500 lines
16 KiB
Fortran
500 lines
16 KiB
Fortran
*> \brief <b> ZGEESX 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 ZGEESX + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgeesx.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/zgeesx.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/zgeesx.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 ZGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W,
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* VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK,
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* BWORK, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER JOBVS, SENSE, SORT
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* INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
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* DOUBLE PRECISION RCONDE, RCONDV
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* ..
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* .. Array Arguments ..
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* LOGICAL BWORK( * )
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* DOUBLE PRECISION RWORK( * )
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* COMPLEX*16 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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*> ZGEESX 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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*> computes a reciprocal condition number for the average of the
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*> selected eigenvalues (RCONDE); and computes a reciprocal condition
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*> number for the right invariant subspace corresponding to the
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*> selected eigenvalues (RCONDV). The leading columns of Z form an
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*> orthonormal basis for this invariant subspace.
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*>
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*> For further explanation of the reciprocal condition numbers RCONDE
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*> and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
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*> these quantities are called s and sep respectively).
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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 procedure) LOGICAL FUNCTION of one COMPLEX*16 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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*> An 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] SENSE
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*> \verbatim
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*> SENSE is CHARACTER*1
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*> Determines which reciprocal condition numbers are computed.
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*> = 'N': None are computed;
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*> = 'E': Computed for average of selected eigenvalues only;
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*> = 'V': Computed for selected right invariant subspace only;
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*> = 'B': Computed for both.
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*> If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The order of the 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*16 array, dimension (LDA, N)
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*> On entry, the N-by-N matrix A.
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*> On exit, A is 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*16 array, dimension (N)
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*> W contains the computed eigenvalues, in the same order
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*> that 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*16 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, and if
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*> JOBVS = 'V', LDVS >= N.
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*> \endverbatim
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*>
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*> \param[out] RCONDE
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*> \verbatim
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*> RCONDE is DOUBLE PRECISION
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*> If SENSE = 'E' or 'B', RCONDE contains the reciprocal
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*> condition number for the average of the selected eigenvalues.
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*> Not referenced if SENSE = 'N' or 'V'.
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*> \endverbatim
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*>
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*> \param[out] RCONDV
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*> \verbatim
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*> RCONDV is DOUBLE PRECISION
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*> If SENSE = 'V' or 'B', RCONDV contains the reciprocal
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*> condition number for the selected right invariant subspace.
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*> Not referenced if SENSE = 'N' or 'E'.
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is COMPLEX*16 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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*> Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM),
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*> where SDIM is the number of selected eigenvalues computed by
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*> this routine. Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also
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*> that an error is only returned if LWORK < max(1,2*N), but if
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*> SENSE = 'E' or 'V' or 'B' this may not be large enough.
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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 upper bound on the optimal size of the
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*> array WORK, returns this value as the first entry of the WORK
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*> array, and no error message related to LWORK is issued by
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*> 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 DOUBLE PRECISION 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; if
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*> JOBVS = 'V', VS contains the transformation 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 some
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*> eigenvalues were too close to separate (the problem
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*> is very ill-conditioned);
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*> = N+2: after reordering, roundoff changed values of some
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*> complex eigenvalues so that leading eigenvalues in
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*> the Schur form no longer satisfy SELECT=.TRUE. This
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*> could also be caused by 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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*> \date November 2011
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*
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*> \ingroup complex16GEeigen
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*
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* =====================================================================
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SUBROUTINE ZGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W,
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$ VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK,
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$ BWORK, INFO )
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*
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* -- LAPACK driver routine (version 3.4.0) --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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* November 2011
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*
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* .. Scalar Arguments ..
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CHARACTER JOBVS, SENSE, SORT
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INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
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DOUBLE PRECISION RCONDE, RCONDV
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* ..
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* .. Array Arguments ..
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LOGICAL BWORK( * )
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DOUBLE PRECISION RWORK( * )
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COMPLEX*16 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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DOUBLE PRECISION ZERO, ONE
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PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
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* ..
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* .. Local Scalars ..
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LOGICAL LQUERY, SCALEA, WANTSB, WANTSE, WANTSN, WANTST,
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$ WANTSV, WANTVS
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INTEGER HSWORK, I, IBAL, ICOND, IERR, IEVAL, IHI, ILO,
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$ ITAU, IWRK, LWRK, MAXWRK, MINWRK
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DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, SMLNUM
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* ..
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* .. Local Arrays ..
