7: error return from ZTGEVC
|
||||
*> =N+8: error return from ZGGBAK (computing VL)
|
||||
*> =N+9: error return from ZGGBAK (computing VR)
|
||||
*> =N+10: error return from ZLASCL (various calls)
|
||||
*> \endverbatim
|
||||
*
|
||||
* Authors:
|
||||
* ========
|
||||
*
|
||||
*> \author Univ. of Tennessee
|
||||
*> \author Univ. of California Berkeley
|
||||
*> \author Univ. of Colorado Denver
|
||||
*> \author NAG Ltd.
|
||||
*
|
||||
*> \ingroup complex16GEeigen
|
||||
*
|
||||
*> \par Further Details:
|
||||
* =====================
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> Balancing
|
||||
*> ---------
|
||||
*>
|
||||
*> This driver calls ZGGBAL to both permute and scale rows and columns
|
||||
*> of A and B. The permutations PL and PR are chosen so that PL*A*PR
|
||||
*> and PL*B*R will be upper triangular except for the diagonal blocks
|
||||
*> A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
|
||||
*> possible. The diagonal scaling matrices DL and DR are chosen so
|
||||
*> that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
|
||||
*> one (except for the elements that start out zero.)
|
||||
*>
|
||||
*> After the eigenvalues and eigenvectors of the balanced matrices
|
||||
*> have been computed, ZGGBAK transforms the eigenvectors back to what
|
||||
*> they would have been (in perfect arithmetic) if they had not been
|
||||
*> balanced.
|
||||
*>
|
||||
*> Contents of A and B on Exit
|
||||
*> -------- -- - --- - -- ----
|
||||
*>
|
||||
*> If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
|
||||
*> both), then on exit the arrays A and B will contain the complex Schur
|
||||
*> form[*] of the "balanced" versions of A and B. If no eigenvectors
|
||||
*> are computed, then only the diagonal blocks will be correct.
|
||||
*>
|
||||
*> [*] In other words, upper triangular form.
|
||||
*> \endverbatim
|
||||
*>
|
||||
* =====================================================================
|
||||
SUBROUTINE ZGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
|
||||
$ VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER JOBVL, JOBVR
|
||||
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
DOUBLE PRECISION RWORK( * )
|
||||
COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
|
||||
$ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
|
||||
$ WORK( * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
DOUBLE PRECISION ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
|
||||
COMPLEX*16 CZERO, CONE
|
||||
PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ),
|
||||
$ CONE = ( 1.0D0, 0.0D0 ) )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL ILIMIT, ILV, ILVL, ILVR, LQUERY
|
||||
CHARACTER CHTEMP
|
||||
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
|
||||
$ IN, IRIGHT, IROWS, IRWORK, ITAU, IWORK, JC, JR,
|
||||
$ LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3
|
||||
DOUBLE PRECISION ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM,
|
||||
$ BNRM1, BNRM2, EPS, SAFMAX, SAFMIN, SALFAI,
|
||||
$ SALFAR, SBETA, SCALE, TEMP
|
||||
COMPLEX*16 X
|
||||
* ..
|
||||
* .. Local Arrays ..
|
||||
LOGICAL LDUMMA( 1 )
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL XERBLA, ZGEQRF, ZGGBAK, ZGGBAL, ZGGHRD, ZHGEQZ,
|
||||
$ ZLACPY, ZLASCL, ZLASET, ZTGEVC, ZUNGQR, ZUNMQR
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
INTEGER ILAENV
|
||||
DOUBLE PRECISION DLAMCH, ZLANGE
|
||||
EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC ABS, DBLE, DCMPLX, DIMAG, INT, MAX
|
||||
* ..
|
||||
* .. Statement Functions ..
|
||||
DOUBLE PRECISION ABS1
|
||||
* ..
|
||||
* .. Statement Function definitions ..
|
||||
ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) )
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
* Decode the input arguments
|
||||
*
|
||||
IF( LSAME( JOBVL, 'N' ) ) THEN
|
||||
IJOBVL = 1
|
||||
ILVL = .FALSE.
