# Arbitrary precision zeros of a function

I read about arbitrary precision libraries. Its an interesting topic, but can you name me some examples in science where this is necessary and actualy? I know the reasons why and would like to see an example topic or a motivation, escpecially for enclosing zeros of nonlinear systems. Thank you.

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(1) Path tracking, aka homotopy continuation. Frequently high precision might be needed in intermediate steps in order to prevent "path jumping". (2) Any number of hybrid symbolic-numeric algorithms that use fast finite precision arithmetic as a surrogate for infinite precision (exact) arithmetic. These algorithms in turn show up in applications e.g. lattice reduction in knapsack solving. –  Daniel Lichtblau Oct 23 '13 at 23:30

Solve[x^2 - 2 == 0, x]


Taken to arbitrary precision, the answer is exactly what Mathematica gives:

{{x -> -Sqrt[2]}, {x -> Sqrt[2]}}


Isn't that more satisfying than a numerical answer like:

N[Solve[x^2 - 2 == 0, x]]
{{x -> -1.41421}, {x -> 1.41421}}


Why might this be useful? Consider an analytic function:

FullSimplify[(x^2 - 2)/(x - Sqrt[2]) (x - 3)]
(-3 + x) (Sqrt[2] + x)


which simplifies very nicely to a polynomial, whereas the corresponding numerical version does not simplify:

FullSimplify[(x^2 - 2)/(x - 1.41421) (x - 3)]
((-3. + x) (-2. + x^2))/(-1.41421 + x)

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I think the question is on arbitrary precision as opposed to e.g. floating point? I think of butterfly effect and weather reporting ... –  asterix314 Oct 23 '13 at 5:49
@asterix314 Finding an example "in science" doesn't really limit the scope of the discussion much! I was trying to think of a simple setting. Another good one would be digits of Pi. Perhaps factoring algorithms. –  bill s Oct 23 '13 at 13:02
@bill_s Your answer showed the use of symbolic results which has nothing to do with arbitrary precision directly. –  asterix314 Oct 24 '13 at 1:59