imtqlv.f

SUBROUTINE IMTQLV (N, D, E, E2, W, IND, IERR, RV1)
C***BEGIN PROLOGUE  IMTQLV
C***PURPOSE  Compute the eigenvalues of a symmetric tridiagonal matrix
C            using the implicit QL method.  Eigenvectors may be computed
C            later.
C***LIBRARY   SLATEC (EISPACK)
C***CATEGORY  D4A5, D4C2A
C***TYPE      SINGLE PRECISION (IMTQLV-S)
C***KEYWORDS  EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR  Smith, B. T., et al.
C***DESCRIPTION
C
C     This subroutine is a variant of  IMTQL1  which is a translation of
C     ALGOL procedure IMTQL1, NUM. MATH. 12, 377-383(1968) by Martin and
C     Wilkinson, 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 of a SYMMETRIC TRIDIAGONAL
C     matrix by the implicit QL method and associates with them
C     their corresponding submatrix indices.
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        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        E2 contains the squares of the corresponding elements of E in
C          its last N-1 positions.  E2(1) is arbitrary.  E2 is a one-
C          dimensional REAL array, dimensioned E2(N).
C
C     On OUTPUT
C
C        D and E are unaltered.
C
C        Elements of E2, corresponding to elements of E regarded as
C          negligible, have been replaced by zero causing the matrix to
C          split into a direct sum of submatrices.  E2(1) is also set
C          to zero.
C
C        W contains the eigenvalues in ascending order.  If an error
C          exit is made, the eigenvalues are correct and ordered for
C          indices 1, 2, ..., IERR-1, but may not be the smallest
C          eigenvalues.  W is a one-dimensional REAL array, dimensioned
C          W(N).
C
C        IND contains the submatrix indices associated with the
C          corresponding eigenvalues in W -- 1 for eigenvalues belonging
C          to the first submatrix from the top, 2 for those belonging to
C          the second submatrix, etc.  IND is a one-dimensional REAL
C          array, dimensioned IND(N).
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 should be correct for indices
C                     1, 2, ..., IERR-1.  These eigenvalues are
C                     ordered, but are not necessarily the smallest.
C
C        RV1 is a one-dimensional REAL array used for temporary storage,
C          dimensioned RV1(N).
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  IMTQLV