DHSEQR - 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 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 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.
(LDH,N)
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) 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 sub
matrix 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 other
wise.
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).
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, DHSEQR 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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