|
||||
ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
|
||||
IJOBVL = 2
|
||||
ILVL = .TRUE.
|
||||
ELSE
|
||||
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
|
||||
*
|
||||
* Test the input arguments
|
||||
*
|
||||
LWKMIN = MAX( 2*N, 1 )
|
||||
LWKOPT = LWKMIN
|
||||
WORK( 1 ) = LWKOPT
|
||||
LQUERY = ( LWORK.EQ.-1 )
|
||||
INFO = 0
|
||||
IF( IJOBVL.LE.0 ) THEN
|
||||
INFO = -1
|
||||
ELSE IF( IJOBVR.LE.0 ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -5
|
||||
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -7
|
||||
ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
|
||||
INFO = -11
|
||||
ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
|
||||
INFO = -13
|
||||
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
|
||||
INFO = -15
|
||||
END IF
|
||||
*
|
||||
IF( INFO.EQ.0 ) THEN
|
||||
NB1 = ILAENV( 1, 'ZGEQRF', ' ', N, N, -1, -1 )
|
||||
NB2 = ILAENV( 1, 'ZUNMQR', ' ', N, N, N, -1 )
|
||||
NB3 = ILAENV( 1, 'ZUNGQR', ' ', N, N, N, -1 )
|
||||
NB = MAX( NB1, NB2, NB3 )
|
||||
LOPT = MAX( 2*N, N*( NB+1 ) )
|
||||
WORK( 1 ) = LOPT
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'ZGEGV ', -INFO )
|
||||
RETURN
|
||||
ELSE IF( LQUERY ) THEN
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
* Quick return if possible
|
||||
*
|
||||
IF( N.EQ.0 )
|
||||
$ RETURN
|
||||
*
|
||||
* Get machine constants
|
||||
*
|
||||
EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
|
||||
SAFMIN = DLAMCH( 'S' )
|
||||
SAFMIN = SAFMIN + SAFMIN
|
||||
SAFMAX = ONE / SAFMIN
|
||||
*
|
||||
* Scale A
|
||||
*
|
||||
ANRM = ZLANGE( 'M', N, N, A, LDA, RWORK )
|
||||
ANRM1 = ANRM
|
||||
ANRM2 = ONE
|
||||
IF( ANRM.LT.ONE ) THEN
|
||||
IF( SAFMAX*ANRM.LT.ONE ) THEN
|
||||
ANRM1 = SAFMIN
|
||||
ANRM2 = SAFMAX*ANRM
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( ANRM.GT.ZERO ) THEN
|
||||
CALL ZLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 10
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Scale B
|
||||
*
|
||||
BNRM = ZLANGE( 'M', N, N, B, LDB, RWORK )
|
||||
BNRM1 = BNRM
|
||||
BNRM2 = ONE
|
||||
IF( BNRM.LT.ONE ) THEN
|
||||
IF( SAFMAX*BNRM.LT.ONE ) THEN
|
||||
BNRM1 = SAFMIN
|
||||
BNRM2 = SAFMAX*BNRM
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( BNRM.GT.ZERO ) THEN
|
||||
CALL ZLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 10
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Permute the matrix to make it more nearly triangular
|
||||
* Also "balance" the matrix.
|
||||
*
|
||||
ILEFT = 1
|
||||
IRIGHT = N + 1
|
||||
IRWORK = IRIGHT + N
|
||||
CALL ZGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ),
|
||||
$ RWORK( IRIGHT ), RWORK( IRWORK ), IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 1
|
||||
GO TO 80
|
||||
END IF
|
||||
*
|
||||
* Reduce B to triangular form, and initialize VL and/or VR
|
||||
*
|
||||
IROWS = IHI + 1 - ILO
|
||||
IF( ILV ) THEN
|
||||
ICOLS = N + 1 - ILO
|
||||
ELSE
|
||||
ICOLS = IROWS
|
||||
END IF
|
||||
ITAU = 1
|
||||
IWORK = ITAU + IROWS
|
||||
CALL ZGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
|
||||
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 2
|
||||
GO TO 80
|
||||
END IF
|
||||
*
|
||||
CALL ZUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
|
||||
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
|
||||
$ LWORK+1-IWORK, IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 3
|
||||
GO TO 80
|
||||
END IF
|
||||
*
|
||||
IF( ILVL ) THEN
|
||||
CALL ZLASET( 'Full', N, N, CZERO, CONE, VL, LDVL )
|
||||
CALL ZLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
|
||||
$ VL( ILO+1, ILO ), LDVL )
|
||||
CALL ZUNGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
|
||||
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
|
||||
$ IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 4
|
||||
GO TO 80
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( ILVR )
|
||||
$ CALL ZLASET( 'Full', N, N, CZERO, CONE, VR, LDVR )
|
||||
*
|
||||
* Reduce to generalized Hessenberg form
|
||||
*
|
||||
IF( ILV ) THEN
|
||||
*
|
||||
* Eigenvectors requested -- work on whole matrix.
