SGGBAL
November 2006
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$-$1 and last IHI+1 to N
elements on the diagonal; and second, applying a diagonal similarity
transformation to rows and columns ILO to IHI to make the rows
and columns as close in norm as possible. Both steps are optional.
Balancing may reduce the 1-norm of the matrices, and improve the
accuracy of the computed eigenvalues and/or eigenvectors in the
generalized eigenvalue problem A*x = lambda*B*x.
involves, first, permuting A and B by similarity transformations to
isolate eigenvalues in the first 1 to ILO$-$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) 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 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) 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.
SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.