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RedHat 9 (Linux i386) - man page for dgebal (redhat section l)

DGEBAL(l)					)					DGEBAL(l)

NAME
       DGEBAL - balance a general real matrix A

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

	   CHARACTER	  JOB

	   INTEGER	  IHI, ILO, INFO, LDA, N

	   DOUBLE	  PRECISION A( LDA, * ), SCALE( * )

PURPOSE
       DGEBAL 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) 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.  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) DOUBLE PRECISION 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 				DGEBAL(l)


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