Series acceleration

[jaje]

I'm presently working on evaluating functions via infinite series. So far, most of the series I have worked on have REALLY slow convergence. Does anyone here know how to apply the convergence acceleration techniques (e.g. Euler-Maclaurin, Wynn epsilon, Levin's transform, etc.) to programs?

I would appreciate any info. on this matter. Thanks.

Jan M. (^_')

[Bill Press]

Jan,

I assume you've already read section 5.1 of Numerical Recipes and want to go beyond that?

Bill P.

[jaje]

I have read them, Mr. Press. However, I would like to try other convergence acceleration methods so that I can compare the evaluation speeds with the routines in the present book.

I may as well mention the reason for asking. I mentioned in an earlier post that I was going to try to implement the Weierstrass P function in C. I came upon this series for the function. See the attached for the formula. (N.B. The prime on the second summation symbol signifies that any n which will make the general term undefined is to be removed.) I tried doing Aitken and Euler's method (not the same as Euler-Maclaurin; Euler-Maclaurin uses an integral to approximate the sum.). There was not much improvement in speed. Can you please help?

Jan M. (^_^)

[Saul Teukolsky]

Hi Jan,

I haven't tried the series you are interested in, but I doubt that series acceleration will help. (A good reference on series acceleration, beyond what's in NR, is Weniger, Computer Physics Reports, vol. 10, p. 189, 1989).

First, are you sure you want to deal with Weierstrass elliptic functions? The Jacobi functions are generally easier to deal with, and the routine in NR is fine for them. You could evaluate Weierstrass functions by converting them to Jacobi functions using the formulas in Abramowitz and Stegun, but these would fail at certain values. I don't know of any public software that evaluates Weierstrass functions in general. There is an algorithm on Netlib that is for special values of the parameters.

Hope this helps.

Saul Teukolsky

[jaje]

Mr. Teukolsky, I really have to deal with Weierstrass` P function. I am aware that identities exist for converting Weierstrass functions to Jacobi functions (see attached for an example. N.B. the lambda is the modular lambda function; e1, e2, and e3 are the values of the Weierstrass P function at the so-called half-periods.) However, as you have said, they do fail at certain values; besides, I want to write something that has as little dependencies as possible.

Could you please explain what you meant when you said that you doubt that the series I gave can be helped by series acceleration?

Thank you very much for taking your time to answer my questions, Sirs.

Jan M. (^_^)

[Saul Teukolsky]

Hi Jan,

Not every series can have its convergence accelerated. The series you are interested in doesn't look like one of the simple ones where you can tell ahead of time that acceleration will work. So you could waste a lot of time trying different methods without success (see the Weniger reference if you want to try the methods there). You may be better off with something like Mathematica, which has the Weierstrass functions built in. (Being an algebraic system as opposed to numerical, Mathematica can use brute force with many significant digits if necessary.)

Good luck.
Saul Teukolskyand in the