shsein(3)

NAME

SHSEIN - use inverse iteration to find specified right

and/or left eigenvectors of a real upper Hessenberg matrix H

SYNOPSIS

SUBROUTINE SHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH,
WR, WI, VL, LDVL, VR, LDVR, MM, M, WORK, IFAILL, IFAILR, INFO )
    CHARACTER      EIGSRC, INITV, SIDE
    INTEGER        INFO, LDH, LDVL, LDVR, M, MM, N
    LOGICAL        SELECT( * )
    INTEGER        IFAILL( * ), IFAILR( * )
    REAL           H( LDH, * ), VL( LDVL, * ), VR( LDVR, *
), WI( * ), WORK( * ), WR( * )

PURPOSE

SHSEIN 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 * 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

(WR,WI):
= 'Q': the eigenvalues were found using SHSEQR;

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 SHSEIN 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, SHSEIN 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 se

lect the real eigenvector corresponding to a real eigenvalue

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 conjugate (WR(j+1),WI(j+1)), either SELECT(j) or SE

LECT(j+1) or both must be set to

N (input) INTEGER

The order of the matrix H. N >= 0.

H (input) REAL 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) REAL array, dimension (N)

WI (input) REAL array, dimension (N) On en

try, the real and imaginary parts of the eigenvalues of H; a com

plex conjugate pair of eigenvalues must be stored in consecutive

elements of WR and WI. On exit, WR may have been altered since

close eigenvalues are perturbed slightly in searching for inde

pendent eigenvectors.

VL (input/output) REAL 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(s) 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. A complex eigenvector corre

sponding to a complex eigenvalue is stored in two consecutive

columns, the first holding the real part and the second the imag

inary part. 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) REAL 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(s) 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. A complex eigenvector corre

sponding to a complex eigenvalue is stored in two consecutive

columns, the first holding the real part and the second the imag

inary part. 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; each selected real eigenvec

tor occupies one column and each selected complex eigenvector oc

cupies two columns.

WORK (workspace) REAL array, dimension ((N+2)*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 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 SIDE = 'R', IFAILL is not refer

enced.

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 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 SIDE = 'L', IFAILR is not refer

enced.

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

SHSEIN(3)