ZGGBAL(l) — LAPACK routine (version 2.0)

NAME

ZGGBAL - balance a pair of general complex matrices (A,B)

SYNOPSIS

SUBROUTINE ZGGBAL(
JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO )

CHARACTER JOB

INTEGER IHI, ILO, INFO, LDA, LDB, N

DOUBLE PRECISION LSCALE( ∗ ), RSCALE( ∗ ), WORK( ∗ )

COMPLEX∗16 A( LDA, ∗ ), B( LDB, ∗ )

PURPOSE

ZGGBAL balances a pair of general complex matrices (A,B). This involves, first, permuting A and B by similarity transformations 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 matrices, and improve the accuracy of the computed eigenvalues and/or eigenvectors 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) COMPLEX∗16 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.

LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).

B (input/output) COMPLEX∗16 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 referenced.

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) DOUBLE PRECISION 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 applied 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) DOUBLE PRECISION 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 RSCALE(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) DOUBLE PRECISION array, dimension (6∗N)

INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.

FURTHER DETAILS

See R.C. WARD, Balancing the generalized eigenvalue problem,
SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.