
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 quasitri
angular 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 quasitri
angular matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors. Option
ally 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 prod
uct 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:ILO1 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.
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 quasitriangular matrix T from the Schur decomposition (the Schur form);
2by2 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 ith and (i+1)th, with WI(i) > 0 and WI(i+1) < 0. If JOB = 'S', the eigen
values 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 2by2 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 orthogo
nal matrix Z of the Schur vectors of H. If COMPZ = 'V': on entry Z must contain
an NbyN 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. Normally Q is the orthogo
nal matrix generated by DORGHR after the call to DGEHRD which formed the Hessen
berg 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 ith argument had an illegal value
> 0: if INFO = i, DHSEQR failed to compute all of the eigenvalues in a total of
30*(IHIILO+1) iterations; elements 1:ilo1 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) 
