sgees(3)

NAME

SGEES - compute for an N-by-N real nonsymmetric matrix A,

the eigenvalues, the real Schur form T, and, optionally, the ma

trix of Schur vectors Z

SYNOPSIS

SUBROUTINE SGEES( JOBVS, SORT, SELECT, N,  A,  LDA,  SDIM,
WR, WI, VS, LDVS, WORK, LWORK, BWORK, INFO )
    CHARACTER     JOBVS, SORT
    INTEGER       INFO, LDA, LDVS, LWORK, N, SDIM
    LOGICAL       BWORK( * )
    REAL           A(  LDA,  *  ), VS( LDVS, * ), WI( * ),
WORK( * ), WR( * )
    LOGICAL       SELECT
    EXTERNAL      SELECT

PURPOSE

SGEES computes for an N-by-N real nonsymmetric matrix A,

the eigenvalues, the real Schur form T, and, optionally, the ma

trix of Schur vectors Z. This gives the Schur factorization A =

Z*T*(Z**T). Optionally, it also orders the eigenvalues on the

diagonal of the real Schur form so that selected eigenvalues are

at the top left. The leading columns of Z then form an orthonor

mal basis for the invariant subspace corresponding to the select

ed eigenvalues.

A matrix is in real Schur form if it is upper quasi-trian

gular with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be stan

dardized in the form

[ a b ]
[ c a ]

where b*c < 0. The eigenvalues of such a block are a +

sqrt(bc).

ARGUMENTS

JOBVS (input) CHARACTER*1

= 'N': Schur vectors are not computed;
= 'V': Schur vectors are computed.

SORT (input) CHARACTER*1

Specifies whether or not to order the eigenvalues

on the diagonal of the Schur form. = 'N': Eigenvalues are not

ordered;
= 'S': Eigenvalues are ordered (see SELECT).

SELECT (input) LOGICAL FUNCTION of two REAL arguments

SELECT must be declared EXTERNAL in the calling

subroutine. If SORT = 'S', SELECT is used to select eigenvalues

to sort to the top left of the Schur form. If SORT = 'N', SELECT

is not referenced. An eigenvalue WR(j)+sqrt(-1)*WI(j) is select

ed if SELECT(WR(j),WI(j)) is true; i.e., if either one of a com

plex conjugate pair of eigenvalues is selected, then both complex

eigenvalues are selected. Note that a selected complex eigenval

ue may no longer satisfy SELECT(WR(j),WI(j)) = .TRUE. after or

dering, since ordering may change the value of complex eigenval

ues (especially if the eigenvalue is ill-conditioned); in this

case INFO is set to N+2 (see INFO below).

N (input) INTEGER

The order of the matrix A. N >= 0.

A (input/output) REAL array, dimension (LDA,N)

On entry, the N-by-N matrix A. On exit, A has

been overwritten by its real Schur form T.

LDA (input) INTEGER

The leading dimension of the array A. LDA >=

max(1,N).

SDIM (output) INTEGER

If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM =

number of eigenvalues (after sorting) for which SELECT is true.

(Complex conjugate pairs for which SELECT is true for either

eigenvalue count as 2.)

WR (output) REAL array, dimension (N)

WI (output) REAL array, dimension (N) WR and

WI contain the real and imaginary parts, respectively, of the

computed eigenvalues in the same order that they appear on the

diagonal of the output Schur form T. Complex conjugate pairs of

eigenvalues will appear consecutively with the eigenvalue having

the positive imaginary part first.

VS (output) REAL array, dimension (LDVS,N)

If JOBVS = 'V', VS contains the orthogonal matrix

Z of Schur vectors. If JOBVS = 'N', VS is not referenced.

LDVS (input) INTEGER

The leading dimension of the array VS. LDVS >= 1;

if JOBVS = 'V', LDVS >= N.

WORK (workspace/output) REAL array, dimension (LWORK)

On exit, if INFO = 0, WORK(1) contains the optimal

LWORK.

LWORK (input) INTEGER

The dimension of the array WORK. LWORK >=

max(1,3*N). For good performance, LWORK must generally be larg

er.

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.

BWORK (workspace) LOGICAL array, dimension (N)

Not referenced if SORT = 'N'.

INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille

gal value.
> 0: if INFO = i, and i is
<= N: the QR algorithm failed to compute all the
eigenvalues; elements 1:ILO-1 and i+1:N of WR and

WI contain those eigenvalues which have converged; if JOBVS =

'V', VS contains the matrix which reduces A to its partially con

verged Schur form. = N+1: the eigenvalues could not be reordered

because some eigenvalues were too close to separate (the problem

is very ill-conditioned); = N+2: after reordering, roundoff

changed values of some complex eigenvalues so that leading eigen

values in the Schur form no longer satisfy SELECT=.TRUE. This

could also be caused by underflow due to scaling.

LAPACK version 3.0 15 June 2000

SGEES(3)