
CGEBAL(l) ) CGEBAL(l)
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 simi
larity transformation to isolate eigenvalues in the first 1 to ILO1 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 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,...,ILO1 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 j, then SCALE(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.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = i, the ith 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 1norms 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.
Modified by TzuYi Chen, Computer Science Division, University of
California at Berkeley, USA
LAPACK version 3.0 15 June 2000 CGEBAL(l) 
