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
```