SGGBAL(3) MathKeisan LAPACK routine SGGBAL(3)
NAME
SGGBAL - 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 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 accu-
racy 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) 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 referenced.
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 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) 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 inter-
changed 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) 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 (lwork)
lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and at
least 1 when JOB = 'N' or 'P'.
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.
LAPACK routine (version 3.1) November 2006 SGGBAL(3)