On Mon, Jun 9, 2008 at 3:03 AM, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
On Monday 09 June 2008, Andy Latto wrote:
On Sun, Jun 8, 2008 at 5:05 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Speaking of math software, can anyone tell me if there exists analytic continuation software? ... . . . and all this to a pre-specified level of numerical accuracy.
I don't think this is possible. In finite tie the software can only evaluate your original function f at a finite number of points. And there are analytic functions that have exactly the given value at that finite number of points, but have any desired value at any other point.
The same argument proves that numerical integration and differentiation are impossible. And yet, if you pick a function you're interested in and ask Maple or Mathematica or whatever to integrate it for you over a given range to a given level of accuracy, they can generally do it pretty well.
Though the above argument shows that there exist pathological functions for which these routines will give very wrong answers. So for any given method of numerical integration or numerical analytic continuation that know a function only by evaluating it at a finite number of poitns, there are implicit assumptions about the nature of the function. A simple trapezoidal rule will work well for numerical integration if the second derivative of the function doesn't get too big. So what I meant to be indirectly suggesting is that a first step towards working out how to do this sort of numerical analytic continuation would be considering what implicit assumptions one wishes to make on the function being continued.
(That doesn't mean that I know of software to do what Dan wants, or that I have a proposal for how to do it so that it works well in practice. I wonder whether it might be more appropriate for what you feed the analytic continuation routine to be a black box that computes not only f but also any requested number of its derivatives.)
That sounds right; it would allow you to expand f as a power series, which is what I think you want for analytic continuation. -- Andy.Latto@pobox.com