NAME

       sgebal.f -

SYNOPSIS

   Functions/Subroutines
       subroutine sgebal (JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
           SGEBAL

Function/Subroutine Documentation

   subroutine sgebal (characterJOB, integerN, real, dimension( lda, * )A, integerLDA, integerILO,
       integerIHI, real, dimension( * )SCALE, integerINFO)
       SGEBAL

       Purpose:

            SGEBAL balances a general real 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.

       Parameters:
           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.

           N

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

           A

                     A is REAL 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

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

           ILO

                     ILO is INTEGER

           IHI

                     IHI is 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

                     SCALE is 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

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

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2013

       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.

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

       Definition at line 161 of file sgebal.f.

Author

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