Root Finding Methods Gaurenteed to Find All Root Between (xmin, xmax)

Discussion in 'Mathematica' started by Ted Ersek, Sep 20, 2010.

  1. Ted Ersek

    Ted Ersek Guest

    In [1] I gave a long response to the thread [FindRoots]

    It seems in [2] Andrzej Kozlowski was saying there are ways to
    obtain the complete set of roots (provably complete) over a
    closed and bounded set provided the function is C2 continuous
    and does not have multiple roots in the domain of definition.

    I suppose he is talking about numeric methods to find all the roots.
    If that is the case I would expect these methods to have practical
    limitations in what they can do? I mean a function could be very
    ill-behaved and still meet the conditions above. Consider f[x] below.

    pnts=Table[ {x,Log[x]}, {x,3.0,250.0,0.01}];
    Part[ pnts, 16000, 2] = -0.05;
    f=Interpolation[ pnts, Method:>Spline ];
    well behaved and having no roots between (3, 250). In fact it is
    C2 continuous, but it does have two root close to 18.99. It actually is
    well behaved except for the small interval between (18.992, 19.006).

    Suppose we wanted to find all the roots between (3,250) of a function
    defined as SomeNumericalAlgorithm[x] and the function always evaluated to
    the same value as the InterpolatingFunction above. How can a numeric
    RootFinding
    algorithm be gaurenteed to find these roots near 18.99 if it doesn't have
    the
    good furtune of taking one or more sample between (18.992, 19.006), or
    sampling higher order derivatives in a interval a bit larger than (18.992,
    19.006)?

    (******* References ********)

    [1] http://forums.wolfram.com/mathgroup/archive/2010/Sep/msg00255.html

    [2] http://forums.wolfram.com/mathgroup/archive/2010/Sep/msg00262.html
     
    Ted Ersek, Sep 20, 2010
    #1
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