dhsein(3F)

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

SYNOPSIS
     SUBROUTINE DHSEIN( 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( * )

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

PURPOSE
     DHSEIN 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 DHSEQR; 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 DHSEIN 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, DHSEIN 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N)
             WI      (input) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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|.