*> \brief \b ZLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZLAEIN + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * 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. * *> \date December 2016 * *> \ingroup complex16OTHERauxiliary * * ===================================================================== SUBROUTINE ZLAEIN( RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK, $ EPS3, SMLNUM, INFO ) * * -- LAPACK auxiliary routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. 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