|
||||
*
|
||||
CALL ZGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
|
||||
$ LDVL, VR, LDVR, IINFO )
|
||||
ELSE
|
||||
CALL ZGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
|
||||
$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO )
|
||||
END IF
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 5
|
||||
GO TO 80
|
||||
END IF
|
||||
*
|
||||
* Perform QZ algorithm
|
||||
*
|
||||
IWORK = ITAU
|
||||
IF( ILV ) THEN
|
||||
CHTEMP = 'S'
|
||||
ELSE
|
||||
CHTEMP = 'E'
|
||||
END IF
|
||||
CALL ZHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
|
||||
$ ALPHA, BETA, VL, LDVL, VR, LDVR, WORK( IWORK ),
|
||||
$ LWORK+1-IWORK, RWORK( IRWORK ), IINFO )
|
||||
IF( IINFO.GE.0 )
|
||||
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
|
||||
INFO = IINFO
|
||||
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
|
||||
INFO = IINFO - N
|
||||
ELSE
|
||||
INFO = N + 6
|
||||
END IF
|
||||
GO TO 80
|
||||
END IF
|
||||
*
|
||||
IF( ILV ) THEN
|
||||
*
|
||||
* Compute Eigenvectors
|
||||
*
|
||||
IF( ILVL ) THEN
|
||||
IF( ILVR ) THEN
|
||||
CHTEMP = 'B'
|
||||
ELSE
|
||||
CHTEMP = 'L'
|
||||
END IF
|
||||
ELSE
|
||||
CHTEMP = 'R'
|
||||
END IF
|
||||
*
|
||||
CALL ZTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
|
||||
$ VR, LDVR, N, IN, WORK( IWORK ), RWORK( IRWORK ),
|
||||
$ IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 7
|
||||
GO TO 80
|
||||
END IF
|
||||
*
|
||||
* Undo balancing on VL and VR, rescale
|
||||
*
|
||||
IF( ILVL ) THEN
|
||||
CALL ZGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ),
|
||||
$ RWORK( IRIGHT ), N, VL, LDVL, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 8
|
||||
GO TO 80
|
||||
END IF
|
||||
DO 30 JC = 1, N
|
||||
TEMP = ZERO
|
||||
DO 10 JR = 1, N
|
||||
TEMP = MAX( TEMP, ABS1( VL( JR, JC ) ) )
|
||||
10 CONTINUE
|
||||
IF( TEMP.LT.SAFMIN )
|
||||
$ GO TO 30
|
||||
TEMP = ONE / TEMP
|
||||
DO 20 JR = 1, N
|
||||
VL( JR, JC ) = VL( JR, JC )*TEMP
|
||||
20 CONTINUE
|
||||
30 CONTINUE
|
||||
END IF
|
||||
IF( ILVR ) THEN
|
||||
CALL ZGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ),
|
||||
$ RWORK( IRIGHT ), N, VR, LDVR, IINFO )
|
||||
IF( IINFO.NE.0 ) THEN
|
||||
INFO = N + 9
|
||||
GO TO 80
|
||||
END IF
|
||||
DO 60 JC = 1, N
|
||||
TEMP = ZERO
|
||||
DO 40 JR = 1, N
|
||||
TEMP = MAX( TEMP, ABS1( VR( JR, JC ) ) )
|
||||
40 CONTINUE
|
||||
IF( TEMP.LT.SAFMIN )
|
||||
$ GO TO 60
|
||||
TEMP = ONE / TEMP
|
||||
DO 50 JR = 1, N
|
||||
VR( JR, JC ) = VR( JR, JC )*TEMP
|
||||
50 CONTINUE
|
||||
60 CONTINUE
|
||||
END IF
|
||||
*
|
||||
* End of eigenvector calculation
|
||||
*
|
||||
END IF
|
||||
*
|
||||
* Undo scaling in alpha, beta
|
||||
*
|
||||
* Note: this does not give the alpha and beta for the unscaled
|
||||
* problem.