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DOUBLE PRECISION DUM( 1 )
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* ..
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* .. External Subroutines ..
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EXTERNAL DLABAD, DLASCL, XERBLA, ZCOPY, ZGEBAK, ZGEBAL,
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$ ZGEHRD, ZHSEQR, ZLACPY, ZLASCL, ZTRSEN, ZUNGHR
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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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DOUBLE PRECISION DLAMCH, ZLANGE
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EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE
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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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WANTVS = LSAME( JOBVS, 'V' )
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WANTST = LSAME( SORT, 'S' )
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WANTSN = LSAME( SENSE, 'N' )
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WANTSE = LSAME( SENSE, 'E' )
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WANTSV = LSAME( SENSE, 'V' )
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WANTSB = LSAME( SENSE, 'B' )
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LQUERY = ( LWORK.EQ.-1 )
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*
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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( .NOT.( WANTSN .OR. WANTSE .OR. WANTSV .OR. WANTSB ) .OR.
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$ ( .NOT.WANTST .AND. .NOT.WANTSN ) ) THEN
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INFO = -4
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ELSE IF( N.LT.0 ) THEN
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INFO = -5
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -7
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ELSE IF( LDVS.LT.1 .OR. ( WANTVS .AND. LDVS.LT.N ) ) THEN
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INFO = -11
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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 real workspace needed at that point in the
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* code, 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 ZHSEQR, 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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* If SENSE = 'E', 'V' or 'B', then the amount of workspace needed
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* depends on SDIM, which is computed by the routine ZTRSEN later
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* in the code.)
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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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LWRK = 1
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ELSE
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MAXWRK = N + N*ILAENV( 1, 'ZGEHRD', ' ', N, 1, N, 0 )
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MINWRK = 2*N
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*
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CALL ZHSEQR( 'S', JOBVS, N, 1, N, A, LDA, W, VS, LDVS,
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$ WORK, -1, IEVAL )
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HSWORK = 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, 'ZUNGHR',
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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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LWRK = MAXWRK
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IF( .NOT.WANTSN )
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$ LWRK = MAX( LWRK, ( N*N )/2 )
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END IF
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WORK( 1 ) = LWRK
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*
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IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
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INFO = -15
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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( 'ZGEESX', -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 = DLAMCH( 'P' )
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SMLNUM = DLAMCH( 'S' )
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BIGNUM = ONE / SMLNUM
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CALL DLABAD( SMLNUM, BIGNUM )
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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 = ZLANGE( '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 ZLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
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*
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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 ZGEBAL( '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 ZGEHRD( 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 ZLACPY( '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 ZUNGHR( 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 ZHSEQR( '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 ZLASCL( '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, transform Schur vectors, and compute
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* reciprocal condition numbers
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* (CWorkspace: if SENSE is not 'N', need 2*SDIM*(N-SDIM)
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* otherwise, need none )
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* (RWorkspace: none)
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*
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CALL ZTRSEN( SENSE, JOBVS, BWORK, N, A, LDA, VS, LDVS, W, SDIM,
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$ RCONDE, RCONDV, WORK( IWRK ), LWORK-IWRK+1,
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$ ICOND )
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IF( .NOT.WANTSN )
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$ MAXWRK = MAX( MAXWRK, 2*SDIM*( N-SDIM ) )
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IF( ICOND.EQ.-14 ) THEN
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*
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* Not enough complex workspace
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*
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INFO = -15
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END IF
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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
|
|
* (CWorkspace: none)
|
|
* (RWorkspace: need N)
|
|
*
|
|
CALL ZGEBAK( 'P', 'R', N, ILO, IHI, RWORK( IBAL ), N, VS, LDVS,
|
|
$ IERR )
|
|
END IF
|
|
*
|
|
IF( SCALEA ) THEN
|
|
*
|
|
* Undo scaling for the Schur form of A
|
|
*
|
|
CALL ZLASCL( 'U', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR )
|
|
CALL ZCOPY( N, A, LDA+1, W, 1 )
|
|
IF( ( WANTSV .OR. WANTSB ) .AND. INFO.EQ.0 ) THEN
|
|
DUM( 1 ) = RCONDV
|
|
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR )
|
|
RCONDV = DUM( 1 )
|
|
END IF
|
|
END IF
|
|
*
|
|
WORK( 1 ) = MAXWRK
|
|
RETURN
|
|
*
|
|
* End of ZGEESX
|
|
*
|
|
END
|