
SHSEIN(l) ) SHSEIN(l)
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 blocktriangular, then the jth eigenvalue can be assumed to
be an eigenvalue of the block containing the jth row/column. This property
allows SHSEIN 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, SHSEIN must always perform inverse iteration using the
whole matrix H.
INITV (input) CHARACTER*1
= 'N': no initial vectors are supplied;
= 'U': usersupplied 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 corre
sponding 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 SELECT(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 entry, the real and imaginary parts
of the eigenvalues of H; a complex 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 independent 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 corresponding to a complex eigenvalue is stored in two con
secutive columns, the first holding the real part and the second the imaginary
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 corresponding to a complex eigenvalue is stored in two con
secutive columns, the first holding the real part and the second the imaginary
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 eigenvec
tors; each selected real eigenvector occupies one column and each selected complex
eigenvector occupies 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 ith column
of VL (corresponding to the eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if
the eigenvector converged satisfactorily. If the ith 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 referenced.
IFAILR (output) INTEGER array, dimension (MM)
If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right eigenvector in the ith col
umn of VR (corresponding to the eigenvalue w(j)) failed to converge; IFAILR(i) = 0
if the eigenvector converged satisfactorily. If the ith 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 referenced.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith 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 version 3.0 15 June 2000 SHSEIN(l) 
