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NAME
DHSEQR - compute 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
SYNOPSIS
SUBROUTINE DHSEQR( 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
DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( *
), WR( * ), Z( LDZ, * )
PURPOSE
DHSEQR 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 factoriza-
tion 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 pre-
vious call to DGEBAL, and then passed to SGEHRD when
the matrix output by DGEBAL is reduced to Hessenberg
form. Otherwise ILO and IHI should be set to 1 and N
respectively. 1 <= ILO <= max(1,IHI); IHI <= N.
H (input/output) DOUBLE PRECISION array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H. On exit,
if JOB = 'S', H contains the upper quasi-triangular
matrix T from the Schur decomposition (the Schur
form); 2-by-2 diagonal blocks (corresponding to com-
plex 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) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension
(N) The real and imaginary parts, respectively, of
the computed eigenvalues. If two eigenvalues are
computed 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) DOUBLE PRECISION 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 submatrix
Z(ILO:IHI,ILO:IHI); on exit Z contains Q*Z. Nor-
mally Q is the orthogonal matrix generated by DORGHR
after the call to DGEHRD 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 otherwise.
WORK (workspace) DOUBLE PRECISION array, dimension (N)
LWORK (input) INTEGER
This argument is currently redundant.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal
value
> 0: if INFO = i, DHSEQR failed to compute all of
the eigenvalues in a total of 30*(IHI-ILO+1) itera-
tions; elements 1:ilo-1 and i+1:n of WR and WI con-
tain those eigenvalues which have been successfully
computed.