SUBROUTINE TQLRAT(N,D,E2,IERR)
C***BEGIN PROLOGUE  TQLRAT
C***DATE WRITTEN   760101   (YYMMDD)
C***REVISION DATE  830518   (YYMMDD)
C***CATEGORY NO.  D4A5,D4C2A
C***KEYWORDS  EIGENVALUES,EIGENVECTORS,EISPACK
C***AUTHOR  SMITH, B. T., ET AL.
C***PURPOSE  Computes eigenvalues of symmetric tridiagonal matrix
C            a rational variant of the QL method.
C***DESCRIPTION
C
C     This subroutine is a translation of the ALGOL procedure TQLRAT,
C     ALGORITHM 464, COMM. ACM 16, 689(1973) by Reinsch.
C
C     This subroutine finds the eigenvalues of a SYMMETRIC
C     TRIDIAGONAL matrix by the rational QL method.
C
C     On Input
C
C        N is the order of the matrix.
C
C        D contains the diagonal elements of the input matrix.
C
C        E2 contains the squares of the subdiagonal elements of the
C          input matrix in its last N-1 positions.  E2(1) is arbitrary.
C
C      On Output
C
C        D contains the eigenvalues in ascending order.  If an
C          error exit is made, the eigenvalues are correct and
C          ordered for indices 1,2,...IERR-1, but may not be
C          the smallest eigenvalues.
C
C        E2 has been destroyed.
C
C        IERR is set to
C          Zero       for normal return,
C          J          if the J-th eigenvalue has not been
C                     determined after 30 iterations.
C
C     Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
C
C     Questions and comments should be directed to B. S. Garbow,
C     APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
C     ------------------------------------------------------------------
C***REFERENCES  B. T. SMITH, J. M. BOYLE, J. J. DONGARRA, B. S. GARBOW,
C                 Y. IKEBE, V. C. KLEMA, C. B. MOLER, *MATRIX EIGEN-
C                 SYSTEM ROUTINES - EISPACK GUIDE*, SPRINGER-VERLAG,
C                 1976.
C***ROUTINES CALLED  PYTHAG
C***END PROLOGUE  TQLRAT