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

CGEBAL(l)					)					CGEBAL(l)

NAME
       CGEBAL - balance a general complex matrix A

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

	   CHARACTER	  JOB

	   INTEGER	  IHI, ILO, INFO, LDA, N

	   REAL 	  SCALE( * )

	   COMPLEX	  A( LDA, * )

PURPOSE
       CGEBAL  balances  a general complex matrix A. This involves, first, permuting A by a simi-
       larity 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) COMPLEX 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 CBAL.

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

LAPACK version 3.0			   15 June 2000 				CGEBAL(l)


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