CGGBAL(l) ) CGGBAL(l)NAME
CGGBAL - balance a pair of general complex matrices (A,B)
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). 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 ele‐
ments on the diagonal; and second, applying a diagonal similarity
transformation to rows and columns ILO to IHI to make the rows and col‐
umns as close in norm as possible. Both steps are optional.
Balancing may reduce the 1-norm of the matrices, and improve the accu‐
racy 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) COMPLEX 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) COMPLEX 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 of the permutations and scaling factors applied to the
left side of A and B. If P(j) is the index of the row inter‐
changed 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 column
interchanged with column j, and D(j) is the scaling factor
applied to column 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.
LAPACK version 3.0 15 June 2000 CGGBAL(l)
