CXML
        	    
LAPACK version 3.0
cggbal(3)

PURPOSE
  CGGBAL - balance a pair of general complex matrices (A,B)

SYNTAX
  SUBROUTINE CGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK,
		     INFO )

      CHARACTER	     JOB

      INTEGER	     IHI, ILO, INFO, LDA, LDB, N

      REAL	     LSCALE( * ), RSCALE( * ), WORK( * )

      COMPLEX	     A( LDA, * ), B( LDB, * )

DESCRIPTION
  CGGBAL balances a pair of general complex matrices (A,B). This involves,
  first, permuting A and B by similarity transformations 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 matrices, and improve the accuracy
  of the computed eigenvalues 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) 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.

  LDA	  (input) INTEGER
	  The leading dimension of the array A. LDA >= max(1,N).

  B	  (input/output) COMPLEX 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) REAL 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 interchanged 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) REAL 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 RSCALE(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 (6*N)

  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.