sggbal(3)

NAME

SGGBAL - balance a pair of general real matrices (A,B)

SYNOPSIS

SUBROUTINE SGGBAL( JOB, N,  A,  LDA,  B,  LDB,  ILO,  IHI,
LSCALE, RSCALE, WORK, INFO )
    CHARACTER      JOB
    INTEGER        IHI, ILO, INFO, LDA, LDB, N
    REAL            A( LDA, * ), B( LDB, * ), LSCALE( * ),
RSCALE( * ), WORK( * )

PURPOSE

SGGBAL balances a pair of general real matrices (A,B).

This involves, first, permuting A and B by similarity transforma

tions to isolate eigenvalues in the first 1 to ILO$-$1 and last

IHI+1 to N elements on the diagonal; and second, applying a diag

onal 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 matrices, and im

prove the accuracy of the computed eigenvalues and/or eigenvec

tors in the generalized eigenvalue problem A*x = lambda*B*x.

ARGUMENTS

JOB (input) CHARACTER*1

Specifies the operations to be performed on A and

B:
= 'N': none: simply set ILO = 1, IHI = N,

LSCALE(I) = 1.0 and RSCALE(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 matrices A and B. 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.

LDA (input) INTEGER

The leading dimension of the array A. LDA >=

max(1,N).

B (input/output) REAL array, dimension (LDB,N)

On entry, the input matrix B. On exit, B is

overwritten by the balanced matrix. If JOB = 'N', B is not ref

erenced.

LDB (input) INTEGER

The leading dimension of the array B. LDB >=

max(1,N).

ILO (output) INTEGER

IHI (output) INTEGER ILO and IHI are set to

integers such that on exit A(i,j) = 0 and B(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.

LSCALE (output) REAL array, dimension (N)

Details of the permutations and scaling factors

applied to the left side of A and B. If P(j) is the index of the

row interchanged with row j, and D(j) is the scaling factor ap

plied to row j, then LSCALE(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.

RSCALE (output) REAL array, dimension (N)

Details of the permutations and scaling factors

applied to the right side of A and B. If P(j) is the index of

the column interchanged with column j, and D(j) is the scaling

factor applied to column j, then LSCALE(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.

WORK (workspace) REAL array, dimension (6*N)

INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille

gal value.

FURTHER DETAILS

See R.C. WARD, Balancing the generalized eigenvalue prob

lem,

SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

LAPACK version 3.0 15 June 2000

SGGBAL(3)