SLATEC Routines --- IMTQL2 ---

*DECK IMTQL2
      SUBROUTINE IMTQL2 (NM, N, D, E, Z, IERR)
C***BEGIN PROLOGUE  IMTQL2
C***PURPOSE  Compute the eigenvalues and eigenvectors of a symmetric
C            tridiagonal matrix using the implicit QL method.
C***LIBRARY   SLATEC (EISPACK)
C***CATEGORY  D4A5, D4C2A
C***TYPE      SINGLE PRECISION (IMTQL2-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 IMTQL2,
C     NUM. MATH. 12, 377-383(1968) by Martin and Wilkinson,
C     as modified in NUM. MATH. 15, 450(1970) by Dubrulle.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 241-248(1971).
C
C     This subroutine finds the eigenvalues and eigenvectors
C     of a SYMMETRIC TRIDIAGONAL matrix by the implicit 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 reduction
C          by  TRED2,  if performed.  This transformation matrix is
C          necessary if you want to obtain the eigenvectors of the full
C          symmetric matrix.  If the eigenvectors of the symmetric
C          tridiagonal matrix are desired, Z must contain the identity
C          matrix.  Z is a two-dimensional REAL array, dimensioned
C          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 full symmetric
C          or symmetric tridiagonal matrix, depending on what it
C          contained on input.  If an error exit is made,  Z contains
C          the eigenvectors associated with the stored 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                     The eigenvalues and eigenvectors should be correct
C                     for indices 1, 2, ..., IERR-1, but the eigenvalues
C                     are not ordered.
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  IMTQL2