SHSEIN(1) LAPACK routine (version 3.2) SHSEIN(1)NAME
SHSEIN - uses 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 eigenvec‐
tor 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-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 SHSEIN to
perform inverse iteration on just one diagonal block. = 'N':
no assumptions are made on the correspondence between eigenval‐
ues and diagonal blocks. In this case, SHSEIN 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/output) LOGICAL array, dimension (N)
Specifies the eigenvectors to be computed. To select the real
eigenvector corresponding to a real eigenvalue WR(j), SELECT(j)
must be set to .TRUE.. To select the complex eigenvector corre‐
sponding 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 .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) 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 conju‐
gate 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 eigen‐
vectors; 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 cor‐
responding to a complex eigenvalue is stored in two consecutive
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 eigen‐
vectors; 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 cor‐
responding to a complex eigenvalue is stored in two consecutive
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 eigenvectors; 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 i-th column of VL (corresponding to the eigenvalue w(j))
failed to converge; IFAILL(i) = 0 if the eigenvector 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 referenced.
IFAILR (output) INTEGER array, dimension (MM)
If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right eigenvec‐
tor 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 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 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 routine (version 3.2) November 2008 SHSEIN(1)
