shseqr(3)

NAME

SHSEQR - compute the eigenvalues of a real upper Hessen

berg matrix H and, optionally, the matrices T and Z from the

Schur decomposition H = Z T Z**T, where T is an upper quasi-tri

angular 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 de

composition 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 or

thogonal matrix Q, so that this routine can give the Schur fac

torization 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 re

turned 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 un

specified 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 re

al and imaginary parts, respectively, of the computed eigenval

ues. 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/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 ille

gal 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(3)