DGEBAL - balance a general real matrix A
SYNOPSIS
SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
CHARACTER JOB
INTEGER IHI, ILO, INFO, LDA, N
DOUBLE PRECISION A( LDA, * ), SCALE( * )
PURPOSE
DGEBAL balances a general real matrix A. This involves,
first, permuting A by a similarity transformation to iso
late eigenvalues in the first 1 to ILO-1 and last IHI+1 to
N elements on the diagonal; and second, applying a diago
nal similarity transformation to rows and columns ILO to
IHI to make the rows and columns as close in norm as pos
sible. Both steps are optional.
Balancing may reduce the 1-norm of the matrix, 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:
= 'N': none: simply set ILO = 1, IHI = N,
SCALE(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 matrix A. 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. See Further Details. LDA
(input) INTEGER The leading dimension of the array
A. LDA >= max(1,N).
ILO (output) INTEGER
IHI (output) INTEGER ILO and IHI are set to
integers such that on exit A(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.
SCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors
column interchanged with row and column j and D(j)
is the scaling factor applied to row and column j,
then SCALE(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.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an ille
gal value.
FURTHER DETAILS
The permutations consist of row and column interchanges
which put the matrix in the form
( T1 X Y )
P A P = ( 0 B Z )
( 0 0 T2 )
where T1 and T2 are upper triangular matrices whose eigen
values lie along the diagonal. The column indices ILO and
IHI mark the starting and ending columns of the submatrix
B. Balancing consists of applying a diagonal similarity
transformation inv(D) * B * D to make the 1-norms of each
row of B and its corresponding column nearly equal. The
output matrix is
( T1 X*D Y )
( 0 inv(D)*B*D inv(D)*Z ).
( 0 0 T2 )
Information about the permutations P and the diagonal
matrix D is returned in the vector SCALE.
This subroutine is based on the EISPACK routine BALANC.
Modified by Tzu-Yi Chen, Computer Science Division, Uni
versity of
California at Berkeley, USA
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