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466 lines
14 KiB
466 lines
14 KiB
2 years ago
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*> \brief \b CHSEIN
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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 CHSEIN + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/chsein.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/chsein.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/chsein.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 CHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, W, VL,
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* LDVL, VR, LDVR, MM, M, WORK, RWORK, IFAILL,
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* IFAILR, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER EIGSRC, INITV, SIDE
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* INTEGER INFO, LDH, LDVL, LDVR, M, MM, N
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* ..
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* .. Array Arguments ..
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* LOGICAL SELECT( * )
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* INTEGER IFAILL( * ), IFAILR( * )
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* REAL RWORK( * )
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* COMPLEX H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ),
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* $ W( * ), 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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*> CHSEIN uses inverse iteration to find specified right and/or left
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*> eigenvectors of a complex upper Hessenberg matrix H.
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*>
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*> The right eigenvector x and the left eigenvector y of the matrix H
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*> corresponding to an eigenvalue w are defined by:
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*>
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*> H * x = w * x, y**h * H = w * y**h
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*>
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*> where y**h denotes the conjugate transpose of the vector y.
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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] SIDE
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*> \verbatim
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*> SIDE is CHARACTER*1
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*> = 'R': compute right eigenvectors only;
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*> = 'L': compute left eigenvectors only;
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*> = 'B': compute both right and left eigenvectors.
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*> \endverbatim
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*>
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*> \param[in] EIGSRC
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*> \verbatim
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*> EIGSRC is CHARACTER*1
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*> Specifies the source of eigenvalues supplied in W:
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*> = 'Q': the eigenvalues were found using CHSEQR; thus, if
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*> H has zero subdiagonal elements, and so is
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*> block-triangular, then the j-th eigenvalue can be
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*> assumed to be an eigenvalue of the block containing
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*> the j-th row/column. This property allows CHSEIN to
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*> perform inverse iteration on just one diagonal block.
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*> = 'N': no assumptions are made on the correspondence
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*> between eigenvalues and diagonal blocks. In this
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*> case, CHSEIN must always perform inverse iteration
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*> using the whole matrix H.
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*> \endverbatim
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*>
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*> \param[in] INITV
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*> \verbatim
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*> INITV is CHARACTER*1
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*> = 'N': no initial vectors are supplied;
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*> = 'U': user-supplied initial vectors are stored in the arrays
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*> VL and/or VR.
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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 LOGICAL array, dimension (N)
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*> Specifies the eigenvectors to be computed. To select the
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*> eigenvector corresponding to the eigenvalue W(j),
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*> SELECT(j) must be set to .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 H. N >= 0.
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*> \endverbatim
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*>
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*> \param[in] H
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*> \verbatim
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*> H is COMPLEX array, dimension (LDH,N)
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*> The upper Hessenberg matrix H.
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*> If a NaN is detected in H, the routine will return with INFO=-6.
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*> \endverbatim
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*>
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*> \param[in] LDH
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*> \verbatim
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*> LDH is INTEGER
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*> The leading dimension of the array H. LDH >= max(1,N).
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*> \endverbatim
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*>
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*> \param[in,out] W
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*> \verbatim
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*> W is COMPLEX array, dimension (N)
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*> On entry, the eigenvalues of H.
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*> On exit, the real parts of W may have been altered since
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*> close eigenvalues are perturbed slightly in searching for
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*> independent eigenvectors.
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*> \endverbatim
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*>
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*> \param[in,out] VL
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*> \verbatim
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*> VL is COMPLEX array, dimension (LDVL,MM)
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*> On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must
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*> contain starting vectors for the inverse iteration for the
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*> left eigenvectors; the starting vector for each eigenvector
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*> must be in the same column in which the eigenvector will be
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*> stored.
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*> On exit, if SIDE = 'L' or 'B', the left eigenvectors
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*> specified by SELECT will be stored consecutively in the
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*> columns of VL, in the same order as their eigenvalues.
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*> If SIDE = 'R', VL is not referenced.
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*> \endverbatim
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*>
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*> \param[in] LDVL
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*> \verbatim
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*> LDVL is INTEGER
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*> The leading dimension of the array VL.
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*> LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.
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*> \endverbatim
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*>
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*> \param[in,out] VR
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*> \verbatim
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*> VR is COMPLEX array, dimension (LDVR,MM)
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*> On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must
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*> contain starting vectors for the inverse iteration for the
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*> right eigenvectors; the starting vector for each eigenvector
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*> must be in the same column in which the eigenvector will be
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*> stored.
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*> On exit, if SIDE = 'R' or 'B', the right eigenvectors
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*> specified by SELECT will be stored consecutively in the
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*> columns of VR, in the same order as their eigenvalues.
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*> If SIDE = 'L', VR is not referenced.
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*> \endverbatim
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*>
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*> \param[in] LDVR
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*> \verbatim
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*> LDVR is INTEGER
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*> The leading dimension of the array VR.
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*> LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.
