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SGGBAL(l)					)					SGGBAL(l)

NAME
       SGGBAL - balance a pair of general real matrices (A,B)

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). 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.

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 scal-
	       ing  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  inter-
	       changes 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 (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.

LAPACK version 3.0			   15 June 2000 				SGGBAL(l)
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