zhsein(3S)

NAME
     ZHSEIN - use inverse iteration to find specified right and/or left
     eigenvectors of a complex upper Hessenberg matrix H

SYNOPSIS
     SUBROUTINE ZHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, W, VL, LDVL,
                        VR, LDVR, MM, M, WORK, RWORK, IFAILL, IFAILR, INFO )

         CHARACTER      EIGSRC, INITV, SIDE

         INTEGER        INFO, LDH, LDVL, LDVR, M, MM, N

         LOGICAL        SELECT( * )

         INTEGER        IFAILL( * ), IFAILR( * )

         DOUBLE         PRECISION RWORK( * )

         COMPLEX*16     H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ), W( * ),
                        WORK( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     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.


ARGUMENTS
     SIDE    (input) CHARACTER*1
             = 'R': compute right eigenvectors only;
             = 'L': compute left eigenvectors only;
             = 'B': compute both right and left eigenvectors.

     EIGSRC  (input) 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.

     INITV   (input) CHARACTER*1
             = 'N': no initial vectors are supplied;
             = 'U': user-supplied initial vectors are stored in the arrays VL
             and/or VR.

     SELECT  (input) 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..

     N       (input) INTEGER
             The order of the matrix H.  N >= 0.

     H       (input) COMPLEX*16 array, dimension (LDH,N)
             The upper Hessenberg matrix H.

     LDH     (input) INTEGER
             The leading dimension of the array H.  LDH >= max(1,N).

     W       (input/output) 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.

     VL      (input/output) 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.

     LDVL    (input) INTEGER
             The leading dimension of the array VL.  LDVL >= max(1,N) if SIDE
             = 'L' or 'B'; LDVL >= 1 otherwise.

     VR      (input/output) 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.

     LDVR    (input) INTEGER
             The leading dimension of the array VR.  LDVR >= max(1,N) if SIDE
             = 'R' or 'B'; LDVR >= 1 otherwise.

     MM      (input) INTEGER
             The number of columns in the arrays VL and/or VR. MM >= M.

     M       (output) INTEGER
             The number of columns in the arrays VL and/or VR required to
             store the eigenvectors (= the number of .TRUE. elements in
             SELECT).

     WORK    (workspace) COMPLEX*16 array, dimension (N*N)

     RWORK   (workspace) DOUBLE PRECISION array, dimension (N)

     IFAILL  (output) 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.

     IFAILR  (output) 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.

     INFO    (output) 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.

FURTHER DETAILS
     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|.


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

     This man page is available only online.