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 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 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 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) 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) REAL 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, 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 2.0  —  08 October 1994