?ggbal

Syntax

FORTRAN 77:

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)

Fortran 95:

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

C:

lapack_int LAPACKE_sggbal( int matrix_order, char job, lapack_int n, float* a, lapack_int lda, float* b, lapack_int ldb, lapack_int* ilo, lapack_int* ihi, float* lscale, float* rscale );

lapack_int LAPACKE_dggbal( int matrix_order, char job, lapack_int n, double* a, lapack_int lda, double* b, lapack_int ldb, lapack_int* ilo, lapack_int* ihi, double* lscale, double* rscale );

lapack_int LAPACKE_cggbal( int matrix_order, char job, lapack_int n, lapack_complex_float* a, lapack_int lda, lapack_complex_float* b, lapack_int ldb, lapack_int* ilo, lapack_int* ihi, float* lscale, float* rscale );

lapack_int LAPACKE_zggbal( int matrix_order, char job, lapack_int n, lapack_complex_double* a, lapack_int lda, lapack_complex_double* b, lapack_int ldb, lapack_int* ilo, lapack_int* ihi, double* lscale, double* rscale );

Include Files

The FORTRAN 77 interfaces are specified in the mkl_lapack.fi and mkl_lapack.h include files, the Fortran 95 interfaces are specified in the lapack.f90 include file, and the C interfaces are specified in the mkl_lapacke.h include file.

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

The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.

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, DIMENSION 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, DIMENSION 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.

Fortran 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 Fortran 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'.