chseqr(3)

NAME

CHSEQR - compute the eigenvalues of a complex upper Hes

senberg 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 CHSEQR( 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        H( LDH, * ), W( * ), WORK( * ), Z( LDZ,
* )

PURPOSE

CHSEQR computes the eigenvalues of a complex upper Hessen

berg 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 uni

tary matrix Q, so that this routine can give the Schur factoriza

tion 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 CGEBAL, and

then passed to CGEHRD when the matrix output by CGEBAL 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 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 array, dimension (N)

The computed eigenvalues. If JOB = 'S', the eigen

values 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 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 CUNGHR after the call to

CGEHRD 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) COMPLEX 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 ille

gal value
> 0: if INFO = i, CHSEQR 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.

LAPACK version 3.0 15 June 2000

CHSEQR(3)