# tql2.f

          SUBROUTINE TQL2 (NM, N, D, E, Z, IERR)
    C***BEGIN PROLOGUE  TQL2
    C***PURPOSE  Compute the eigenvalues and eigenvectors of symmetric
    C            tridiagonal matrix.
    C***LIBRARY   SLATEC (EISPACK)
    C***CATEGORY  D4A5, D4C2A
    C***TYPE      SINGLE PRECISION (TQL2-S)
    C***KEYWORDS  EIGENVALUES, EIGENVECTORS, EISPACK
    C***AUTHOR  Smith, B. T., et al.
    C***DESCRIPTION
    C
    C     This subroutine is a translation of the ALGOL procedure TQL2,
    C     NUM. MATH. 11, 293-306(1968) by Bowdler, Martin, Reinsch, and
    C     Wilkinson.
    C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 227-240(1971).
    C
    C     This subroutine finds the eigenvalues and eigenvectors
    C     of a SYMMETRIC TRIDIAGONAL matrix by the QL method.
    C     The eigenvectors of a FULL SYMMETRIC matrix can also
    C     be found if  TRED2  has been used to reduce this
    C     full matrix to tridiagonal form.
    C
    C     On Input
    C
    C        NM must be set to the row dimension of the two-dimensional
    C          array parameter, Z, as declared in the calling program
    C          dimension statement.  NM is an INTEGER variable.
    C
    C        N is the order of the matrix.  N is an INTEGER variable.
    C          N must be less than or equal to NM.
    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        E contains the subdiagonal elements of the symmetric
    C          tridiagonal matrix in its last N-1 positions.  E(1) is
    C          arbitrary.  E is a one-dimensional REAL array, dimensioned
    C          E(N).
    C
    C        Z contains the transformation matrix produced in the
    C          reduction by  TRED2, if performed.  If the eigenvectors
    C          of the tridiagonal matrix are desired, Z must contain
    C          the identity matrix.  Z is a two-dimensional REAL array,
    C          dimensioned Z(NM,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 but
    C          unordered for indices 1, 2, ..., IERR-1.
    C
    C        E has been destroyed.
    C
    C        Z contains orthonormal eigenvectors of the symmetric
    C          tridiagonal (or full) matrix.  If an error exit is made,
    C          Z contains the eigenvectors associated with the stored
    C          eigenvalues.
    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***ROUTINES CALLED  PYTHAG
    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  TQL2