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# cggbal

```
NAME
CGGBAL - balance a pair of general complex matrices (A,B)
for the generalized eigenvalue problem A*X = lambda*B*X

SYNOPSIS
SUBROUTINE CGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE,
RSCALE, WORK, INFO )

CHARACTER      JOB

INTEGER        IHI, ILO, INFO, LDA, LDB, N

REAL           LSCALE( * ), RSCALE( * ), WORK( * )

COMPLEX        A( LDA, * ), B( LDB, * )

PURPOSE
CGGBAL balances a pair of general complex matrices (A,B) for
the generalized eigenvalue problem A*X = lambda*B*X.  This
involves, first, permuting A and B by similarity transforma-
tions to isolate eigenvalues in the first 1 to ILO-1 and
last IHI+1 to N elements on the diagonal; and second, apply-
ing a diagonal similarity
transformation to rows and columns ILO to IHI to make the
rows and columns as close in norm as possible.  Both steps
are optional.

Balancing may reduce the 1-norm of the matrices, and improve
the accuracy of the computed eigenvalues and/or eigenvec-
tors.

ARGUMENTS
JOB     (input) CHARACTER*1
Specifies the operations to be performed on A and B:
= 'N':  none:  simply set ILO = 1, IHI = N,
LSCALE(I) = 1.0 and RSCALE(I) = 1.0 for i=1,...,N; =
'P':  permute only;
= 'S':  scale only;
= 'B':  both permute and scale.

N       (input) INTEGER
The order of matrices A and B.  N >= 0.

A       (input/output) COMPLEX array, dimension (LDA,N)
On entry, the input matrix A.  On exit, A is
overwritten by the balanced matrix.

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

B       (input/output) COMPLEX array, dimension (LDB,N)
On entry, the input matrix B.  On exit, B is
overwritten by the balanced matrix.

LDB     (input) INTEGER
The leading dimension of the matrix B. LDB >=
max(1,N).

ILO     (output) INTEGER
IHI     (output) INTEGER ILO and IHI are set to
integers such that on exit A(i,j) = 0 and B(i,j) = 0
if i > j and j = 1,...,ILO-1 or i = IHI+1,...,N.  If
JOB = 'N' or 'S', ILO = 1 and IHI = N.

LSCALE  (output) REAL array, dimension (N)
Details of the permutations and scaling factors
applied to the left side of A and B.  If P(j) is the
index of the row interchanged with row j, and D(j)
is the scaling factor applied to row j, then
LSCALE(j) = P(j)    for J = 1,...,ILO-1 = D(j)
for J = ILO,...,IHI = P(j)    for J = IHI+1,...,N.
The order in which the interchanges are made is N to
IHI+1, then 1 to ILO-1.

RSCALE  (output) REAL array, dimension (N)
Details of the permutations and scaling factors
applied to the right side of A and B.  If P(j) is
the index of the row interchanged with row j, and
D(j) is the scaling factor applied to row j, then
RSCALE(j) = P(j)    for J = 1,...,ILO-1 = D(j)
for J = ILO,...,IHI = P(j)    for J = IHI+1,...,N.
The order in which the interchanges are made is N to
IHI+1, then 1 to ILO-1.

WORK    (workspace) REAL array, dimension (6*N)

INFO    (output) INTEGER
= 0:  successful exit.
< 0:  if INFO = -i, the i-th argument had an illegal
value.

FURTHER DETAILS
See R.C. WARD, Balancing the generalized eigenvalue problem,
SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.
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