CHSEQR - compute the eigenvalues of a complex upper Hes
senberg matrix H, and, optionally, the matrices T and Z
from the Schur decomposition H = Z T Z**H, where T is an
upper triangular matrix (the Schur form), and Z is the
unitary matrix of Schur vectors
SYNOPSIS
SUBROUTINE CHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z,
LDZ, WORK, LWORK, INFO )
CHARACTER COMPZ, JOB
INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
COMPLEX H( LDH, * ), W( * ), WORK( * ), Z( LDZ,
* )
PURPOSE
CHSEQR computes the eigenvalues of a complex upper Hessen
berg matrix H, and, optionally, the matrices T and Z from
the Schur decomposition H = Z T Z**H, where T is an upper
triangular matrix (the Schur form), and Z is the unitary
matrix of Schur vectors. Optionally Z may be postmulti
plied into an input unitary matrix Q, so that this routine
can give the Schur factorization of a matrix A which has
been reduced to the Hessenberg form H by the unitary
matrix Q: A = Q*H*Q**H = (QZ)*T*(QZ)**H.
ARGUMENTS
JOB (input) CHARACTER*1
= 'E': compute eigenvalues only;
= 'S': compute eigenvalues and the Schur form T.
COMPZ (input) CHARACTER*1
= 'N': no Schur vectors are computed;
= 'I': Z is initialized to the unit matrix and the
matrix Z of Schur vectors of H is returned; = 'V':
Z must contain an unitary matrix Q on entry, and
the product Q*Z is returned.
N (input) INTEGER
The order of the matrix H. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER It is assumed that H is
already upper triangular in rows and columns
1:ILO-1 and IHI+1:N. ILO and IHI are normally set
by a previous call to CGEBAL, and then passed to
CGEHRD when the matrix output by CGEBAL is reduced
to Hessenberg form. Otherwise ILO and IHI should
be set to 1 and N respectively. 1 <= ILO <= IHI
<= N, if N > 0; ILO=1 and IHI=0, if N=0.
On entry, the upper Hessenberg matrix H. On exit,
if JOB = 'S', H contains the upper triangular
matrix T from the Schur decomposition (the Schur
form). If JOB = 'E', the contents of H are unspec
ified on exit.
LDH (input) INTEGER
The leading dimension of the array H. LDH >=
max(1,N).
W (output) COMPLEX array, dimension (N)
The computed eigenvalues. If JOB = 'S', the eigen
values are stored in the same order as on the
diagonal of the Schur form returned in H, with
W(i) = H(i,i).
Z (input/output) COMPLEX array, dimension (LDZ,N)
If COMPZ = 'N': Z is not referenced.
If COMPZ = 'I': on entry, Z need not be set, and
on exit, Z contains the unitary matrix Z of the
Schur vectors of H. If COMPZ = 'V': on entry Z
must contain an N-by-N matrix Q, which is assumed
to be equal to the unit matrix except for the sub
matrix Z(ILO:IHI,ILO:IHI); on exit Z contains Q*Z.
Normally Q is the unitary matrix generated by
CUNGHR after the call to CGEHRD which formed the
Hessenberg matrix H.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >=
max(1,N) if COMPZ = 'I' or 'V'; LDZ >= 1 other
wise.
WORK (workspace/output) COMPLEX array, dimension
(LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >=
max(1,N).
If LWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of
the WORK array, returns this value as the first
entry of the WORK array, and no error message
related to LWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille
gal value
> 0: if INFO = i, CHSEQR failed to compute all
ations; elements 1:ilo-1 and i+1:n of W contain
those eigenvalues which have been successfully
computed.
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