zggesx(3)

NAME

ZGGESX - compute for a pair of N-by-N complex nonsymmetric

matrices (A,B), the generalized eigenvalues, the complex Schur

form (S,T),

SYNOPSIS

SUBROUTINE ZGGESX( JOBVSL, JOBVSR, SORT, DELCTG, SENSE, N,
A,  LDA,  B,  LDB,  SDIM,  ALPHA,  BETA,  VSL, LDVSL, VSR, LDVSR,
RCONDE, RCONDV, WORK, LWORK, RWORK, IWORK, LIWORK, BWORK, INFO )
    CHARACTER      JOBVSL, JOBVSR, SENSE, SORT
    INTEGER        INFO, LDA, LDB, LDVSL,  LDVSR,  LIWORK,
LWORK, N, SDIM
    LOGICAL        BWORK( * )
    INTEGER        IWORK( * )
    DOUBLE          PRECISION  RCONDE(  2  ), RCONDV( 2 ),
RWORK( * )
    COMPLEX*16     A( LDA, * ), ALPHA( * ), B( LDB,  *  ),
BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), WORK( * )
    LOGICAL        DELCTG
    EXTERNAL       DELCTG

PURPOSE

ZGGESX computes for a pair of N-by-N complex nonsymmetric

matrices (A,B), the generalized eigenvalues, the complex Schur

form (S,T), and, optionally, the left and/or right matrices of

Schur vectors (VSL and VSR). This gives the generalized Schur

factorization

(A,B) = ( (VSL) S (VSR)**H, (VSL) T (VSR)**H )

where (VSR)**H is the conjugate-transpose of VSR.

Optionally, it also orders the eigenvalues so that a se

lected cluster of eigenvalues appears in the leading diagonal

blocks of the upper triangular matrix S and the upper triangular

matrix T; computes a reciprocal condition number for the average

of the selected eigenvalues (RCONDE); and computes a reciprocal

condition number for the right and left deflating subspaces cor

responding to the selected eigenvalues (RCONDV). The leading

columns of VSL and VSR then form an orthonormal basis for the

corresponding left and right eigenspaces (deflating subspaces).

A generalized eigenvalue for a pair of matrices (A,B) is a

scalar w or a ratio alpha/beta = w, such that A - w*B is singu

lar. It is usually represented as the pair (alpha,beta), as

there is a reasonable interpretation for beta=0 or for both being

zero.

A pair of matrices (S,T) is in generalized complex Schur

form if T is upper triangular with non-negative diagonal and S is

upper triangular.

ARGUMENTS

JOBVSL (input) CHARACTER*1

= 'N': do not compute the left Schur vectors;
= 'V': compute the left Schur vectors.

JOBVSR (input) CHARACTER*1

= 'N': do not compute the right Schur vectors;
= 'V': compute the right Schur vectors.

SORT (input) CHARACTER*1

Specifies whether or not to order the eigenvalues

on the diagonal of the generalized Schur form. = 'N': Eigenval

ues are not ordered;
= 'S': Eigenvalues are ordered (see DELZTG).

DELZTG (input) LOGICAL FUNCTION of two COMPLEX*16 argu

ments

DELZTG must be declared EXTERNAL in the calling

subroutine. If SORT = 'N', DELZTG is not referenced. If SORT =

'S', DELZTG is used to select eigenvalues to sort to the top left

of the Schur form. Note that a selected complex eigenvalue may

no longer satisfy DELZTG(ALPHA(j),BETA(j)) = .TRUE. after order

ing, since ordering may change the value of complex eigenvalues

(especially if the eigenvalue is ill-conditioned), in this case

INFO is set to N+3 see INFO below).

SENSE (input) CHARACTER

Determines which reciprocal condition numbers are

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

ues only;
= 'V' : Computed for selected deflating subspaces

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

or 'B', SORT must equal 'S'.

N (input) INTEGER

The order of the matrices A, B, VSL, and VSR. N

>= 0.

A (input/output) COMPLEX*16 array, dimension (LDA,

N)

On entry, the first of the pair of matrices. On

exit, A has been overwritten by its generalized Schur form S.

LDA (input) INTEGER

The leading dimension of A. LDA >= max(1,N).

B (input/output) COMPLEX*16 array, dimension (LDB,

N)

On entry, the second of the pair of matrices. On

exit, B has been overwritten by its generalized Schur form T.

LDB (input) INTEGER

The leading dimension of B. LDB >= max(1,N).

SDIM (output) INTEGER

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

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

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

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

On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized

eigenvalues. ALPHA(j) and BETA(j),j=1,...,N are the diagonals

of the complex Schur form (S,T). BETA(j) will be non-negative

real.

Note: the quotients ALPHA(j)/BETA(j) may easily

over- or underflow, and BETA(j) may even be zero. Thus, the user

should avoid naively computing the ratio alpha/beta. However,

ALPHA will be always less than and usually comparable with

norm(A) in magnitude, and BETA always less than and usually com

parable with norm(B).

VSL (output) COMPLEX*16 array, dimension (LDVSL,N)

If JOBVSL = 'V', VSL will contain the left Schur

vectors. Not referenced if JOBVSL = 'N'.

LDVSL (input) INTEGER

The leading dimension of the matrix VSL. LDVSL

>=1, and if JOBVSL = 'V', LDVSL >= N.

VSR (output) COMPLEX*16 array, dimension (LDVSR,N)

If JOBVSR = 'V', VSR will contain the right Schur

vectors. Not referenced if JOBVSR = 'N'.

LDVSR (input) INTEGER

The leading dimension of the matrix VSR. LDVSR >=

1, and if JOBVSR = 'V', LDVSR >= N.

RCONDE (output) DOUBLE PRECISION array, dimension ( 2 )

If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2)

contain the reciprocal condition numbers for the average of the

selected eigenvalues. Not referenced if SENSE = 'N' or 'V'.

RCONDV (output) DOUBLE PRECISION array, dimension ( 2 )

If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2)

contain the reciprocal condition number for the selected deflat

ing subspaces. 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 >= 2*N.

If SENSE = 'E', 'V', or 'B', LWORK >= MAX(2*N, 2*SDIM*(N-SDIM)).

RWORK (workspace) DOUBLE PRECISION array, dimension (

8*N )

Real workspace.

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

WORK)

Not referenced if SENSE = 'N'. On exit, if INFO =

0, IWORK(1) returns the optimal LIWORK.

LIWORK (input) INTEGER

The dimension of the array WORK. LIWORK >= N+2.

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.
= 1,...,N: The QZ iteration failed. (A,B) are not

in Schur form, but ALPHA(j) and BETA(j) should be correct for

j=INFO+1,...,N. > N: =N+1: other than QZ iteration failed in

ZHGEQZ
=N+2: after reordering, roundoff changed values of

some complex eigenvalues so that leading eigenvalues in the Gen

eralized Schur form no longer satisfy DELZTG=.TRUE. This could

also be caused due to scaling. =N+3: reordering failed in ZT

GSEN.

LAPACK version 3.0 15 June 2000

ZGGESX(3)