 ZHSEIN(3) uses 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( * )

## 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|.