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*> \brief \b ZHSEIN
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZHSEIN + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhsein.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhsein.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhsein.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, W, VL,
* LDVL, VR, LDVR, MM, M, WORK, RWORK, IFAILL,
* IFAILR, INFO )
*
* .. Scalar Arguments ..
* CHARACTER EIGSRC, INITV, SIDE
* INTEGER INFO, LDH, LDVL, LDVR, M, MM, N
* ..
* .. Array Arguments ..
* LOGICAL SELECT( * )
* INTEGER IFAILL( * ), IFAILR( * )
* DOUBLE PRECISION RWORK( * )
* COMPLEX*16 H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ),
* $ W( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZHSEIN uses inverse iteration to find specified right and/or left
*> eigenvectors of a complex upper Hessenberg matrix H.
*>
*> The right eigenvector x and the left eigenvector y of the matrix H
*> corresponding to an eigenvalue w are defined by:
*>
*> H * x = w * x, y**h * H = w * y**h
*>
*> where y**h denotes the conjugate transpose of the vector y.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'R': compute right eigenvectors only;
*> = 'L': compute left eigenvectors only;
*> = 'B': compute both right and left eigenvectors.
*> \endverbatim
*>
*> \param[in] EIGSRC
*> \verbatim
*> EIGSRC is CHARACTER*1
*> Specifies the source of eigenvalues supplied in W:
*> = 'Q': the eigenvalues were found using ZHSEQR; thus, if
*> H has zero subdiagonal elements, and so is
*> block-triangular, then the j-th eigenvalue can be
*> assumed to be an eigenvalue of the block containing
*> the j-th row/column. This property allows ZHSEIN to
*> perform inverse iteration on just one diagonal block.
*> = 'N': no assumptions are made on the correspondence
*> between eigenvalues and diagonal blocks. In this
*> case, ZHSEIN must always perform inverse iteration
*> using the whole matrix H.
*> \endverbatim
*>
*> \param[in] INITV
*> \verbatim
*> INITV is CHARACTER*1
*> = 'N': no initial vectors are supplied;
*> = 'U': user-supplied initial vectors are stored in the arrays
*> VL and/or VR.
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*> SELECT is LOGICAL array, dimension (N)
*> Specifies the eigenvectors to be computed. To select the
*> eigenvector corresponding to the eigenvalue W(j),
*> SELECT(j) must be set to .TRUE..
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H. N >= 0.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*> H is COMPLEX*16 array, dimension (LDH,N)
*> The upper Hessenberg matrix H.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the array H. LDH >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] W
*> \verbatim
*> W is COMPLEX*16 array, dimension (N)
*> On entry, the eigenvalues of H.
*> On exit, the real parts of W may have been altered since
*> close eigenvalues are perturbed slightly in searching for
*> independent eigenvectors.
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*> VL is COMPLEX*16 array, dimension (LDVL,MM)
*> On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must
*> contain starting vectors for the inverse iteration for the
*> left eigenvectors; the starting vector for each eigenvector
*> must be in the same column in which the eigenvector will be
*> stored.
*> On exit, if SIDE = 'L' or 'B', the left eigenvectors
*> specified by SELECT will be stored consecutively in the
*> columns of VL, in the same order as their eigenvalues.
*> If SIDE = 'R', VL is not referenced.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the array VL.
*> LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.
*> \endverbatim
*>
*> \param[in,out] VR
*> \verbatim
*> VR is COMPLEX*16 array, dimension (LDVR,MM)
*> On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must
*> contain starting vectors for the inverse iteration for the
*> right eigenvectors; the starting vector for each eigenvector
*> must be in the same column in which the eigenvector will be
*> stored.
*> On exit, if SIDE = 'R' or 'B', the right eigenvectors
*> specified by SELECT will be stored consecutively in the
*> columns of VR, in the same order as their eigenvalues.
*> If SIDE = 'L', VR is not referenced.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the array VR.
*> LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*> MM is INTEGER
*> The number of columns in the arrays VL and/or VR. MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The number of columns in the arrays VL and/or VR required to
*> store the eigenvectors (= the number of .TRUE. elements in
*> SELECT).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX*16 array, dimension (N*N)
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] IFAILL
*> \verbatim
*> IFAILL is INTEGER array, dimension (MM)
*> If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left
*> eigenvector in the i-th column of VL (corresponding to the
*> eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the
*> eigenvector converged satisfactorily.
*> If SIDE = 'R', IFAILL is not referenced.
*> \endverbatim
*>
*> \param[out] IFAILR
*> \verbatim
*> IFAILR is INTEGER array, dimension (MM)
*> If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right
*> eigenvector in the i-th column of VR (corresponding to the
*> eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the
*> eigenvector converged satisfactorily.
*> If SIDE = 'L', IFAILR is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, i is the number of eigenvectors which
*> failed to converge; see IFAILL and IFAILR for further
*> details.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup complex16OTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Each eigenvector is normalized so that the element of largest
*> magnitude has magnitude 1; here the magnitude of a complex number
*> (x,y) is taken to be |x|+|y|.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE ZHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, W, VL,
$ LDVL, VR, LDVR, MM, M, WORK, RWORK, IFAILL,
$ IFAILR, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER EIGSRC, INITV, SIDE
INTEGER INFO, LDH, LDVL, LDVR, M, MM, N
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
INTEGER IFAILL( * ), IFAILR( * )
DOUBLE PRECISION RWORK( * )
COMPLEX*16 H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ),
$ W( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX*16 ZERO
PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ) )
DOUBLE PRECISION RZERO
PARAMETER ( RZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL BOTHV, FROMQR, LEFTV, NOINIT, RIGHTV
INTEGER I, IINFO, K, KL, KLN, KR, KS, LDWORK
DOUBLE PRECISION EPS3, HNORM, SMLNUM, ULP, UNFL
COMPLEX*16 CDUM, WK
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, ZLANHS
EXTERNAL LSAME, DLAMCH, ZLANHS
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZLAEIN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DIMAG, MAX
* ..
* .. Statement Functions ..
DOUBLE PRECISION CABS1
* ..
* .. Statement Function definitions ..
CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
* ..
* .. Executable Statements ..
*
* Decode and test the input parameters.
*
BOTHV = LSAME( SIDE, 'B' )
RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV
LEFTV = LSAME( SIDE, 'L' ) .OR. BOTHV
*
FROMQR = LSAME( EIGSRC, 'Q' )
*
NOINIT = LSAME( INITV, 'N' )
*
* Set M to the number of columns required to store the selected
* eigenvectors.
*
M = 0
DO 10 K = 1, N
IF( SELECT( K ) )
$ M = M + 1
10 CONTINUE
*
INFO = 0
IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
INFO = -1
ELSE IF( .NOT.FROMQR .AND. .NOT.LSAME( EIGSRC, 'N' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOINIT .AND. .NOT.LSAME( INITV, 'U' ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN
INFO = -10
ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN
INFO = -12
ELSE IF( MM.LT.M ) THEN
INFO = -13
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZHSEIN', -INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( N.EQ.0 )
$ RETURN
*
* Set machine-dependent constants.
*
UNFL = DLAMCH( 'Safe minimum' )
ULP = DLAMCH( 'Precision' )
SMLNUM = UNFL*( N / ULP )
*
LDWORK = N
*
KL = 1
KLN = 0
IF( FROMQR ) THEN
KR = 0
ELSE
KR = N
END IF
KS = 1
*
DO 100 K = 1, N
IF( SELECT( K ) ) THEN
*
* Compute eigenvector(s) corresponding to W(K).
*
IF( FROMQR ) THEN
*
* 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 = ZLANHS( 'I', KR-KL+1, H( KL, KL ), LDH, RWORK )
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 ZLAEIN( .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 ZLAEIN( .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 ZHSEIN
*
END
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