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# zhsein

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

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

CHARACTER      EIGSRC, INITV, JOB

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( * )

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 = w y'

where y' denotes the conjugate transpose of the vector y.

ARGUMENTS
JOB     (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 eigen-
values are perturbed slightly in searching for
independent eigenvectors.

VL      (input/output) COMPLEX*16 array, dimension (LDVL,MM)
On entry, if INITV = 'U' and JOB = 'L' or 'B', VL
must contain starting vectors for the inverse itera-
tion 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
JOB = '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
JOB = 'R', VL is not referenced.

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

VR      (input/output) COMPLEX*16 array, dimension (LDVR,MM)
On entry, if INITV = 'U' and JOB = 'R' or 'B', VR
must contain starting vectors for the inverse itera-
tion 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
JOB = '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 JOB = 'L', VR is not referenced.

LDVR    (input) INTEGER
The leading dimension of the array VR.  LDVR >=
max(1,N) if JOB = '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 JOB = '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 satisfac-
torily.  If JOB = 'R', IFAILL is not referenced.

IFAILR  (output) INTEGER array, dimension (MM)
If JOB = '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 satisfac-
torily.  If JOB = '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 larg-
est magnitude has magnitude 1; here the magnitude of a com-
plex number (x,y) is taken to be |x|+|y|.
```