# bisect.f

          SUBROUTINE BISECT (N, EPS1, D, E, E2, LB, UB, MM, M, W, IND, IERR,
         +   RV4, RV5)
    C***BEGIN PROLOGUE  BISECT
    C***PURPOSE  Compute the eigenvalues of a symmetric tridiagonal matrix
    C            in a given interval using Sturm sequencing.
    C***LIBRARY   SLATEC (EISPACK)
    C***CATEGORY  D4A5, D4C2A
    C***TYPE      SINGLE PRECISION (BISECT-S)
    C***KEYWORDS  EIGENVALUES, EISPACK
    C***AUTHOR  Smith, B. T., et al.
    C***DESCRIPTION
    C
    C     This subroutine is a translation of the bisection technique
    C     in the ALGOL procedure TRISTURM by Peters and Wilkinson.
    C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 418-439(1971).
    C
    C     This subroutine finds those eigenvalues of a TRIDIAGONAL
    C     SYMMETRIC matrix which lie in a specified interval,
    C     using bisection.
    C
    C     On INPUT
    C
    C        N is the order of the matrix.  N is an INTEGER variable.
    C
    C        EPS1 is an absolute error tolerance for the computed
    C          eigenvalues.  If the input EPS1 is non-positive,
    C          it is reset for each submatrix to a default value,
    C          namely, minus the product of the relative machine
    C          precision and the 1-norm of the submatrix.
    C          EPS1 is a REAL variable.
    C
    C        D contains the diagonal elements of the input matrix.
    C          D is a one-dimensional REAL array, dimensioned D(N).
    C
    C        E contains the subdiagonal elements of the input matrix
    C          in its last N-1 positions.  E(1) is arbitrary.
    C          E is a one-dimensional REAL array, dimensioned E(N).
    C
    C        E2 contains the squares of the corresponding elements of E.
    C          E2(1) is arbitrary.  E2 is a one-dimensional REAL array,
    C          dimensioned E2(N).
    C
    C        LB and UB define the interval to be searched for eigenvalues.
    C          If LB is not less than UB, no eigenvalues will be found.
    C          LB and UB are REAL variables.
    C
    C        MM should be set to an upper bound for the number of
    C          eigenvalues in the interval.  WARNING - If more than
    C          MM eigenvalues are determined to lie in the interval,
    C          an error return is made with no eigenvalues found.
    C          MM is an INTEGER variable.
    C
    C     On OUTPUT
    C
    C        EPS1 is unaltered unless it has been reset to its
    C          (last) default value.
    C
    C        D and E are unaltered.
    C
    C        Elements of E2, corresponding to elements of E regarded
    C          as negligible, have been replaced by zero causing the
    C          matrix to split into a direct sum of submatrices.
    C          E2(1) is also set to zero.
    C
    C        M is the number of eigenvalues determined to lie in (LB,UB).
    C          M is an INTEGER variable.
    C
    C        W contains the M eigenvalues in ascending order.
    C          W is a one-dimensional REAL array, dimensioned W(MM).
    C
    C        IND contains in its first M positions the submatrix indices
    C          associated with the corresponding eigenvalues in W --
    C          1 for eigenvalues belonging to the first submatrix from
    C          the top, 2 for those belonging to the second submatrix, etc.
    C          IND is an one-dimensional INTEGER array, dimensioned IND(MM).
    C
    C        IERR is an INTEGER flag set to
    C          Zero       for normal return,
    C          3*N+1      if M exceeds MM.  In this case, M contains the
    C                     number of eigenvalues determined to lie in
    C                     (LB,UB).
    C
    C        RV4 and RV5 are one-dimensional REAL arrays used for temporary
    C          storage, dimensioned RV4(N) and RV5(N).
    C
    C     The ALGOL procedure STURMCNT contained in TRISTURM
    C     appears in BISECT in-line.
    C
    C     Note that subroutine TQL1 or IMTQL1 is generally faster than
    C     BISECT, if more than N/4 eigenvalues are to be found.
    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  R1MACH
    C***REVISION HISTORY  (YYMMDD)
    C   760101  DATE WRITTEN
    C   890531  Changed all specific intrinsics to generic.  (WRB)
    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  BISECT