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NAME SGGBAL - balance a pair of general real matrices (A,B) for the generalized eigenvalue problem A*X = lambda*B*X 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) for the generalized eigenvalue problem A*X = lambda*B*X. This involves, first, permuting A and B by similarity transforma- tions to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and second, apply- ing 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 eigenvec- tors. 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 about the permutations and scaling factors applied to the left side of A and B. If P(j) is the index of the row and column interchanged with row and column j, and D(j) is the scaling factor applied to row and 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 inter- changes are made is N to IHI+1, then 1 to ILO-1. RSCALE (output) REAL array, dimension (N) Details about the permutations and scaling factors applied to the right side of A and B. If P(j) is the index of the row and column interchanged with row and column j, and D(j) is the scaling factor applied to row and 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 (6*N) 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.