.TH DGGBAL 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) " .SH NAME DGGBAL - a pair of general real matrices (A,B) .SH SYNOPSIS .TP 19 SUBROUTINE DGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO ) .TP 19 .ti +4 CHARACTER JOB .TP 19 .ti +4 INTEGER IHI, ILO, INFO, LDA, LDB, N .TP 19 .ti +4 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), LSCALE( * ), RSCALE( * ), WORK( * ) .SH PURPOSE DGGBAL 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. .br .SH ARGUMENTS .TP 8 JOB (input) CHARACTER*1 Specifies the operations to be performed on A and B: .br = \(aqN\(aq: none: simply set ILO = 1, IHI = N, LSCALE(I) = 1.0 and RSCALE(I) = 1.0 for i = 1,...,N. = \(aqP\(aq: permute only; .br = \(aqS\(aq: scale only; .br = \(aqB\(aq: both permute and scale. .TP 8 N (input) INTEGER The order of the matrices A and B. N >= 0. .TP 8 A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the input matrix A. On exit, A is overwritten by the balanced matrix. If JOB = \(aqN\(aq, A is not referenced. .TP 8 LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). .TP 8 B (input/output) DOUBLE PRECISION array, dimension (LDB,N) On entry, the input matrix B. On exit, B is overwritten by the balanced matrix. If JOB = \(aqN\(aq, B is not referenced. .TP 8 LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). .TP 8 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 = \(aqN\(aq or \(aqS\(aq, ILO = 1 and IHI = N. .TP 8 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. .TP 8 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 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. .TP 8 WORK (workspace) REAL array, dimension (lwork) lwork must be at least max(1,6*N) when JOB = \(aqS\(aq or \(aqB\(aq, and at least 1 when JOB = \(aqN\(aq or \(aqP\(aq. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value. .SH FURTHER DETAILS See R.C. WARD, Balancing the generalized eigenvalue problem, SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.