Contents


NAME

     dgebal - balance a general real matrix A

SYNOPSIS

     SUBROUTINE DGEBAL(JOB, N, A, LDA, ILO, IHI, SCALE, INFO)

     CHARACTER * 1 JOB
     INTEGER N, LDA, ILO, IHI, INFO
     DOUBLE PRECISION A(LDA,*), SCALE(*)

     SUBROUTINE DGEBAL_64(JOB, N, A, LDA, ILO, IHI, SCALE, INFO)

     CHARACTER * 1 JOB
     INTEGER*8 N, LDA, ILO, IHI, INFO
     DOUBLE PRECISION A(LDA,*), SCALE(*)

  F95 INTERFACE
     SUBROUTINE GEBAL(JOB, [N], A, [LDA], ILO, IHI, SCALE, [INFO])

     CHARACTER(LEN=1) :: JOB
     INTEGER :: N, LDA, ILO, IHI, INFO
     REAL(8), DIMENSION(:) :: SCALE
     REAL(8), DIMENSION(:,:) :: A

     SUBROUTINE GEBAL_64(JOB, [N], A, [LDA], ILO, IHI, SCALE, [INFO])

     CHARACTER(LEN=1) :: JOB
     INTEGER(8) :: N, LDA, ILO, IHI, INFO
     REAL(8), DIMENSION(:) :: SCALE
     REAL(8), DIMENSION(:,:) :: A

  C INTERFACE
     #include <sunperf.h>

     void dgebal(char job, int n, double *a, int lda,  int  *ilo,
               int *ihi, double *scale, int *info);

     void dgebal_64(char job, long n, double *a, long  lda,  long
               *ilo, long *ihi, double *scale, long *info);

PURPOSE

     dgebal 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 ele-
     ments on the diagonal; and second, applying a diagonal simi-
     larity 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 eigenvec-
     tors.

ARGUMENTS

     JOB (input)
               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) The order of the matrix A.  N >= 0.

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

     ILO (output)
               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.

     IHI (output)
               See the description for ILO.

     SCALE (output)
               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)
               = 0:  successful exit.
               < 0:  if INFO = -i, the i-th argument had an ille-
               gal 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  eigen-
     values  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,  Univer-
     sity of
       California at Berkeley, USA