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420 lines
13 KiB
420 lines
13 KiB
*> \brief <b> CGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices</b>
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download CGEES + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgees.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/cgees.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/cgees.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 CGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, W, VS,
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* LDVS, WORK, LWORK, RWORK, BWORK, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER JOBVS, SORT
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* INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
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* ..
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* .. Array Arguments ..
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* LOGICAL BWORK( * )
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* REAL RWORK( * )
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* COMPLEX A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )
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* ..
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* .. Function Arguments ..
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* LOGICAL SELECT
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* EXTERNAL SELECT
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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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*> CGEES computes for an N-by-N complex nonsymmetric matrix A, the
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*> eigenvalues, the Schur form T, and, optionally, the matrix of Schur
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*> vectors Z. This gives the Schur factorization A = Z*T*(Z**H).
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*>
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*> Optionally, it also orders the eigenvalues on the diagonal of the
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*> Schur form so that selected eigenvalues are at the top left.
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*> The leading columns of Z then form an orthonormal basis for the
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*> invariant subspace corresponding to the selected eigenvalues.
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*>
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*> A complex matrix is in Schur form if it is upper triangular.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] JOBVS
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*> \verbatim
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*> JOBVS is CHARACTER*1
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*> = 'N': Schur vectors are not computed;
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*> = 'V': Schur vectors are computed.
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*> \endverbatim
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*>
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*> \param[in] SORT
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*> \verbatim
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*> SORT is CHARACTER*1
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*> Specifies whether or not to order the eigenvalues on the
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*> diagonal of the Schur form.
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*> = 'N': Eigenvalues are not ordered:
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*> = 'S': Eigenvalues are ordered (see SELECT).
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*> \endverbatim
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*>
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*> \param[in] SELECT
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*> \verbatim
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*> SELECT is a LOGICAL FUNCTION of one COMPLEX argument
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*> SELECT must be declared EXTERNAL in the calling subroutine.
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*> If SORT = 'S', SELECT is used to select eigenvalues to order
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*> to the top left of the Schur form.
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*> IF SORT = 'N', SELECT is not referenced.
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*> The eigenvalue W(j) is selected if SELECT(W(j)) is true.
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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 matrix A. 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 N-by-N matrix A.
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*> On exit, A has been overwritten by its Schur form T.
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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 the array A. LDA >= max(1,N).
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*> \endverbatim
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*>
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*> \param[out] SDIM
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*> \verbatim
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*> SDIM is INTEGER
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*> If SORT = 'N', SDIM = 0.
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*> If SORT = 'S', SDIM = number of eigenvalues for which
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*> SELECT is true.
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*> \endverbatim
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*>
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*> \param[out] W
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*> \verbatim
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*> W is COMPLEX array, dimension (N)
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*> W contains the computed eigenvalues, in the same order that
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*> they appear on the diagonal of the output Schur form T.
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*> \endverbatim
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*>
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*> \param[out] VS
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*> \verbatim
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*> VS is COMPLEX array, dimension (LDVS,N)
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*> If JOBVS = 'V', VS contains the unitary matrix Z of Schur
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*> vectors.
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*> If JOBVS = 'N', VS is not referenced.
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*> \endverbatim
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*>
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*> \param[in] LDVS
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*> \verbatim
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*> LDVS is INTEGER
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*> The leading dimension of the array VS. LDVS >= 1; if
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*> JOBVS = 'V', LDVS >= 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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*>
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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 (N)
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*> \endverbatim
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*>
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*> \param[out] BWORK
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*> \verbatim
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*> BWORK is LOGICAL array, dimension (N)
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*> Not referenced if SORT = 'N'.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit
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*> < 0: if INFO = -i, the i-th argument had an illegal value.
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*> > 0: if INFO = i, and i is
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*> <= N: the QR algorithm failed to compute all the
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*> eigenvalues; elements 1:ILO-1 and i+1:N of W
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*> contain those eigenvalues which have converged;
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*> if JOBVS = 'V', VS contains the matrix which
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*> reduces A to its partially converged Schur form.
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*> = N+1: the eigenvalues could not be reordered because
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*> some eigenvalues were too close to separate (the
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*> problem is very ill-conditioned);
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*> = N+2: after reordering, roundoff changed values of
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*> some complex eigenvalues so that leading
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*> eigenvalues in the Schur form no longer satisfy
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*> SELECT = .TRUE.. This could also be caused by
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*> underflow due to scaling.
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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 CGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, W, VS,
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$ LDVS, WORK, LWORK, RWORK, BWORK, 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 JOBVS, SORT
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INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
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* ..
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* .. Array Arguments ..
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LOGICAL BWORK( * )
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REAL RWORK( * )
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COMPLEX A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )
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* ..
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* .. Function Arguments ..
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LOGICAL SELECT
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EXTERNAL SELECT
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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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* ..
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* .. Local Scalars ..
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LOGICAL LQUERY, SCALEA, WANTST, WANTVS
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INTEGER HSWORK, I, IBAL, ICOND, IERR, IEVAL, IHI, ILO,
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$ ITAU, IWRK, MAXWRK, MINWRK
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REAL ANRM, BIGNUM, CSCALE, EPS, S, SEP, SMLNUM
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* ..
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* .. Local Arrays ..
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REAL DUM( 1 )
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* ..
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* .. External Subroutines ..
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EXTERNAL CCOPY, CGEBAK, CGEBAL, CGEHRD, CHSEQR, CLACPY,
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$ CLASCL, CTRSEN, CUNGHR, 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 LSAME, ILAENV, CLANGE, SLAMCH
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, SQRT
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* ..
