zgegs(3)

NAME

ZGEGS - routine is deprecated and has been replaced by

routine ZGGES

SYNOPSIS

SUBROUTINE  ZGEGS(  JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, INFO )
    CHARACTER     JOBVSL, JOBVSR
    INTEGER       INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
    DOUBLE        PRECISION RWORK( * )
    COMPLEX*16    A( LDA, * ), ALPHA( * ), B(  LDB,  *  ),
BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), WORK( * )

PURPOSE

This routine is deprecated and has been replaced by rou

tine ZGGES. ZGEGS computes for a pair of N-by-N complex nonsym

metric matrices A, B: the generalized eigenvalues (alpha, beta),

the complex 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 ZGEGV 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 unitary

matrix and both on the right by another unitary matrix, these two

unitary matrices being chosen so as to bring the pair of matrices

into upper triangular form with the diagonal elements of B being

non-negative real numbers (this is also called complex Schur

form.)

The left and right Schur vectors are the columns of VSL

and VSR, respectively, where VSL and VSR are the unitary matrices
which reduce A and B to Schur form:

Schur form of (A,B) = ( (VSL)**H A (VSR), (VSL)**H 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) COMPLEX*16 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.

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 whose

generalized eigenvalues and (optionally) Schur vectors are to be

computed. On exit, the generalized Schur form of B.

LDB (input) INTEGER

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

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), j=1,...,N and BETA(j), j=1,...,N are

the diagonals of the complex Schur form (A,B) output by ZGEGS.

The 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. (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) COMPLEX*16 array, dimension (LDVSR,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) 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). For good performance, LWORK must generally be larg

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

blocksizes (for ZGEQRF, ZUNMQR, and CUNGQR.) Then compute: NB

-- MAX of the blocksizes for ZGEQRF, ZUNMQR, and CUNGQR; the op

timal LWORK is 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.

RWORK (workspace) DOUBLE PRECISION array, dimension

(3*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: errors that usually indicate LAPACK prob

lems:
=N+1: error return from ZGGBAL
=N+2: error return from ZGEQRF
=N+3: error return from ZUNMQR
=N+4: error return from ZUNGQR
=N+5: error return from ZGGHRD
=N+6: error return from ZHGEQZ (other than failed

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

LAPACK version 3.0 15 June 2000

ZGEGS(3)