SHSEQR(l) ) SHSEQR(l)
NAME
SHSEQR - 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 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 orthogo-
nal 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.
H (input/output) REAL 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) REAL array, dimension (N)
WI (output) REAL array, dimension (N) The real and imaginary parts, respectively, of the computed eigenvalues. If two eigen-
values 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) 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 submatrix
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 otherwise.
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 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, SHSEQR failed to compute all of the eigenvalues 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 SHSEQR(l)