SHSEIN(3) LAPACK routine of NEC Numeric Library Collection SHSEIN(3)
NAME
SHSEIN
SYNOPSIS
SUBROUTINE SHSEIN (SIDE, EIGSRC, INITV, SELECT, N, H, LDH, WR, WI, VL,
LDVL, VR, LDVR, MM, M, WORK, IFAILL, IFAILR, INFO)
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)
SIDE is CHARACTER*1
= 'R': compute right eigenvectors only;
= 'L': compute left eigenvectors only;
= 'B': compute both right and left eigenvectors.
EIGSRC (input)
EIGSRC is 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 eigenvalues and diagonal blocks. In this
case, SHSEIN must always perform inverse iteration
using the whole matrix H.
INITV (input)
INITV is 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)
SELECT is 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 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
.TRUE.; then on exit SELECT(j) is .TRUE. and SELECT(j+1) is
.FALSE..
N (input)
N is INTEGER
The order of the matrix H. N >= 0.
H (input)
H is REAL array, dimension (LDH,N)
The upper Hessenberg matrix H.
LDH (input)
LDH is INTEGER
The leading dimension of the array H. LDH >= max(1,N).
WR (input/output)
WR is REAL array, dimension (N)
WI (input)
WI is 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)
VL is 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 consecutive columns, the first holding the real
part and the second the imaginary part.
If SIDE = 'R', VL is not referenced.
LDVL (input)
LDVL is INTEGER
The leading dimension of the array VL.
LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.
VR (input/output)
VR is 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 consecutive columns, the first holding the real
part and the second the imaginary part.
If SIDE = 'L', VR is not referenced.
LDVR (input)
LDVR is INTEGER
The leading dimension of the array VR.
LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.
MM (input)
MM is INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.
M (output)
M is 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 (output)
WORK is REAL array, dimension ((N+2)*N)
IFAILL (output)
IFAILL is 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)
IFAILR is 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 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 referenced.
INFO (output)
INFO is 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 31 October 2017 SHSEIN(3)