Some of the funsters' flow work fits here: Giving a numerical definition of the half-flow, or the t-th flow, for a function, assuming it's close to an analytically flowable function. Simple examples: model f(x) = x^2 + x as the formal power series near x=0, with half-flow F(.5,x) = x + .5x^2 + ... (the rest of the power series can be computed from commutative composition, F(.5,f(x))=f(F(.5,x))). And also near x=infinity, model f(x) as near g(x) = x^2, with half flow G(.5,x) = x^sqrt2. For intermediate values like x=1, iterate f forward or backward to get close to 0 or infinity. The numerical results of the two methods almost match, but not quite. Another, even more surprising example, is f(x) = sqrt2^x. This has fixed points at x = 2 & 4, and is well-behaved (nearly linear, smooth, etc.) near the fixed points and in the interval between. Using the procedure "iterate forward or backward to get near the fixed point, model F(.5,-) with the square-root of the derivative, uniterate", evaluate the half-flow at x=3. Which fixed point to use? The two results agree to about 25 decimals, but then differ. Third example: Modular Dilogarithms (see http://richard.schroeppel.name:8015/dilog-paper-020402) "defines" a modular version of the dilog function Li2(z), (with power series z + z^2/4 + z^3/9 + ... + z^n/n^2 + ..., Li2(1)=zeta(2)=pi^2/6), as "something that matches the dilog functional equations", and does a computer search to find examples. Each of these defines a function as the outcome of a calculation, and the first two examples can even prove the limits exist. But the failure to be consistent over the choice of fixed points, or to be unique, undermines the believability of the answer. Rich -----
At 08:41 AM 7/2/2013, James Propp wrote:
For some thoughts about the enterprise of computing quantities one hasn't rigorously defined, see http://mathoverflow.net/questions/135536/procedure-based-as-opposed-to-defin... (which I hope will elicit interesting responses over the next few days).
Jim Propp
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