dgges(3)

NAME

DGGES - compute for a pair of N-by-N real nonsymmetric ma

trices (A,B),

SYNOPSIS

SUBROUTINE DGGES( JOBVSL, JOBVSR, SORT, DELCTG, N, A, LDA,
B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
LWORK, BWORK, INFO )
    CHARACTER     JOBVSL, JOBVSR, SORT
    INTEGER        INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N,
SDIM
    LOGICAL       BWORK( * )
    DOUBLE        PRECISION A( LDA, * ), ALPHAI( * ),  ALPHAR( * ), B( LDB, * ), BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, *
), WORK( * )
    LOGICAL       DELCTG
    EXTERNAL      DELCTG

PURPOSE

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

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

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

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

factorization

(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.The leading columns of VSL and VSR then form an

orthonormal basis for the corresponding left and right

eigenspaces (deflating subspaces).

(If only the generalized eigenvalues are needed, use the

driver DGGEV instead, which is faster.)

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 both being ze

ro.

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 in the ill-conditioned case, a selected

complex eigenvalue may no longer satisfy DELZTG(ALPHAR(j),AL

PHAI(j), BETA(j)) = .TRUE. after ordering. INFO is to be set to

N+2 in this case.

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.

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+16.

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.
= 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.

LAPACK version 3.0 15 June 2000

DGGES(3)