DGGBAL - balance a pair of general real matrices (A,B)
SYNOPSIS
SUBROUTINE DGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI,
LSCALE, RSCALE, WORK, INFO )
CHARACTER JOB
INTEGER IHI, ILO, INFO, LDA, LDB, N
DOUBLE PRECISION A( LDA, * ), B( LDB, * ),
LSCALE( * ), RSCALE( * ), WORK( * )
PURPOSE
DGGBAL 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 1-norm 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension
(LDB,N)
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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 col
umn 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) DOUBLE PRECISION array, dimension
(6*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille
gal value.
FURTHER DETAILS
See R.C. WARD, Balancing the generalized eigenvalue prob
lem,
SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.
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