SGEBAL(l)								 )								 SGEBAL(l)

NAME

       SGEBAL - balance a general real matrix A

SYNOPSIS

       SUBROUTINE SGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )

	   CHARACTER	  JOB

	   INTEGER	  IHI, ILO, INFO, LDA, N

	   REAL 	  A( LDA, * ), SCALE( * )

PURPOSE

       SGEBAL  balances  a  general  real matrix A. This involves, first, permuting A by a similarity transformation 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  col-
       umns 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 matrix, and improve the accuracy of the computed eigenvalues and/or eigenvectors.

ARGUMENTS

       JOB     (input) CHARACTER*1
	       Specifies the operations to be performed on A:
	       = 'N':  none:  simply set ILO = 1, IHI = N, SCALE(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 matrix A.  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.  See Further
	       Details.  LDA	 (input) INTEGER The leading dimension of the array A.	LDA >= max(1,N).

       ILO     (output) INTEGER
	       IHI     (output) INTEGER ILO and IHI are set to integers such that on exit A(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.

       SCALE   (output) REAL array, dimension (N)
	       Details of the permutations and scaling factors applied to A.  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 SCALE(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.

       INFO    (output) INTEGER
	       = 0:  successful exit.
	       < 0:  if INFO = -i, the i-th argument had an illegal value.

FURTHER DETAILS

       The permutations consist of row and column interchanges which put the matrix in the form

		  ( T1	 X   Y	)
	  P A P = (  0	 B   Z	)
		  (  0	 0   T2 )

       where  T1  and T2 are upper triangular matrices whose eigenvalues lie along the diagonal.  The column indices ILO and IHI mark the starting
       and ending columns of the submatrix B. Balancing consists of applying a diagonal similarity transformation inv(D) *  B  *  D  to  make  the
       1-norms of each row of B and its corresponding column nearly equal.  The output matrix is

	  ( T1	   X*D		Y    )
	  (  0	inv(D)*B*D  inv(D)*Z ).
	  (  0	    0		T2   )

       Information about the permutations P and the diagonal matrix D is returned in the vector SCALE.

       This subroutine is based on the EISPACK routine BALANC.

       Modified by Tzu-Yi Chen, Computer Science Division, University of
	 California at Berkeley, USA

LAPACK version 3.0						   15 June 2000 							 SGEBAL(l)