sgeesx(3)

NAME

SGEESX - 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 SGEESX( JOBVS, SORT, SELECT, SENSE, N, A,  LDA,
SDIM,  WR,  WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO )
    CHARACTER      JOBVS, SENSE, SORT
    INTEGER        INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM
    REAL           RCONDE, RCONDV
    LOGICAL        BWORK( * )
    INTEGER        IWORK( * )
    REAL            A(  LDA,  * ), VS( LDVS, * ), WI( * ),
WORK( * ), WR( * )
    LOGICAL        SELECT
    EXTERNAL       SELECT

PURPOSE

SGEESX 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; computes a reciprocal condition number for the

average of the selected eigenvalues (RCONDE); and computes a re

ciprocal condition number for the right invariant subspace corre

sponding to the selected eigenvalues (RCONDV). The leading

columns of Z form an orthonormal basis for this invariant sub

space.

For further explanation of the reciprocal condition num

bers RCONDE and RCONDV, see Section 4.10 of the LAPACK Users'

Guide (where these quantities are called s and sep respectively).

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

triangular with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be

standardized 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 are.

Note that a selected complex eigenvalue may no longer satisfy SE

LECT(WR(j),WI(j)) = .TRUE. after ordering, since ordering may

change the value of complex eigenvalues (especially if the eigen

value is ill-conditioned); in this case INFO may be set to N+3

(see INFO below).

SENSE (input) CHARACTER*1

Determines which reciprocal condition numbers are

computed. = 'N': None are computed;
= 'E': Computed for average of selected eigenval

ues only;
= 'V': Computed for selected right invariant sub

space only;
= 'B': Computed for both. If SENSE = 'E', 'V' or

'B', SORT must equal 'S'.

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 is

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 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,

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

RCONDE (output) REAL

If SENSE = 'E' or 'B', RCONDE contains the recip

rocal condition number for the average of the selected eigenval

ues. Not referenced if SENSE = 'N' or 'V'.

RCONDV (output) REAL

If SENSE = 'V' or 'B', RCONDV contains the recip

rocal condition number for the selected right invariant subspace.

Not referenced if SENSE = 'N' or 'E'.

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(1,3*N). Also, if SENSE = 'E' or 'V' or 'B', LWORK >=

N+2*SDIM*(N-SDIM), where SDIM is the number of selected eigenval

ues computed by this routine. Note that N+2*SDIM*(N-SDIM) <=

N+N*N/2. For good performance, LWORK must generally be larger.

IWORK (workspace/output) INTEGER array, dimension (LI

WORK)

Not referenced if SENSE = 'N' or 'E'. On exit, if

INFO = 0, IWORK(1) returns the optimal LIWORK.

LIWORK (input) INTEGER

The dimension of the array IWORK. LIWORK >= 1; if

SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).

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 transformation which reduces A to its par

tially converged 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 lead

ing eigenvalues 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

SGEESX(3)