dgegs(3)

NAME

DGEGS - routine is deprecated and has been replaced by

routine DGGES

SYNOPSIS

SUBROUTINE  DGEGS(  JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, INFO )
    CHARACTER     JOBVSL, JOBVSR
    INTEGER       INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
    DOUBLE        PRECISION A( LDA, * ), ALPHAI( * ),  ALPHAR( * ), B( LDB, * ), BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, *
), WORK( * )

PURPOSE

This routine is deprecated and has been replaced by rou

tine DGGES. DGEGS computes for a pair of N-by-N real nonsymmet

ric matrices A, B: the generalized eigenvalues (alphar +/- al

phai*i, beta), the real Schur form (A, B), and optionally left

and/or right Schur vectors (VSL and VSR).

(If only the generalized eigenvalues are needed, use the

driver DGEGV instead.)

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

roughly speaking, a scalar w or a ratio alpha/beta = w, such

that A - w*B is singular. It is usually represented as the pair

(alpha,beta), as there is a reasonable interpretation for beta=0,

and even for both being zero. A good beginning reference is the

book, "Matrix Computations", by G. Golub & C. van Loan (Johns

Hopkins U. Press)

The (generalized) Schur form of a pair of matrices is the

result of multiplying both matrices on the left by one orthogonal

matrix and both on the right by another orthogonal matrix, these

two orthogonal matrices being chosen so as to bring the pair of

matrices into (real) Schur form.

A pair of matrices A, B is in generalized real Schur form

if B is upper triangular with non-negative diagonal and A 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 A will be "standardized" by making the corresponding

elements of B have the form:

[ a 0 ]
[ 0 b ]

and the pair of corresponding 2-by-2 blocks in A and B

will have a complex conjugate pair of generalized eigenvalues.

The left and right Schur vectors are the columns of VSL

and VSR, respectively, where VSL and VSR are the orthogonal ma

trices which reduce A and B to Schur form:

Schur form of (A,B) = ( (VSL)**T A (VSR), (VSL)**T B (VSR)

)

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.

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 whose

generalized eigenvalues and (optionally) Schur vectors are to be

computed. On exit, the generalized Schur form of A. Note: to

avoid overflow, the Frobenius norm of the matrix A should be less

than the overflow threshold.

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 whose

generalized eigenvalues and (optionally) Schur vectors are to be

computed. On exit, the generalized Schur form of B. Note: to

avoid overflow, the Frobenius norm of the matrix B should be less

than the overflow threshold.

LDB (input) INTEGER

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

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, j=1,...,N and

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

(A,B) 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 alpha/beta. 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. (See "Purpose", above.) 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. (See "Purpose", above.) 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 >=

max(1,4*N). For good performance, LWORK must generally be larg

er. To compute the optimal value of LWORK, call ILAENV to get

blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute: NB

-- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR The opti

mal LWORK is 2*N + N*(NB+1).

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.

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: errors that usually indicate

LAPACK problems:
=N+1: error return from DGGBAL
=N+2: error return from DGEQRF
=N+3: error return from DORMQR
=N+4: error return from DORGQR
=N+5: error return from DGGHRD
=N+6: error return from DHGEQZ (other than failed

iteration) =N+7: error return from DGGBAK (computing VSL)
=N+8: error return from DGGBAK (computing VSR)
=N+9: error return from DLASCL (various places)

LAPACK version 3.0 15 June 2000

DGEGS(3)