*> \brief \b ZLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE ZLAEIN( RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK,
* EPS3, SMLNUM, INFO )
*
* .. Scalar Arguments ..
* LOGICAL NOINIT, RIGHTV
* INTEGER INFO, LDB, LDH, N
* DOUBLE PRECISION EPS3, SMLNUM
* COMPLEX*16 W
* ..
* .. Array Arguments ..
* DOUBLE PRECISION RWORK( * )
* COMPLEX*16 B( LDB, * ), H( LDH, * ), V( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZLAEIN uses inverse iteration to find a right or left eigenvector
*> corresponding to the eigenvalue W of a complex upper Hessenberg
*> matrix H.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] RIGHTV
*> \verbatim
*> RIGHTV is LOGICAL
*> = .TRUE. : compute right eigenvector;
*> = .FALSE.: compute left eigenvector.
*> \endverbatim
*>
*> \param[in] NOINIT
*> \verbatim
*> NOINIT is LOGICAL
*> = .TRUE. : no initial vector supplied in V
*> = .FALSE.: initial vector supplied in V.
*> \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] W
*> \verbatim
*> W is COMPLEX*16
*> The eigenvalue of H whose corresponding right or left
*> eigenvector is to be computed.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*> V is COMPLEX*16 array, dimension (N)
*> On entry, if NOINIT = .FALSE., V must contain a starting
*> vector for inverse iteration; otherwise V need not be set.
*> On exit, V contains the computed eigenvector, normalized so
*> that the component of largest magnitude has magnitude 1; here
*> the magnitude of a complex number (x,y) is taken to be
*> |x| + |y|.
*> \endverbatim
*>
*> \param[out] B
*> \verbatim
*> B is COMPLEX*16 array, dimension (LDB,N)
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[in] EPS3
*> \verbatim
*> EPS3 is DOUBLE PRECISION
*> A small machine-dependent value which is used to perturb
*> close eigenvalues, and to replace zero pivots.
*> \endverbatim
*>
*> \param[in] SMLNUM
*> \verbatim
*> SMLNUM is DOUBLE PRECISION
*> A machine-dependent value close to the underflow threshold.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> = 1: inverse iteration did not converge; V is set to the
*> last iterate.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complex16OTHERauxiliary
*
* =====================================================================
SUBROUTINE ZLAEIN( RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK,
$ EPS3, SMLNUM, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL NOINIT, RIGHTV
INTEGER INFO, LDB, LDH, N
DOUBLE PRECISION EPS3, SMLNUM
COMPLEX*16 W
* ..
* .. Array Arguments ..
DOUBLE PRECISION RWORK( * )
COMPLEX*16 B( LDB, * ), H( LDH, * ), V( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, TENTH
PARAMETER ( ONE = 1.0D+0, TENTH = 1.0D-1 )
COMPLEX*16 ZERO
PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
CHARACTER NORMIN, TRANS
INTEGER I, IERR, ITS, J
DOUBLE PRECISION GROWTO, NRMSML, ROOTN, RTEMP, SCALE, VNORM
COMPLEX*16 CDUM, EI, EJ, TEMP, X
* ..
* .. External Functions ..
INTEGER IZAMAX
DOUBLE PRECISION DZASUM, DZNRM2
COMPLEX*16 ZLADIV
EXTERNAL IZAMAX, DZASUM, DZNRM2, ZLADIV
* ..
* .. External Subroutines ..
EXTERNAL ZDSCAL, ZLATRS
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DIMAG, MAX, SQRT
* ..
* .. Statement Functions ..
DOUBLE PRECISION CABS1
* ..
* .. Statement Function definitions ..
CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
* ..
* .. Executable Statements ..
*
INFO = 0
*
* GROWTO is the threshold used in the acceptance test for an
* eigenvector.
*
ROOTN = SQRT( DBLE( N ) )
GROWTO = TENTH / ROOTN
NRMSML = MAX( ONE, EPS3*ROOTN )*SMLNUM
*
* Form B = H - W*I (except that the subdiagonal elements are not
* stored).
