chsein(3)

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( * )

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-trian

gular, then the j-th eigenvalue can be assumed to be an eigenval

ue of the block containing 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 be

tween eigenvalues and diagonal blocks. In this case, CHSEIN 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 se

lect the eigenvector corresponding to the eigenvalue W(j), SE

LECT(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 per

turbed 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 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 refer

enced.

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 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. ele

ments 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 eigen

vector 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 eigen

vector 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 ille

gal value
> 0: if INFO = i, i is the number of eigenvectors

which failed to converge; see IFAILL and IFAILR for further de

tails.

FURTHER DETAILS

Each eigenvector is normalized so that the element of

largest magnitude has magnitude 1; here the magnitude of a com

plex number (x,y) is taken to be |x|+|y|.

LAPACK version 3.0 15 June 2000

CHSEIN(3)