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# cgegv

```
NAME
CGEGV - a pair of N-by-N complex nonsymmetric matrices A, B

SYNOPSIS
SUBROUTINE CGEGV( 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

REAL          RWORK( * )

COMPLEX       A( LDA, * ), ALPHA( * ), B( LDB, * ),
BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
WORK( * )

PURPOSE
For a pair of N-by-N complex nonsymmetric matrices A, B:

compute the generalized eigenvalues (alpha, beta)
compute the left and/or right generalized eigenvectors
(VL and VR)

The second action is optional -- see the description of
JOBVL and JOBVR below.

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 interpre-
tation 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 general-
ized eigenvalue  w  for a pair of matrices (A,B) is a vector
r  such that  (A - w B) r = 0 .  A left generalized eigen-
vector is a vector l  such that  (A - w B)**H l = 0 .

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 eigenvectors.

N       (input) INTEGER
The number of rows and columns in the matrices A, B,
VL, and VR.  N >= 0.

A       (input/workspace) COMPLEX 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 con-
tents will have been destroyed.  (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/workspace) COMPLEX 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 con-
tents will have been destroyed.  (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 array, dimension (N)
BETA    (output) COMPLEX array, dimension (N) On
exit, ALPHA(j)/BETA(j), j=1,...,N, will be the gen-
eralized 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 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
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 array, dimension (LDVR,N)
If JOBVL = 'V', the right 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
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 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 gen-
erally be larger.  To compute the optimal value of
LWORK, call ILAENV to get blocksizes (for CGEQRF,
CUNMQR, and CUNGQR.)  Then compute: NB  -- MAX of
the blocksizes for CGEQRF, CUNMQR, and CUNGQR; The
optimal LWORK is  MAX( 2*N, N*(NB+1) ).

RWORK   (workspace/output) REAL 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:  errors
that usually indicate LAPACK problems:
=N+1: error return from CGGBAL
=N+2: error return from CGEQRF
=N+3: error return from CUNMQR
=N+4: error return from CUNGQR
=N+5: error return from CGGHRD
=N+6: error return from CHGEQZ (other than failed
iteration) =N+7: error return from CTGEVC
=N+8: error return from CGGBAK (computing VL)
=N+9: error return from CGGBAK (computing VR)
=N+10: error return from CLASCL (various calls)

FURTHER DETAILS
Balancing

---------

This driver calls CGGBAL 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 entries close to one
(except for the entries that start out zero.)

After the eigenvalues and eigenvectors of the balanced
matrices have been computed, CGGBAK transforms the eigenvec-
tors back to what they would have been (in perfect arith-
metic) if they had not been balanced.

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

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