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DGGBAL(1)		 LAPACK routine (version 3.2)		     DGGBAL(1)

NAME
       DGGBAL - balances 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 accuracy of the computed eigen‐
       values 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) REAL array, dimension (lwork)
	       lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and at
	       least 1 when JOB = 'N' or 'P'.

       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 routine (version 3.2)	 November 2008			     DGGBAL(1)
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