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352 lines
9.4 KiB
352 lines
9.4 KiB
2 years ago
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*> \brief \b CLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.
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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 CLAEIN + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/claein.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/claein.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/claein.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 CLAEIN( RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK,
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* EPS3, SMLNUM, INFO )
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*
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* .. Scalar Arguments ..
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* LOGICAL NOINIT, RIGHTV
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* INTEGER INFO, LDB, LDH, N
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* REAL EPS3, SMLNUM
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* COMPLEX W
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* ..
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* .. Array Arguments ..
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* REAL RWORK( * )
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* COMPLEX B( LDB, * ), H( LDH, * ), V( * )
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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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*> CLAEIN uses inverse iteration to find a right or left eigenvector
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*> corresponding to the eigenvalue W of a complex upper Hessenberg
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*> matrix H.
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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] RIGHTV
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*> \verbatim
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*> RIGHTV is LOGICAL
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*> = .TRUE. : compute right eigenvector;
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*> = .FALSE.: compute left eigenvector.
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*> \endverbatim
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*>
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*> \param[in] NOINIT
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*> \verbatim
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*> NOINIT is LOGICAL
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*> = .TRUE. : no initial vector supplied in V
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*> = .FALSE.: initial vector supplied in V.
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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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*> \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] W
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*> \verbatim
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*> W is COMPLEX
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*> The eigenvalue of H whose corresponding right or left
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*> eigenvector is to be computed.
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*> \endverbatim
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*>
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*> \param[in,out] V
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*> \verbatim
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*> V is COMPLEX array, dimension (N)
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*> On entry, if NOINIT = .FALSE., V must contain a starting
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*> vector for inverse iteration; otherwise V need not be set.
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*> On exit, V contains the computed eigenvector, normalized so
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*> that the component of largest magnitude has magnitude 1; here
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*> the magnitude of a complex number (x,y) is taken to be
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*> |x| + |y|.
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*> \endverbatim
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*>
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*> \param[out] B
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*> \verbatim
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*> B is COMPLEX array, dimension (LDB,N)
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*> \endverbatim
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*>
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*> \param[in] LDB
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*> \verbatim
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*> LDB is INTEGER
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*> The leading dimension of the array B. LDB >= max(1,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[in] EPS3
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*> \verbatim
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*> EPS3 is REAL
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*> A small machine-dependent value which is used to perturb
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*> close eigenvalues, and to replace zero pivots.
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*> \endverbatim
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*>
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*> \param[in] SMLNUM
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*> \verbatim
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*> SMLNUM is REAL
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*> A machine-dependent value close to the underflow threshold.
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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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*> = 1: inverse iteration did not converge; V is set to the
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*> last iterate.
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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 complexOTHERauxiliary
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*
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* =====================================================================
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SUBROUTINE CLAEIN( RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK,
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$ EPS3, SMLNUM, INFO )
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*
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* -- LAPACK auxiliary 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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LOGICAL NOINIT, RIGHTV
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INTEGER INFO, LDB, LDH, N
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REAL EPS3, SMLNUM
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COMPLEX W
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* ..
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* .. Array Arguments ..
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REAL RWORK( * )
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COMPLEX B( LDB, * ), H( LDH, * ), V( * )
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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 ONE, TENTH
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PARAMETER ( ONE = 1.0E+0, TENTH = 1.0E-1 )
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COMPLEX ZERO
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PARAMETER ( ZERO = ( 0.0E+0, 0.0E+0 ) )
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* ..
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* .. Local Scalars ..
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CHARACTER NORMIN, TRANS
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INTEGER I, IERR, ITS, J
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REAL GROWTO, NRMSML, ROOTN, RTEMP, SCALE, VNORM
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COMPLEX CDUM, EI, EJ, TEMP, X
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* ..
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* .. External Functions ..
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INTEGER ICAMAX
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REAL SCASUM, SCNRM2
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COMPLEX CLADIV
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EXTERNAL ICAMAX, SCASUM, SCNRM2, CLADIV
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* ..
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* .. External Subroutines ..
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EXTERNAL CLATRS, CSSCAL
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, AIMAG, MAX, REAL, SQRT
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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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INFO = 0
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*
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* GROWTO is the threshold used in the acceptance test for an
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* eigenvector.
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*
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ROOTN = SQRT( REAL( N ) )
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GROWTO = TENTH / ROOTN
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NRMSML = MAX( ONE, EPS3*ROOTN )*SMLNUM
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*
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* Form B = H - W*I (except that the subdiagonal elements are not
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* stored).
