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dgebal examples
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Man Page Details
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NAME
       DGEBAL - 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, permut-
       ing 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.


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.

