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        ZGEBAL - balance a general complex matrix A


            CHARACTER      JOB
            INTEGER        IHI, ILO, INFO, LDA, N
            DOUBLE         PRECISION SCALE( * )
            COMPLEX*16     A( LDA, * )


        ZGEBAL  balances a general complex matrix A. This involves, first, per‐
        muting 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 sec‐
        ond, 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.


        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*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.  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.
        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


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