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*
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DO 20 J = 1, N
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DO 10 I = 1, J - 1
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B( I, J ) = H( I, J )
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10 CONTINUE
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B( J, J ) = H( J, J ) - W
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20 CONTINUE
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*
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IF( NOINIT ) THEN
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*
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* Initialize V.
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*
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DO 30 I = 1, N
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V( I ) = EPS3
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30 CONTINUE
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ELSE
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*
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* Scale supplied initial vector.
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*
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VNORM = SCNRM2( N, V, 1 )
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CALL CSSCAL( N, ( EPS3*ROOTN ) / MAX( VNORM, NRMSML ), V, 1 )
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END IF
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*
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IF( RIGHTV ) THEN
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*
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* LU decomposition with partial pivoting of B, replacing zero
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* pivots by EPS3.
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*
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DO 60 I = 1, N - 1
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EI = H( I+1, I )
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IF( CABS1( B( I, I ) ).LT.CABS1( EI ) ) THEN
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*
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* Interchange rows and eliminate.
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*
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X = CLADIV( B( I, I ), EI )
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B( I, I ) = EI
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DO 40 J = I + 1, N
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TEMP = B( I+1, J )
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B( I+1, J ) = B( I, J ) - X*TEMP
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B( I, J ) = TEMP
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40 CONTINUE
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ELSE
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*
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* Eliminate without interchange.
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*
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IF( B( I, I ).EQ.ZERO )
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$ B( I, I ) = EPS3
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X = CLADIV( EI, B( I, I ) )
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IF( X.NE.ZERO ) THEN
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DO 50 J = I + 1, N
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B( I+1, J ) = B( I+1, J ) - X*B( I, J )
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50 CONTINUE
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END IF
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END IF
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60 CONTINUE
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IF( B( N, N ).EQ.ZERO )
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$ B( N, N ) = EPS3
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*
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TRANS = 'N'
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*
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ELSE
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*
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* UL decomposition with partial pivoting of B, replacing zero
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* pivots by EPS3.
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*
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DO 90 J = N, 2, -1
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EJ = H( J, J-1 )
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IF( CABS1( B( J, J ) ).LT.CABS1( EJ ) ) THEN
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*
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* Interchange columns and eliminate.
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*
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X = CLADIV( B( J, J ), EJ )
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B( J, J ) = EJ
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DO 70 I = 1, J - 1
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TEMP = B( I, J-1 )
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B( I, J-1 ) = B( I, J ) - X*TEMP
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B( I, J ) = TEMP
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70 CONTINUE
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ELSE
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*
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* Eliminate without interchange.
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*
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IF( B( J, J ).EQ.ZERO )
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$ B( J, J ) = EPS3
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X = CLADIV( EJ, B( J, J ) )
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IF( X.NE.ZERO ) THEN
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DO 80 I = 1, J - 1
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B( I, J-1 ) = B( I, J-1 ) - X*B( I, J )
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80 CONTINUE
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END IF
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END IF
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90 CONTINUE
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IF( B( 1, 1 ).EQ.ZERO )
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$ B( 1, 1 ) = EPS3
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*
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TRANS = 'C'
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*
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END IF
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*
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NORMIN = 'N'
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DO 110 ITS = 1, N
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*
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* Solve U*x = scale*v for a right eigenvector
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* or U**H *x = scale*v for a left eigenvector,
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* overwriting x on v.
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*
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CALL CLATRS( 'Upper', TRANS, 'Nonunit', NORMIN, N, B, LDB, V,
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$ SCALE, RWORK, IERR )
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NORMIN = 'Y'
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*
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* Test for sufficient growth in the norm of v.
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*
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VNORM = SCASUM( N, V, 1 )
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IF( VNORM.GE.GROWTO*SCALE )
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$ GO TO 120
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*
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* Choose new orthogonal starting vector and try again.
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*
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RTEMP = EPS3 / ( ROOTN+ONE )
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V( 1 ) = EPS3
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DO 100 I = 2, N
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V( I ) = RTEMP
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100 CONTINUE
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V( N-ITS+1 ) = V( N-ITS+1 ) - EPS3*ROOTN
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110 CONTINUE
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*
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* Failure to find eigenvector in N iterations.
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*
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INFO = 1
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*
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120 CONTINUE
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*
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* Normalize eigenvector.
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*
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I = ICAMAX( N, V, 1 )
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CALL CSSCAL( N, ONE / CABS1( V( I ) ), V, 1 )
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*
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RETURN
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*
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* End of CLAEIN
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*
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END
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