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DGGBAL(l)			       )			     DGGBAL(l)

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

SYNOPSIS
       SUBROUTINE DGGBAL( JOB,	N,  A,	LDA, B, LDB, ILO, IHI, LSCALE, RSCALE,
			  WORK, INFO )

	   CHARACTER	  JOB

	   INTEGER	  IHI, ILO, INFO, LDA, LDB, N

	   DOUBLE	  PRECISION A( LDA, * ), B( LDB, *  ),	LSCALE(	 *  ),
			  RSCALE( * ), WORK( * )

PURPOSE
       DGGBAL  balances	 a pair of general real matrices (A,B). This involves,
       first, permuting A and B by similarity transformations to  isolate  ei‐
       genvalues 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	 accu‐
       racy 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) 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 inter‐
	       changed 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.

       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.

       WORK    (workspace) DOUBLE PRECISION 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			     DGGBAL(l)
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