RATQR(3F)

NAME
     RATQR, SRATQR   -  EISPACK routine.  This subroutine finds the
     algebraically smallest or largest eigenvalues of a SYMMETRIC TRIDIAGONAL
     matrix by the rational QR method with Newton corrections.


SYNOPSYS
          subroutine  ratqr(n,eps1,d,e,e2,m,w,ind,bd,type,idef,ierr)
          integer          n, m, ind(n), idef, ierr
          double precision eps1
          double precision d(n), e(n), e2(n), w(n), bd(n)
          logical          type

          subroutine sratqr(n,eps1,d,e,e2,m,w,ind,bd,type,idef,ierr)
          integer          n, m, ind(n), idef, ierr
          real             eps1
          real             d(n), e(n), e2(n), w(n), bd(n)
          logical          type



DESCRIPTION
     On Input

     N is the order of the matrix.

     EPS1 is a theoretical absolute error tolerance for the computed
     eigenvalues.  If the input EPS1 is non-positive, or indeed smaller than
     its default value, it is reset at each iteration to the respective
     default value, namely, the product of the relative machine precision and
     the magnitude of the current eigenvalue iterate.  The theoretical
     absolute error in the K-th eigenvalue is usually not greater than K times
     EPS1.

     D contains the diagonal elements of the input matrix.

     E contains the subdiagonal elements of the input matrix in its last N-1
     positions.  E(1) is arbitrary.

     E2 contains the squares of the corresponding elements of E. E2(1) is
     arbitrary.

     M is the number of eigenvalues to be found.

     IDEF should be set to 1 if the input matrix is known to be positive
     definite, to -1 if the input matrix is known to be negative definite, and
     to 0 otherwise.

     TYPE should be set to .TRUE. if the smallest eigenvalues are to be found,
     and to .FALSE. If the largest eigenvalues are to be found.  On Output

     EPS1 is unaltered unless it has been reset to its (last) default value.
     D and E are unaltered (unless W overwrites D). ELEMENTS of E2,
     corresponding to elements of E regarded as negligible, have been replaced
     by zero causing the matrix to split into a direct sum of submatrices.
     E2(1) is set to 0.0e0 if the smallest eigenvalues have been found, and to
     2.0e0 if the largest eigenvalues have been found.  E2 is otherwise
     unaltered (unless overwritten by BD).

     W contains the M algebraically smallest eigenvalues in ascending order,
     or the M largest eigenvalues in descending order.  If an error exit is
     made because of an incorrect specification of IDEF, no eigenvalues are
     found.  If the Newton iterates for a particular eigenvalue are not
     monotone, the best estimate obtained is returned and IERR is set.  W may
     coincide with D.

     IND contains in its first M positions the submatrix indices associated
     with the corresponding eigenvalues in W -- 1 for eigenvalues belonging to
     the first submatrix from the top, 2 for those belonging to the second
     submatrix, etc.

     BD contains refined bounds for the theoretical errors of the
     corresponding eigenvalues in W.  These bounds are usually within the
     tolerance specified by EPS1.  BD may coincide with E2.

     IERR is set to Zero       for normal return, 6*N+1      if  IDEF  is set
     to 1 and  type  to .TRUE.
        when the matrix is NOT positive definite, or
        if  IDEF  is set to -1 and  type  to .FALSE.
        when the matrix is NOT negative definite, 5*N+K      if successive
     iterates to the K-th eigenvalue
        are NOT monotone increasing, where K refers
        to the last such occurrence.  Note that subroutine TRIDIB is generally
     faster and more accurate than RATQR if the eigenvalues are clustered.
     Questions and comments should be directed to B. S. Garbow, APPLIED
     MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY