NAME

zhseqr - compute the eigenvalues of a complex upper Hessenberg matrix H, and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z∗∗H, where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix of Schur vectors

SYNOPSIS

SUBROUTINE ZHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK, LWORK, INFO )

CHARACTER COMPZ, JOB

INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N

COMPLEX∗16 H( LDH, ∗ ), W( ∗ ), WORK( ∗ ), Z( LDZ, ∗ )

#include <sunperf.h>

void zhseqr(char job, char compz, int n, int ilo, int ihi, doublecomplex ∗h, int ldh, doublecomplex ∗w, doublecomplex ∗zz, int ldz, int ∗info);

PURPOSE

ZHSEQR computes the eigenvalues of a complex upper Hessenberg matrix H, and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z∗∗H, where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix of Schur vectors. 
 
Optionally Z may be postmultiplied into an input unitary 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 unitary matrix Q:  A = Q∗H∗Q∗∗H = (QZ)∗T∗(QZ)∗∗H.
 

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 unitary 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 ZGEBAL, and then passed to CGEHRD when the matrix output by ZGEBAL 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) COMPLEX∗16 array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H. On exit, if JOB = ’S’, H contains the upper triangular matrix T from the Schur decomposition (the Schur form). 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).

W (output) COMPLEX∗16 array, dimension (N)
The computed eigenvalues. If JOB = ’S’, the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with W(i) = H(i,i).

Z (input/output) COMPLEX∗16 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 unitary 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 unitary matrix generated by ZUNGHR after the call to ZGEHRD 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) COMPLEX∗16 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, ZHSEQR failed to compute all the eigenvalues in a total of 30∗(IHI-ILO+1) iterations; elements 1:ilo-1 and i+1:n of W contain those eigenvalues which have been successfully computed.

SunOS 5.0  —  Last change: 10 Dec 1998