DGGBAL(3)      LAPACK routine of NEC Numeric Library Collection      DGGBAL(3)



NAME
       DGGBAL

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



PURPOSE
            DGGBAL balances a pair of general real 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)
                     JOB is 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)
                     N is INTEGER
                     The order of the matrices A and B.  N >= 0.

           A         (input/output)
                     A is 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.

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

           B         (input/output)
                     B is DOUBLE PRECISION 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)
                     LDB is INTEGER
                     The leading dimension of the array B. LDB >= max(1,N).

           ILO       (output)
                     ILO is INTEGER

           IHI       (output)
                     IHI is 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)
                     LSCALE is DOUBLE PRECISION 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)
                     RSCALE is DOUBLE PRECISION 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
                       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.

           WORK      (output)
                     WORK is DOUBLE PRECISION array, dimension (lwork)
                     lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and
                     at least 1 when JOB = 'N' or 'P'.

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



LAPACK routine                  31 October 2017                      DGGBAL(3)