# tqlrat.f

          SUBROUTINE TQLRAT (N, D, E2, IERR)
    C***BEGIN PROLOGUE  TQLRAT
    C***PURPOSE  Compute the eigenvalues of symmetric tridiagonal matrix
    C            using a rational variant of the QL method.
    C***LIBRARY   SLATEC (EISPACK)
    C***CATEGORY  D4A5, D4C2A
    C***TYPE      SINGLE PRECISION (TQLRAT-S)
    C***KEYWORDS  EIGENVALUES OF A SYMMETRIC TRIDIAGONAL MATRIX, EISPACK,
    C             QL METHOD
    C***AUTHOR  Smith, B. T., et al.
    C***DESCRIPTION
    C
    C     This subroutine is a translation of the ALGOL procedure TQLRAT.
    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.  N is an INTEGER variable.
    C
    C        D contains the diagonal elements of the symmetric tridiagonal
    C          matrix.  D is a one-dimensional REAL array, dimensioned D(N).
    C
    C        E2 contains the squares of the subdiagonal elements of the
    C          symmetric tridiagonal matrix in its last N-1 positions.
    C          E2(1) is arbitrary.  E2 is a one-dimensional REAL array,
    C          dimensioned E2(N).
    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 an INTEGER flag 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
    C***REFERENCES  B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow,
    C                 Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen-
    C                 system Routines - EISPACK Guide, Springer-Verlag,
    C                 1976.
    C               C. H. Reinsch, Eigenvalues of a real, symmetric, tri-
    C                 diagonal matrix, Algorithm 464, Communications of the
    C                 ACM 16, 11 (November 1973), pp. 689.
    C***ROUTINES CALLED  PYTHAG, R1MACH
    C***REVISION HISTORY  (YYMMDD)
    C   760101  DATE WRITTEN
    C   890831  Modified array declarations.  (WRB)
    C   890831  REVISION DATE from Version 3.2
    C   891214  Prologue converted to Version 4.0 format.  (BAB)
    C   920501  Reformatted the REFERENCES section.  (WRB)
    C***END PROLOGUE  TQLRAT