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# LAPACK-3

## man page of LAPACK-3

### LAPACK-3: uses inverse iteration to find specified right and/or left eigenvectors of a complex upper Hessenberg matrix H

```NAME
LAPACK-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|.

LAPACK routine (version 3.2)     March 2011                       LAPACK-3(3)
```