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# dhsein

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

SYNOPSIS
SUBROUTINE DHSEIN( JOB, EIGSRC, INITV, SELECT, N, H, LDH,
WR, WI, VL, LDVL, VR, LDVR, MM, M, WORK,
IFAILL, IFAILR, INFO )

CHARACTER      EIGSRC, INITV, JOB

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 = 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
(WR,WI):
= 'Q': the eigenvalues were found using DHSEQR;
thus, if H has zero subdiagonal entries, 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 eigen-
value 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 con-
jugate (WR(j+1),WI(j+1)), either SELECT(j) or
SELECT(j+1) or both must be set to .TRUE.; then on
exit SELECT(j) is .TRUE. and SELECT(j+1) is .FALSE..

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 eigen-
values must be stored in consecutive elements of WR
and WI.  On exit, WR may have been altered since
close eigenvalues are perturbed slightly in search-
ing for independent eigenvectors.

(LDVL,MM)
VL      (input/output) DOUBLE PRECISION array, dimension
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(s)
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 eigen-
values. 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 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.

(LDVR,MM)
VR      (input/output) DOUBLE PRECISION array, dimension
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(s)
in which the eigenvector will be stored.  On exit,
if JOB = 'R' or 'B', the right eigenvectors speci-
fied by SELECT will be stored consecutively in the
columns of VR, in the same order as their eigen-
values. 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 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; 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 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 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 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 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 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|.
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