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# cgebal

```
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 ele-
ments on the diagonal; and second, applying a diagonal simi-
larity 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 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) 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 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 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 CBAL.
```