
ZHSEIN(l) ) ZHSEIN(l)
NAME
ZHSEIN  use inverse iteration to find specified right and/or left eigenvectors of a com
plex 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( * )
PURPOSE
ZHSEIN uses inverse iteration to find specified right and/or left eigenvectors of a com
plex 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 blocktriangular, then the jth eigenvalue can be assumed to
be an eigenvalue of the block containing the jth 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': usersupplied 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 indepen
dent 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 eigenvec
tors (= 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 ith 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 refer
enced.
IFAILR (output) INTEGER array, dimension (MM)
If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right eigenvector in the ith col
umn 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 refer
enced.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith 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.
LAPACK version 3.0 15 June 2000 ZHSEIN(l) 