*
DO 20 J = 1, N
DO 10 I = 1, J - 1
B( I, J ) = H( I, J )
10 CONTINUE
B( J, J ) = H( J, J ) - W
20 CONTINUE
*
IF( NOINIT ) THEN
*
* Initialize V.
*
DO 30 I = 1, N
V( I ) = EPS3
30 CONTINUE
ELSE
*
* Scale supplied initial vector.
*
VNORM = DZNRM2( N, V, 1 )
CALL ZDSCAL( N, ( EPS3*ROOTN ) / MAX( VNORM, NRMSML ), V, 1 )
END IF
*
IF( RIGHTV ) THEN
*
* LU decomposition with partial pivoting of B, replacing zero
* pivots by EPS3.
*
DO 60 I = 1, N - 1
EI = H( I+1, I )
IF( CABS1( B( I, I ) ).LT.CABS1( EI ) ) THEN
*
* Interchange rows and eliminate.
*
X = ZLADIV( B( I, I ), EI )
B( I, I ) = EI
DO 40 J = I + 1, N
TEMP = B( I+1, J )
B( I+1, J ) = B( I, J ) - X*TEMP
B( I, J ) = TEMP
40 CONTINUE
ELSE
*
* Eliminate without interchange.
*
IF( B( I, I ).EQ.ZERO )
$ B( I, I ) = EPS3
X = ZLADIV( EI, B( I, I ) )
IF( X.NE.ZERO ) THEN
DO 50 J = I + 1, N
B( I+1, J ) = B( I+1, J ) - X*B( I, J )
50 CONTINUE
END IF
END IF
60 CONTINUE
IF( B( N, N ).EQ.ZERO )
$ B( N, N ) = EPS3
*
TRANS = 'N'
*
ELSE
*
* UL decomposition with partial pivoting of B, replacing zero
* pivots by EPS3.
*
DO 90 J = N, 2, -1
EJ = H( J, J-1 )
IF( CABS1( B( J, J ) ).LT.CABS1( EJ ) ) THEN
*
* Interchange columns and eliminate.
*
X = ZLADIV( B( J, J ), EJ )
B( J, J ) = EJ
DO 70 I = 1, J - 1
TEMP = B( I, J-1 )
B( I, J-1 ) = B( I, J ) - X*TEMP
B( I, J ) = TEMP
70 CONTINUE
ELSE
*
* Eliminate without interchange.
*
IF( B( J, J ).EQ.ZERO )
$ B( J, J ) = EPS3
X = ZLADIV( EJ, B( J, J ) )
IF( X.NE.ZERO ) THEN
DO 80 I = 1, J - 1
B( I, J-1 ) = B( I, J-1 ) - X*B( I, J )
80 CONTINUE
END IF
END IF
90 CONTINUE
IF( B( 1, 1 ).EQ.ZERO )
$ B( 1, 1 ) = EPS3
*
TRANS = 'C'
*
END IF
*
NORMIN = 'N'
DO 110 ITS = 1, N
*
* Solve U*x = scale*v for a right eigenvector
* or U**H *x = scale*v for a left eigenvector,
* overwriting x on v.
*
CALL ZLATRS( 'Upper', TRANS, 'Nonunit', NORMIN, N, B, LDB, V,
$ SCALE, RWORK, IERR )
NORMIN = 'Y'
*
* Test for sufficient growth in the norm of v.
*
VNORM = DZASUM( N, V, 1 )
IF( VNORM.GE.GROWTO*SCALE )
$ GO TO 120
*
* Choose new orthogonal starting vector and try again.
*
RTEMP = EPS3 / ( ROOTN+ONE )
V( 1 ) = EPS3
DO 100 I = 2, N
V( I ) = RTEMP
100 CONTINUE
V( N-ITS+1 ) = V( N-ITS+1 ) - EPS3*ROOTN
110 CONTINUE
*
* Failure to find eigenvector in N iterations.
*
INFO = 1
*
120 CONTINUE
*
* Normalize eigenvector.
*
I = IZAMAX( N, V, 1 )
CALL ZDSCAL( N, ONE / CABS1( V( I ) ), V, 1 )
*
RETURN
*
* End of ZLAEIN
*
END