SGEBAL(l)		LAPACK routine (version	1.1)		    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 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 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	simi-
  larity 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.