|
||||
*
|
||||
* Un-scaling is limited to avoid underflow in alpha and beta
|
||||
* if they are significant.
|
||||
*
|
||||
DO 70 JC = 1, N
|
||||
ABSAR = ABS( DBLE( ALPHA( JC ) ) )
|
||||
ABSAI = ABS( DIMAG( ALPHA( JC ) ) )
|
||||
ABSB = ABS( DBLE( BETA( JC ) ) )
|
||||
SALFAR = ANRM*DBLE( ALPHA( JC ) )
|
||||
SALFAI = ANRM*DIMAG( ALPHA( JC ) )
|
||||
SBETA = BNRM*DBLE( BETA( JC ) )
|
||||
ILIMIT = .FALSE.
|
||||
SCALE = ONE
|
||||
*
|
||||
* Check for significant underflow in imaginary part of ALPHA
|
||||
*
|
||||
IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE.
|
||||
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN
|
||||
ILIMIT = .TRUE.
|
||||
SCALE = ( SAFMIN / ANRM1 ) / MAX( SAFMIN, ANRM2*ABSAI )
|
||||
END IF
|
||||
*
|
||||
* Check for significant underflow in real part of ALPHA
|
||||
*
|
||||
IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE.
|
||||
$ MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN
|
||||
ILIMIT = .TRUE.
|
||||
SCALE = MAX( SCALE, ( SAFMIN / ANRM1 ) /
|
||||
$ MAX( SAFMIN, ANRM2*ABSAR ) )
|
||||
END IF
|
||||
*
|
||||
* Check for significant underflow in BETA
|
||||
*
|
||||
IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE.
|
||||
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN
|
||||
ILIMIT = .TRUE.
|
||||
SCALE = MAX( SCALE, ( SAFMIN / BNRM1 ) /
|
||||
$ MAX( SAFMIN, BNRM2*ABSB ) )
|
||||
END IF
|
||||
*
|
||||
* Check for possible overflow when limiting scaling
|
||||
*
|
||||
IF( ILIMIT ) THEN
|
||||
TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ),
|
||||
$ ABS( SBETA ) )
|
||||
IF( TEMP.GT.ONE )
|
||||
$ SCALE = SCALE / TEMP
|
||||
IF( SCALE.LT.ONE )
|
||||
$ ILIMIT = .FALSE.
|
||||
END IF
|
||||
*
|
||||
* Recompute un-scaled ALPHA, BETA if necessary.
|
||||
*
|
||||
IF( ILIMIT ) THEN
|
||||
SALFAR = ( SCALE*DBLE( ALPHA( JC ) ) )*ANRM
|
||||
SALFAI = ( SCALE*DIMAG( ALPHA( JC ) ) )*ANRM
|
||||
SBETA = ( SCALE*BETA( JC ) )*BNRM
|
||||
END IF
|
||||
ALPHA( JC ) = DCMPLX( SALFAR, SALFAI )
|
||||
BETA( JC ) = SBETA
|
||||
70 CONTINUE
|
||||
*
|
||||
80 CONTINUE
|
||||
WORK( 1 ) = LWKOPT
|
||||
*
|
||||
RETURN
|
||||
*
|
||||
* End of ZGEGV
|
||||
*
|
||||
END
|
||||
Loading…
Reference in New Issue