zgeesx(3)

NAME

ZGEESX - compute for an N-by-N complex nonsymmetric matrix

A, the eigenvalues, the Schur form T, and, optionally, the matrix

of Schur vectors Z

SYNOPSIS

SUBROUTINE ZGEESX( JOBVS, SORT, SELECT, SENSE, N, A,  LDA,
SDIM, W, VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK, BWORK, INFO )
    CHARACTER      JOBVS, SENSE, SORT
    INTEGER        INFO, LDA, LDVS, LWORK, N, SDIM
    DOUBLE         PRECISION RCONDE, RCONDV
    LOGICAL        BWORK( * )
    DOUBLE         PRECISION RWORK( * )
    COMPLEX*16     A( LDA, * ), VS( LDVS, *  ),  W(  *  ),
WORK( * )
    LOGICAL        SELECT
    EXTERNAL       SELECT

PURPOSE

ZGEESX computes for an N-by-N complex nonsymmetric matrix

A, the eigenvalues, the Schur form T, and, optionally, the matrix

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

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

diagonal of the Schur form so that selected eigenvalues are at

the top left; computes a reciprocal condition number for the av

erage of the selected eigenvalues (RCONDE); and computes a recip

rocal 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 complex matrix is in Schur form if it is upper triangu

lar.

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 one COMPLEX*16 argu

ment

SELECT must be declared EXTERNAL in the calling

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

to order to the top left of the Schur form. If SORT = 'N', SE

LECT is not referenced. An eigenvalue W(j) is selected if SE

LECT(W(j)) is true.

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) COMPLEX*16 array, dimension (LDA,

N)

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

overwritten by its 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 for which SELECT is true.

W (output) COMPLEX*16 array, dimension (N)

W contains the computed eigenvalues, in the same

order that they appear on the diagonal of the output Schur form

T.

VS (output) COMPLEX*16 array, dimension (LDVS,N)

If JOBVS = 'V', VS contains the unitary 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) DOUBLE PRECISION

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) DOUBLE PRECISION

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) COMPLEX*16 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(1,2*N). Also, if SENSE = 'E' or 'V' or 'B', LWORK >=

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

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

For good performance, LWORK must generally be larger.

RWORK (workspace) DOUBLE PRECISION array, dimension (N)

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 W con

tain those eigenvalues which have converged; if JOBVS = 'V', VS

contains the transformation 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

ZGEESX(3)