zgegv(3)

NAME

ZGEGV - routine is deprecated and has been replaced by

routine ZGGEV

SYNOPSIS

SUBROUTINE  ZGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA,
BETA, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
    CHARACTER     JOBVL, JOBVR
    INTEGER       INFO, LDA, LDB, LDVL, LDVR, LWORK, N
    DOUBLE        PRECISION RWORK( * )
    COMPLEX*16    A( LDA, * ), ALPHA( * ), B(  LDB,  *  ),
BETA( * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )

PURPOSE

This routine is deprecated and has been replaced by rou

tine ZGGEV. ZGEGV computes for a pair of N-by-N complex nonsym

metric matrices A and B, the generalized eigenvalues (alpha, be

ta), and optionally, the left and/or right generalized eigenvec

tors (VL and VR).

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)

A right generalized eigenvector corresponding to a gener

alized eigenvalue w for a pair of matrices (A,B) is a vector r

such that (A - w B) r = 0 . A left generalized eigenvector is a

vector l such that l**H * (A - w B) = 0, where l**H is the
conjugate-transpose of l.

Note: this routine performs "full balancing" on A and B -

see "Further Details", below.

ARGUMENTS

JOBVL (input) CHARACTER*1

= 'N': do not compute the left generalized eigen

vectors;
= 'V': compute the left generalized eigenvectors.

JOBVR (input) CHARACTER*1

= 'N': do not compute the right generalized

eigenvectors;
= 'V': compute the right generalized eigenvec

tors.

N (input) INTEGER

The order of the matrices A, B, VL, and VR. 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) generalized eigenvectors

are to be computed. On exit, the contents will have been de

stroyed. (For a description of the contents of A on exit, see

"Further Details", below.)

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) generalized eigenvectors

are to be computed. On exit, the contents will have been de

stroyed. (For a description of the contents of B on exit, see

"Further Details", below.)

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.

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).

VL (output) COMPLEX*16 array, dimension (LDVL,N)

If JOBVL = 'V', the left generalized eigenvectors.

(See "Purpose", above.) Each eigenvector will be scaled so the

largest component will have abs(real part) + abs(imag. part) = 1,

*except* that for eigenvalues with alpha=beta=0, a zero vector

will be returned as the corresponding eigenvector. Not refer

enced if JOBVL = 'N'.

LDVL (input) INTEGER

The leading dimension of the matrix VL. LDVL >= 1,

and if JOBVL = 'V', LDVL >= N.

VR (output) COMPLEX*16 array, dimension (LDVR,N)

If JOBVR = 'V', the right generalized eigenvec

tors. (See "Purpose", above.) Each eigenvector will be scaled

so the largest component will have abs(real part) + abs(imag.

part) = 1, *except* that for eigenvalues with alpha=beta=0, a ze

ro vector will be returned as the corresponding eigenvector. Not

referenced if JOBVR = 'N'.

LDVR (input) INTEGER

The leading dimension of the matrix VR. LDVR >= 1,

and if JOBVR = 'V', LDVR >= 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 MAX( 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.

RWORK (workspace/output) DOUBLE PRECISION array, dimen

sion (8*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. No eigenvec

tors have been calculated, but ALPHA(j) and BETA(j) should be

correct for j=INFO+1,...,N. > N: errors that usually indicate

LAPACK problems:
=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 ZTGEVC
=N+8: error return from ZGGBAK (computing VL)
=N+9: error return from ZGGBAK (computing VR)
=N+10: error return from ZLASCL (various calls)

FURTHER DETAILS

Balancing
--------

This driver calls ZGGBAL to both permute and scale rows

and columns of A and B. The permutations PL and PR are chosen so

that PL*A*PR and PL*B*R will be upper triangular except for the

diagonal blocks A(i:j,i:j) and B(i:j,i:j), with i and j as close

together as possible. The diagonal scaling matrices DL and DR

are chosen so that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have

elements close to one (except for the elements that start out ze

ro.)

After the eigenvalues and eigenvectors of the balanced ma

trices have been computed, ZGGBAK transforms the eigenvectors

back to what they would have been (in perfect arithmetic) if they

had not been balanced.

Contents of A and B on Exit
-------- -- - --- - -- ---

If any eigenvectors are computed (either JOBVL='V' or JOB

VR='V' or both), then on exit the arrays A and B will contain the

complex Schur form[*] of the "balanced" versions of A and B. If

no eigenvectors are computed, then only the diagonal blocks will

be correct.

[*] In other words, upper triangular form.

LAPACK version 3.0 15 June 2000

ZGEGV(3)