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*> \endverbatim
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*>
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*> \param[in] MM
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*> \verbatim
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*> MM is INTEGER
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*> The number of columns in the arrays VL and/or VR. MM >= M.
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*> \endverbatim
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*>
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*> \param[out] M
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*> \verbatim
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*> M is INTEGER
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*> The number of columns in the arrays VL and/or VR required to
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*> store the eigenvectors (= the number of .TRUE. elements in
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*> SELECT).
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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 (N*N)
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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] IFAILL
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*> \verbatim
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*> IFAILL is INTEGER array, dimension (MM)
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*> If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left
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*> eigenvector in the i-th column of VL (corresponding to the
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*> eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the
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*> eigenvector converged satisfactorily.
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*> If SIDE = 'R', IFAILL is not referenced.
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*> \endverbatim
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*>
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*> \param[out] IFAILR
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*> \verbatim
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*> IFAILR is INTEGER array, dimension (MM)
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*> If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right
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*> eigenvector in the i-th column of VR (corresponding to the
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*> eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the
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*> eigenvector converged satisfactorily.
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*> If SIDE = 'L', IFAILR is not referenced.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit
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*> < 0: if INFO = -i, the i-th argument had an illegal value
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*> > 0: if INFO = i, i is the number of eigenvectors which
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*> failed to converge; see IFAILL and IFAILR for further
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*> details.
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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 complexOTHERcomputational
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*
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*> \par Further Details:
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* =====================
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*>
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*> \verbatim
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*>
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*> Each eigenvector is normalized so that the element of largest
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*> magnitude has magnitude 1; here the magnitude of a complex number
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*> (x,y) is taken to be |x|+|y|.
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*> \endverbatim
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*>
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* =====================================================================
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SUBROUTINE CHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, W, VL,
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$ LDVL, VR, LDVR, MM, M, WORK, RWORK, IFAILL,
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$ IFAILR, INFO )
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*
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* -- LAPACK computational 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 EIGSRC, INITV, SIDE
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INTEGER INFO, LDH, LDVL, LDVR, M, MM, N
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* ..
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* .. Array Arguments ..
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LOGICAL SELECT( * )
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INTEGER IFAILL( * ), IFAILR( * )
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REAL RWORK( * )
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COMPLEX H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ),
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$ W( * ), 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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COMPLEX ZERO
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PARAMETER ( ZERO = ( 0.0E+0, 0.0E+0 ) )
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REAL RZERO
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PARAMETER ( RZERO = 0.0E+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL BOTHV, FROMQR, LEFTV, NOINIT, RIGHTV
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INTEGER I, IINFO, K, KL, KLN, KR, KS, LDWORK
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REAL EPS3, HNORM, SMLNUM, ULP, UNFL
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COMPLEX CDUM, WK
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* ..
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* .. External Functions ..
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LOGICAL LSAME, SISNAN
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REAL CLANHS, SLAMCH
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EXTERNAL LSAME, CLANHS, SLAMCH, SISNAN
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* ..
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* .. External Subroutines ..
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EXTERNAL CLAEIN, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, AIMAG, MAX, REAL
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* ..
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* .. Statement Functions ..
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REAL CABS1
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* ..
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* .. Statement Function definitions ..
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CABS1( CDUM ) = ABS( REAL( CDUM ) ) + ABS( AIMAG( CDUM ) )
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* ..
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* .. Executable Statements ..
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*
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* Decode and test the input parameters.
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*
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BOTHV = LSAME( SIDE, 'B' )
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RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV
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LEFTV = LSAME( SIDE, 'L' ) .OR. BOTHV
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*
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FROMQR = LSAME( EIGSRC, 'Q' )
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*
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NOINIT = LSAME( INITV, 'N' )
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*
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* Set M to the number of columns required to store the selected
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* eigenvectors.
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*
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M = 0
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DO 10 K = 1, N
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IF( SELECT( K ) )
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$ M = M + 1
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10 CONTINUE
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*
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INFO = 0
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IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
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INFO = -1
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ELSE IF( .NOT.FROMQR .AND. .NOT.LSAME( EIGSRC, 'N' ) ) THEN
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INFO = -2
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ELSE IF( .NOT.NOINIT .AND. .NOT.LSAME( INITV, 'U' ) ) THEN
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INFO = -3
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ELSE IF( N.LT.0 ) THEN
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INFO = -5
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ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
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INFO = -7
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ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN
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INFO = -10
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ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN
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INFO = -12
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ELSE IF( MM.LT.M ) THEN
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INFO = -13
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'CHSEIN', -INFO )
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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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* Set machine-dependent constants.
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*
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UNFL = SLAMCH( 'Safe minimum' )
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ULP = SLAMCH( 'Precision' )
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SMLNUM = UNFL*( N / ULP )
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*
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LDWORK = N
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*
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KL = 1
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KLN = 0
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IF( FROMQR ) THEN
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KR = 0
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ELSE
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KR = N
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END IF
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KS = 1
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*
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DO 100 K = 1, N
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IF( SELECT( K ) ) THEN
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*
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* Compute eigenvector(s) corresponding to W(K).
