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NAME
chsein - use inverse iteration to find specified right
and/or left eigenvectors of a complex upper Hessenberg
matrix H
SYNOPSIS
SUBROUTINE CHSEIN( 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( * )
REAL RWORK( * )
COMPLEX H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ), W( * ),
WORK( * )
#include <sunperf.h>
void chsein(char side, char eigsrc, char initv, int *select,
int n, complex *h, int ldh, complex *w, complex
*vl, int ldvl, complex *vr, int ldvr, int mm, int
*m, int *ifaill, int *ifailr, int *info);
PURPOSE
CHSEIN 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 CHSEQR;
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 con-
taining the j-th row/column. This property allows
CHSEIN to perform inverse iteration on just one
diagonal block. = 'N': no assumptions are made on
the correspondence between eigenvalues and diago-
nal blocks. In this case, CHSEIN must always per-
form 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 eigen-
value W(j), SELECT(j) must be set to .TRUE..
N (input) INTEGER
The order of the matrix H. N >= 0.
H (input) COMPLEX 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 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 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 eigenvec-
tors specified by SELECT will be stored consecu-
tively 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 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 eigenvec-
tors specified by SELECT will be stored consecu-
tively 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 other-
wise.
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 array, dimension (N*N)
RWORK (workspace) REAL 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 con-
verged 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 con-
verged satisfactorily. If SIDE = 'L', IFAILR is
not referenced.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille-
gal 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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