DHSEQR(l) ) DHSEQR(l)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 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 DGEBAL, and then passed
to SGEHRD when the matrix output by DGEBAL is reduced to Hes‐
senberg 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.
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 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) 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 subma‐
trix Z(ILO:IHI,ILO:IHI); on exit Z contains Q*Z. Normally 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/output) DOUBLE PRECISION 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 illegal value
> 0: if INFO = i, DHSEQR failed to compute all of the eigen‐
values in a total of 30*(IHI-ILO+1) iterations; elements
1:ilo-1 and i+1:n of WR and WI contain those eigenvalues which
have been successfully computed.
LAPACK version 3.0 15 June 2000 DHSEQR(l)
