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# LAPACK-3

## man page of LAPACK-3

### LAPACK-3: computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, and optionally, the

```NAME
LAPACK-3  - computes for a pair of N-by-N complex nonsymmetric matrices
(A,B), the generalized eigenvalues, and  optionally,  the  left  and/or
right generalized eigenvectors

SYNOPSIS
SUBROUTINE ZGGEV( 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
ZGGEV  computes  for  a  pair  of  N-by-N complex nonsymmetric matrices
(A,B), the generalized eigenvalues, and  optionally,  the  left  and/or
right generalized eigenvectors.
A generalized eigenvalue for a pair of matrices (A,B) is a scalar
lambda or a ratio alpha/beta = lambda, such that A - lambda*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.
The right generalized eigenvector v(j) corresponding to the
generalized eigenvalue lambda(j) of (A,B) satisfies
A * v(j) = lambda(j) * B * v(j).
The left generalized eigenvector u(j) corresponding to the
generalized eigenvalues lambda(j) of (A,B) satisfies
u(j)**H * A = lambda(j) * u(j)**H * B
where u(j)**H is the conjugate-transpose of u(j).

ARGUMENTS
JOBVL   (input) CHARACTER*1
= 'N':  do not compute the left generalized eigenvectors;
= 'V':  compute the left generalized eigenvectors.

JOBVR   (input) CHARACTER*1
= 'N':  do not compute the right generalized eigenvectors;
= 'V':  compute the right generalized eigenvectors.

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 matrix A in the pair (A,B).
On exit, A has been overwritten.

LDA     (input) INTEGER
The leading dimension of A.  LDA >= max(1,N).

B       (input/output) COMPLEX*16 array, dimension (LDB, N)
On entry, the matrix B in the pair (A,B).
On exit, B has been overwritten.

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 comparable with norm(B).

VL      (output) COMPLEX*16 array, dimension (LDVL,N)
If JOBVL = 'V', the left generalized eigenvectors u(j) are
stored one after another in the columns of VL, in the same
order as their eigenvalues.
Each eigenvector is scaled so the largest component has
abs(real part) + abs(imag. part) = 1.
Not referenced 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 eigenvectors v(j) are
stored one after another in the columns of VR, in the same
order as their eigenvalues.
Each eigenvector is scaled so the largest component has
abs(real part) + abs(imag. part) = 1.
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 (MAX(1,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 larger.
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, dimension (8*N)

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value.
=1,...,N:
The QZ iteration failed.  No eigenvectors have been
calculated, but ALPHA(j) and BETA(j) should be
correct for j=INFO+1,...,N.
> N:  =N+1: other then QZ iteration failed in DHGEQZ,
=N+2: error return from DTGEVC.

LAPACK driver routine (version 3.March 2011                       LAPACK-3(3)
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