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* .. Executable Statements ..
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*
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* Test the input arguments
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*
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INFO = 0
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LQUERY = ( LWORK.EQ.-1 )
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WANTVS = LSAME( JOBVS, 'V' )
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WANTST = LSAME( SORT, 'S' )
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IF( ( .NOT.WANTVS ) .AND. ( .NOT.LSAME( JOBVS, 'N' ) ) ) THEN
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INFO = -1
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ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
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INFO = -2
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ELSE IF( N.LT.0 ) THEN
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INFO = -4
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -6
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ELSE IF( LDVS.LT.1 .OR. ( WANTVS .AND. LDVS.LT.N ) ) THEN
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INFO = -10
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END IF
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*
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* Compute workspace
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* (Note: Comments in the code beginning "Workspace:" describe the
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* minimal amount of workspace needed at that point in the code,
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* as well as the preferred amount for good performance.
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* CWorkspace refers to complex workspace, and RWorkspace to real
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* workspace. NB refers to the optimal block size for the
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* immediately following subroutine, as returned by ILAENV.
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* HSWORK refers to the workspace preferred by CHSEQR, as
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* calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
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* the worst case.)
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*
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IF( INFO.EQ.0 ) THEN
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IF( N.EQ.0 ) THEN
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MINWRK = 1
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MAXWRK = 1
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ELSE
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MAXWRK = N + N*ILAENV( 1, 'CGEHRD', ' ', N, 1, N, 0 )
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MINWRK = 2*N
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*
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CALL CHSEQR( 'S', JOBVS, N, 1, N, A, LDA, W, VS, LDVS,
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$ WORK, -1, IEVAL )
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HSWORK = INT( WORK( 1 ) )
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*
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IF( .NOT.WANTVS ) THEN
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MAXWRK = MAX( MAXWRK, HSWORK )
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ELSE
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MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'CUNGHR',
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$ ' ', N, 1, N, -1 ) )
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MAXWRK = MAX( MAXWRK, HSWORK )
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END IF
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END IF
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WORK( 1 ) = MAXWRK
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*
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IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
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INFO = -12
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END IF
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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( 'CGEES ', -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 ) THEN
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SDIM = 0
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RETURN
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END IF
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*
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* Get machine constants
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*
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EPS = SLAMCH( 'P' )
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SMLNUM = SLAMCH( 'S' )
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BIGNUM = ONE / SMLNUM
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SMLNUM = SQRT( SMLNUM ) / 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, DUM )
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SCALEA = .FALSE.
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IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
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SCALEA = .TRUE.
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CSCALE = SMLNUM
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ELSE IF( ANRM.GT.BIGNUM ) THEN
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SCALEA = .TRUE.
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CSCALE = BIGNUM
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END IF
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IF( SCALEA )
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$ CALL CLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
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*
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* Permute the matrix to make it more nearly triangular
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* (CWorkspace: none)
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* (RWorkspace: need N)
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*
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IBAL = 1
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CALL CGEBAL( 'P', N, A, LDA, ILO, IHI, RWORK( IBAL ), IERR )
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*
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* Reduce to upper Hessenberg form
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* (CWorkspace: need 2*N, prefer N+N*NB)
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* (RWorkspace: none)
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*
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ITAU = 1
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IWRK = N + ITAU
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CALL CGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
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$ LWORK-IWRK+1, IERR )
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*
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IF( WANTVS ) THEN
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*
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* Copy Householder vectors to VS
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*
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CALL CLACPY( 'L', N, N, A, LDA, VS, LDVS )
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*
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* Generate unitary matrix in VS
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* (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
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* (RWorkspace: none)
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*
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CALL CUNGHR( N, ILO, IHI, VS, LDVS, WORK( ITAU ), WORK( IWRK ),
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$ LWORK-IWRK+1, IERR )
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END IF
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*
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SDIM = 0
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*
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* Perform QR iteration, accumulating Schur vectors in VS if desired
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* (CWorkspace: need 1, prefer HSWORK (see comments) )
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* (RWorkspace: none)
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*
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IWRK = ITAU
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CALL CHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, W, VS, LDVS,
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$ WORK( IWRK ), LWORK-IWRK+1, IEVAL )
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IF( IEVAL.GT.0 )
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$ INFO = IEVAL
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*
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* Sort eigenvalues if desired
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*
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IF( WANTST .AND. INFO.EQ.0 ) THEN
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IF( SCALEA )
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$ CALL CLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, W, N, IERR )
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DO 10 I = 1, N
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BWORK( I ) = SELECT( W( I ) )
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10 CONTINUE
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*
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* Reorder eigenvalues and transform Schur vectors
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* (CWorkspace: none)
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* (RWorkspace: none)
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*
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CALL CTRSEN( 'N', JOBVS, BWORK, N, A, LDA, VS, LDVS, W, SDIM,
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$ S, SEP, WORK( IWRK ), LWORK-IWRK+1, ICOND )
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END IF
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*
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IF( WANTVS ) THEN
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*
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* Undo balancing
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* (CWorkspace: none)
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* (RWorkspace: need N)
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*
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CALL CGEBAK( 'P', 'R', N, ILO, IHI, RWORK( IBAL ), N, VS, LDVS,
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$ IERR )
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END IF
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*
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IF( SCALEA ) THEN
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*
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* Undo scaling for the Schur form of A
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*
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CALL CLASCL( 'U', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR )
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CALL CCOPY( N, A, LDA+1, W, 1 )
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END IF
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*
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WORK( 1 ) = MAXWRK
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RETURN
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*
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* End of CGEES
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*
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END
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