CGEBAL(3F)CGEBAL(3F)NAME
CGEBAL - balance a general complex matrix A
SYNOPSIS
SUBROUTINE CGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
CHARACTER JOB
INTEGER IHI, ILO, INFO, LDA, N
REAL SCALE( * )
COMPLEX A( LDA, * )
PURPOSE
CGEBAL balances a general complex matrix A. This involves, first,
permuting A by a similarity transformation 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 matrix, and improve the accuracy
of the computed eigenvalues and/or eigenvectors.
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) 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. 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) REAL array, dimension (N)
Details of the permutations and scaling factors applied to A. 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
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CGEBAL(3F)CGEBAL(3F)
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 illegal 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 eigenvalues 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 CBAL.
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