?ggbal

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?ggbal

Balances a pair of general real or complex matrices.

Syntax

call sggbal(job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)

call dggbal(job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)

call cggbal(job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)

call zggbal(job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)

call ggbal(a, b [,ilo] [,ihi] [,lscale] [,rscale] [,job] [,info])

Include Files

mkl.fi, lapack.f90

Description

The routine balances a pair of general real/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 = λ*B*x.

Input Parameters

job

CHARACTER*1. Specifies the operations to be performed on A and B. Must be 'N' or 'P' or 'S' or 'B'.

If job = 'N ', then no operations are done; simply set ilo =1, ihi=n, lscale(i) =1.0 and rscale(i)=1.0 for

i = 1,..., n.

If job = 'P', then permute only.

If job = 'S', then scale only.

If job = 'B', then both permute and scale.

n

INTEGER. The order of the matrices A and B (n≥ 0).

a, b

REAL for sggbal

DOUBLE PRECISION for dggbal

COMPLEX for cggbal

DOUBLE COMPLEX for zggbal.

Arrays:

a(lda,*) contains the matrix A. The second dimension of a must be at least max(1, n).

b(ldb,*) contains the matrix B. The second dimension of b must be at least max(1, n).

If job = 'N', a and b are not referenced.

lda

INTEGER. The leading dimension of a; at least max(1, n).

ldb

INTEGER. The leading dimension of b; at least max(1, n).

work

REAL for single precision flavors

DOUBLE PRECISION for double precision flavors.

Workspace array, size at least max(1, 6n) when job = 'S'or 'B', or at least 1 when job = 'N'or 'P'.

Output Parameters

a, b

Overwritten by the balanced matrices A and B, respectively.

ilo, ihi

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', then ilo = 1 and ihi = n.

lscale, rscale

REAL for single precision flavors

DOUBLE PRECISION for double precision flavors.

Arrays, size at least max(1, n).

lscale contains 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.

rscale contains 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.

info

INTEGER.

If info = 0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine ggbal interface are the following:

a: Holds the matrix A of size (n,n).
b: Holds the matrix B of size (n,n).
lscale: Holds the vector of length (n).
rscale: Holds the vector of length (n).
ilo: Default value for this argument is ilo = 1.
ihi: Default value for this argument is ihi = n.
job: Must be 'B', 'S', 'P', or 'N'. The default value is 'B'.

Parent topic: Generalized Nonsymmetric Eigenvalue Problems: LAPACK Computational Routines

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?ggbal