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# sggbal

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

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

CHARACTER      JOB

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

REAL           A( LDA, * ), B( LDB, * ), LSCALE( * ),
RSCALE( * ), WORK( * )

PURPOSE
SGGBAL balances a pair of general real 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 the matrices A and B.  N >= 0.

A       (input/output) REAL array, dimension (LDA,N)
On entry, the input matrix A.  On exit,  A is
overwritten by the balanced matrix.  If JOB = 'N', A
is not referenced.

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

B       (input/output) REAL array, dimension (LDB,N)
On entry, the input matrix B.  On exit,  B is
overwritten by the balanced matrix.  If JOB = 'N', B
is not referenced.

LDB     (input) INTEGER
The leading dimension of the array 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 about the permutations and scaling factors
applied to the left side of A and B.  If P(j) is the
index of the row and column interchanged with row
and column j, and D(j) is the scaling factor applied
to row and column 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 inter-
changes are made is N to IHI+1, then 1 to ILO-1.

RSCALE  (output) REAL array, dimension (N)
Details about the permutations and scaling factors
applied to the right side of A and B.  If P(j) is
the index of the row and column interchanged with
row and column j, and D(j) is the scaling factor
applied to row and column 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.

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.
```