SHSEQR - compute the eigenvalues of a real upper Hessen
berg matrix H and, optionally, the matrices T and Z from
the Schur decomposition H = Z T Z**T, where T is an upper
quasi-triangular matrix (the Schur form), and Z is the
orthogonal matrix of Schur vectors
SYNOPSIS
SUBROUTINE SHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR,
WI, Z, LDZ, WORK, LWORK, INFO )
CHARACTER COMPZ, JOB
INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
REAL H( LDH, * ), WI( * ), WORK( * ), WR( *
), Z( LDZ, * )
PURPOSE
SHSEQR computes the eigenvalues of a real upper Hessenberg
matrix H and, optionally, the matrices T and Z from the
Schur decomposition H = Z T Z**T, where T is an upper
quasi-triangular matrix (the Schur form), and Z is the
orthogonal matrix of Schur vectors. Optionally Z may be
postmultiplied into an input orthogonal 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
orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
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 orthogonal 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 SGEBAL, and then passed to
SGEHRD when the matrix output by SGEBAL 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 quasi-triangu
lar matrix T from the Schur decomposition (the
Schur form); 2-by-2 diagonal blocks (corresponding
to complex conjugate pairs of eigenvalues) are
returned in standard form, with H(i,i) =
H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0. If JOB =
'E', the contents of H are unspecified on exit.
LDH (input) INTEGER
The leading dimension of the array H. LDH >=
max(1,N).
WR (output) REAL array, dimension (N)
WI (output) REAL array, dimension (N) The
real and imaginary parts, respectively, of the
computed eigenvalues. If two eigenvalues are com
puted as a complex conjugate pair, they are stored
in consecutive elements of WR and WI, say the i-th
and (i+1)th, with WI(i) > 0 and WI(i+1) < 0. If
JOB = 'S', the eigenvalues are stored in the same
order as on the diagonal of the Schur form
returned in H, with WR(i) = H(i,i) and, if
H(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) =
sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).
Z (input/output) REAL 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 orthogonal 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 orthogonal matrix generated by
SORGHR after the call to SGEHRD 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) REAL 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
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, SHSEQR failed to compute all of
the eigenvalues in a total of 30*(IHI-ILO+1) iter
ations; elements 1:ilo-1 and i+1:n of WR and WI
contain those eigenvalues which have been success
fully computed.
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