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zgebal examples
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Man Page Details
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 Online html documentation available at 
            http://www.netlib.org/lapack/explore-html/ 

 Definition:
 ===========

       SUBROUTINE ZGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
 
       .. Scalar Arguments ..
       CHARACTER          JOB
       INTEGER            IHI, ILO, INFO, LDA, N
       ..
       .. Array Arguments ..
       DOUBLE PRECISION   SCALE( * )
       COMPLEX*16         A( LDA, * )
       ..
  

 Purpose:
 =============

  ZGEBAL balances a general complex 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:
 ==========

   [in] JOB

          JOB is 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.

   [in] N

          N is INTEGER
          The order of the matrix A.  N >= 0.

   [in,out] A

          A is COMPLEX*16 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.

   [in] LDA

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

   [out] ILO

   [out] IHI

          ILO and IHI are set to INTEGER 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.

   [out] SCALE

          SCALE is 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.

   [out] INFO

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

 Authors:
 ========

   Univ. of Tennessee 
   Univ. of California Berkeley 
   Univ. of Colorado Denver 
   NAG Ltd. 

 November 2011


 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

