dggesx(3)

NAME

DGGESX - compute for a pair of N-by-N real nonsymmetric

matrices (A,B), the generalized eigenvalues, the real Schur form

(S,T), and,

SYNOPSIS

SUBROUTINE DGGESX( JOBVSL, JOBVSR, SORT, DELCTG, SENSE, N,
A,  LDA, B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, 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 A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ), RCONDE( 2 ), RCONDV( 2 ), VSL(
LDVSL, * ), VSR( LDVSR, * ), WORK( * )
    LOGICAL        DELCTG
    EXTERNAL       DELCTG

PURPOSE

DGGESX computes for a pair of N-by-N real nonsymmetric ma

trices (A,B), the generalized eigenvalues, the real Schur form

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

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

ization

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

Optionally, it also orders the eigenvalues so that a se

lected cluster of eigenvalues appears in the leading diagonal

blocks of the upper quasi-triangular matrix S and the upper tri

angular matrix T; computes a reciprocal condition number for the

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

ciprocal condition number for the right and left deflating sub

spaces corresponding 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 sub

spaces).

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 real Schur form

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

block upper triangular with 1-by-1 and 2-by-2 blocks. 1-by-1

blocks correspond to real generalized eigenvalues, while 2-by-2

blocks of S will be "standardized" by making the corresponding

elements of T have the form:

[ a 0 ]
[ 0 b ]

and the pair of corresponding 2-by-2 blocks in S and T

will have a complex conjugate pair of generalized eigenvalues.

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 three DOUBLE PRECISION

arguments

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. An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j)

is selected if DELZTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e.

if either one of a complex conjugate pair of eigenvalues is se

lected, then both complex eigenvalues are selected. Note that a

selected complex eigenvalue may no longer satisfy DELZTG(AL

PHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering, since order

ing may change the value of complex eigenvalues (especially if

the eigenvalue is ill-conditioned), in this case INFO is set to

N+3.

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) DOUBLE PRECISION 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) DOUBLE PRECISION 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.

(Complex conjugate pairs for which DELZTG is true for either

eigenvalue count as 2.)

ALPHAR (output) DOUBLE PRECISION array, dimension (N)

ALPHAI (output) DOUBLE PRECISION array, dimension

(N) BETA (output) DOUBLE PRECISION array, dimension (N) On ex

it, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the

generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i and BE

TA(j),j=1,...,N are the diagonals of the complex Schur form

(S,T) that would result if the 2-by-2 diagonal blocks of the real

Schur form of (A,B) were further reduced to triangular form using

2-by-2 complex unitary transformations. If ALPHAI(j) is zero,

then the j-th eigenvalue is real; if positive, then the j-th and

(j+1)-st eigenvalues are a complex conjugate pair, with AL

PHAI(j+1) negative.

Note: the quotients ALPHAR(j)/BETA(j) and AL

PHAI(j)/BETA(j) may easily over- or underflow, and BETA(j) may

even be zero. Thus, the user should avoid naively computing the

ratio. However, ALPHAR and ALPHAI will be always less than and

usually comparable with norm(A) in magnitude, and BETA always

less than and usually comparable with norm(B).

VSL (output) DOUBLE PRECISION array, dimension (LD

VSL,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) DOUBLE PRECISION array, dimension (LD

VSR,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 numbers for the selected deflat

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

WORK (workspace/output) DOUBLE PRECISION array, dimen

sion (LWORK)

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

LWORK.

LWORK (input) INTEGER

The dimension of the array WORK. LWORK >=

8*(N+1)+16. If SENSE = 'E', 'V', or 'B', LWORK >= MAX(

8*(N+1)+16, 2*SDIM*(N-SDIM) ).

IWORK (workspace) INTEGER array, dimension (LIWORK)

Not referenced if SENSE = 'N'.

LIWORK (input) INTEGER

The dimension of the array WORK. LIWORK >= N+6.

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 ALPHAR(j), ALPHAI(j), and BETA(j) should be

correct for j=INFO+1,...,N. > N: =N+1: other than QZ iteration

failed in DHGEQZ
=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 DT

GSEN.

Further details ===============

An approximate (asymptotic) bound on the average

absolute error of the selected eigenvalues is

EPS * norm((A, B)) / RCONDE( 1 ).

An approximate (asymptotic) bound on the maximum

angular error in the computed deflating subspaces is

EPS * norm((A, B)) / RCONDV( 2 ).

See LAPACK User's Guide, section 4.11 for more in

formation.

LAPACK version 3.0 15 June 2000

DGGESX(3)