On 15/04/2011 2:25 AM, Michel Talon wrote:
> Hello,
>
> i have discovered 2 programs included in maxima, but whose function is not
> very clear to me.
>
> One is levin, it has been mentioned recently here. Acccording to the
> comments it is used to improve convergence of "numerical" sums, such as:
> levin_u_sum(1/n2,n,1,10);
>
> However i have tried to add a parameter, such as:
> g: rat(levin_u_sum(t^n/n2,n,1,10));
> which results in something looking like a Pad? approximant in the variable
> t. I have checked for various values of t like 1/3, 1/5 that this formula
> reproduces what one gets by directly substituting t=1/3 ... before computing
> levin, so the formula seems correct. Is this an intended use of levin,
> and does this produce an alternative way to get Pad? approximants - which
> would be interesting because the pade command in maxima saturates very fast?
I looked at Levin transforms many years ago for an undergrad project.
In his original 1973 paper, Levin [1] briefly discusses rational
approximations,
and in a later paper [2], which I no longer have, uses the related
d-transformation.
It seems that the rational approximations generated by the u-transform
are not
(always?) Pade approximates.[3]
There is quite a bit more literature.
Referencea
1. D. Levin, Development of Non-Linear Transformations for Improving
Convergence
of Sequences, Intern. J. Computer Math. B3:371--388, 1973
2. A Sidi, D Levin, Rational Approximations from the d-Transformation
IMA J Numer
Anal (1982) 2 (2): 153-167.
3. Tarun Sheel, Rational approximants generated by pade approximation
and u-transform,
M Sc dissertation, University of Dhaka, 1992.
http://perso.uclouvain.be/tarun.sheel/docs/Sheel_MS_Thesis.pdf