
SGGBAL(l) ) SGGBAL(l)
NAME
SGGBAL  balance a pair of general real matrices (A,B)
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). This involves, first, permuting A
and B by similarity transformations to isolate eigenvalues in the first 1 to ILO$$1 and
last IHI+1 to N elements on the diagonal; and second, applying 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 1norm of the matrices, and improve the accuracy of the computed
eigenvalues and/or eigenvectors in the generalized eigenvalue problem A*x = lambda*B*x.
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,...,ILO1 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 scal
ing factor applied to row j, then LSCALE(j) = P(j) for J = 1,...,ILO1 = 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 ILO1.
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 column interchanged with column j, and D(j) is the
scaling factor applied to column j, then LSCALE(j) = P(j) for J = 1,...,ILO1 =
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 ILO1.
WORK (workspace) REAL array, dimension (6*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value.
FURTHER DETAILS
See R.C. WARD, Balancing the generalized eigenvalue problem,
SIAM J. Sci. Stat. Comp. 2 (1981), 141152.
LAPACK version 3.0 15 June 2000 SGGBAL(l) 
