SHSEIN (3lapack)

use inverse iteration to find specified right and/or left eigenvectors of a real upper Hessenberg matrix H

SYNOPSIS

  SUBROUTINE SHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, WR, WI, VL,
                     LDVL, VR, LDVR, MM, M, WORK, IFAILL, IFAILR, INFO )

      CHARACTER      EIGSRC, INITV, SIDE

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

      LOGICAL        SELECT( * )

      INTEGER        IFAILL( * ), IFAILR( * )

      REAL           H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ), WI( * ),
                     WORK( * ), WR( * )

PURPOSE

  SHSEIN uses inverse iteration to find specified right and/or left
  eigenvectors of a real 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 (WR,WI):
          = 'Q': the eigenvalues were found using SHSEQR; 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 SHSEIN 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, SHSEIN 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/output) LOGICAL array, dimension (N)
          Specifies the eigenvectors to be computed. To select the real
          eigenvector corresponding to a real eigenvalue WR(j), SELECT(j)
          must be set to .TRUE.. To select the complex eigenvector
          corresponding to a complex eigenvalue (WR(j),WI(j)), with complex
          conjugate (WR(j+1),WI(j+1)), either SELECT(j) or SELECT(j+1) or
          both must be set to

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

  H       (input) REAL array, dimension (LDH,N)
          The upper Hessenberg matrix H.

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

  WR      (input/output) REAL array, dimension (N)
          WI      (input) REAL array, dimension (N) On entry, the real and
          imaginary parts of the eigenvalues of H; a complex conjugate pair
          of eigenvalues must be stored in consecutive elements of WR and WI.
          On exit, WR may have been altered since close eigenvalues are
          perturbed slightly in searching for independent eigenvectors.

  VL      (input/output) REAL 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(s) 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. A complex eigenvector
          corresponding to a complex eigenvalue is stored in two consecutive
          columns, the first holding the real part and the second the
          imaginary part.  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) REAL 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(s) 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. A complex eigenvector
          corresponding to a complex eigenvalue is stored in two consecutive
          columns, the first holding the real part and the second the
          imaginary part.  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; each selected real eigenvector occupies one
          column and each selected complex eigenvector occupies two columns.

  WORK    (workspace) REAL array, dimension ((N+2)*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 the i-th and (i+1)th columns of VL hold a
          complex eigenvector, then IFAILL(i) and IFAILL(i+1) are set to the
          same value.  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 the i-th and (i+1)th columns of VR hold a
          complex eigenvector, then IFAILR(i) and IFAILR(i+1) are set to the
          same value.  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|.