sgebal(3)

NAME

SGEBAL - balance a general real matrix A

SYNOPSIS

SUBROUTINE SGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
    CHARACTER      JOB
    INTEGER        IHI, ILO, INFO, LDA, N
    REAL           A( LDA, * ), SCALE( * )

PURPOSE

SGEBAL balances a general real 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 option

al.

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) REAL 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 ref

erenced. See Further Details. LDA (input) INTEGER The lead

ing 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 inter

changed with row and column j and D(j) is the scaling factor ap

plied 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. Balanc

ing consists of applying a diagonal similarity transformation

inv(D) * B * D to make the 1-norms of each row of B and its cor

responding 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 ma

trix 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

LAPACK version 3.0 15 June 2000

SGEBAL(3)