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*
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IF( FROMQR ) THEN
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||
|
|
*
|
||
|
|
* If affiliation of eigenvalues is known, check whether
|
||
|
|
* the matrix splits.
|
||
|
|
*
|
||
|
|
* Determine KL and KR such that 1 <= KL <= K <= KR <= N
|
||
|
|
* and H(KL,KL-1) and H(KR+1,KR) are zero (or KL = 1 or
|
||
|
|
* KR = N).
|
||
|
|
*
|
||
|
|
* Then inverse iteration can be performed with the
|
||
|
|
* submatrix H(KL:N,KL:N) for a left eigenvector, and with
|
||
|
|
* the submatrix H(1:KR,1:KR) for a right eigenvector.
|
||
|
|
*
|
||
|
|
DO 20 I = K, KL + 1, -1
|
||
|
|
IF( H( I, I-1 ).EQ.ZERO )
|
||
|
|
$ GO TO 30
|
||
|
|
20 CONTINUE
|
||
|
|
30 CONTINUE
|
||
|
|
KL = I
|
||
|
|
IF( K.GT.KR ) THEN
|
||
|
|
DO 40 I = K, N - 1
|
||
|
|
IF( H( I+1, I ).EQ.ZERO )
|
||
|
|
$ GO TO 50
|
||
|
|
40 CONTINUE
|
||
|
|
50 CONTINUE
|
||
|
|
KR = I
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
IF( KL.NE.KLN ) THEN
|
||
|
|
KLN = KL
|
||
|
|
*
|
||
|
|
* Compute infinity-norm of submatrix H(KL:KR,KL:KR) if it
|
||
|
|
* has not ben computed before.
|
||
|
|
*
|
||
|
|
HNORM = CLANHS( 'I', KR-KL+1, H( KL, KL ), LDH, RWORK )
|
||
|
|
IF( SISNAN( HNORM ) ) THEN
|
||
|
|
INFO = -6
|
||
|
|
RETURN
|
||
|
|
ELSE IF( (HNORM.GT.RZERO) ) THEN
|
||
|
|
EPS3 = HNORM*ULP
|
||
|
|
ELSE
|
||
|
|
EPS3 = SMLNUM
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
* Perturb eigenvalue if it is close to any previous
|
||
|
|
* selected eigenvalues affiliated to the submatrix
|
||
|
|
* H(KL:KR,KL:KR). Close roots are modified by EPS3.
|
||
|
|
*
|
||
|
|
WK = W( K )
|
||
|
|
60 CONTINUE
|
||
|
|
DO 70 I = K - 1, KL, -1
|
||
|
|
IF( SELECT( I ) .AND. CABS1( W( I )-WK ).LT.EPS3 ) THEN
|
||
|
|
WK = WK + EPS3
|
||
|
|
GO TO 60
|
||
|
|
END IF
|
||
|
|
70 CONTINUE
|
||
|
|
W( K ) = WK
|
||
|
|
*
|
||
|
|
IF( LEFTV ) THEN
|
||
|
|
*
|
||
|
|
* Compute left eigenvector.
|
||
|
|
*
|
||
|
|
CALL CLAEIN( .FALSE., NOINIT, N-KL+1, H( KL, KL ), LDH,
|
||
|
|
$ WK, VL( KL, KS ), WORK, LDWORK, RWORK, EPS3,
|
||
|
|
$ SMLNUM, IINFO )
|
||
|
|
IF( IINFO.GT.0 ) THEN
|
||
|
|
INFO = INFO + 1
|
||
|
|
IFAILL( KS ) = K
|
||
|
|
ELSE
|
||
|
|
IFAILL( KS ) = 0
|
||
|
|
END IF
|
||
|
|
DO 80 I = 1, KL - 1
|
||
|
|
VL( I, KS ) = ZERO
|
||
|
|
80 CONTINUE
|
||
|
|
END IF
|
||
|
|
IF( RIGHTV ) THEN
|
||
|
|
*
|
||
|
|
* Compute right eigenvector.
|
||
|
|
*
|
||
|
|
CALL CLAEIN( .TRUE., NOINIT, KR, H, LDH, WK, VR( 1, KS ),
|
||
|
|
$ WORK, LDWORK, RWORK, EPS3, SMLNUM, IINFO )
|
||
|
|
IF( IINFO.GT.0 ) THEN
|
||
|
|
INFO = INFO + 1
|
||
|
|
IFAILR( KS ) = K
|
||
|
|
ELSE
|
||
|
|
IFAILR( KS ) = 0
|
||
|
|
END IF
|
||
|
|
DO 90 I = KR + 1, N
|
||
|
|
VR( I, KS ) = ZERO
|
||
|
|
90 CONTINUE
|
||
|
|
END IF
|
||
|
|
KS = KS + 1
|
||
|
|
END IF
|
||
|
|
100 CONTINUE
|
||
|
|
*
|
||
|
|
RETURN
|
||
|
|
*
|
||
|
|
* End of CHSEIN
|
||
|
|
*
|
||
|
|
END
|