Pade Approximant

Question

I have some questions about Pade approximants.

1. Given a divergent power series $ \sum_{n >0} a(n)x^{n} $ can we use a Pade Approximant to it $ R(x)$ so we can obtain a SUM of the series for every $x$ ?

2. Given a Taylor power series $ \sum_{n >0} c(n)x^{n}=O(x^{a}) $ for positive $x >0$ for some positve but unknown $a$ can we obtain the value of $a$ by approximating the power series by a Pade Approximant?

3. Can we compute a Borel transform of a series $ \sum_{n>0} \frac{a_n}{n!}x^{n} $ by a Pade Approximant?

4. Can we use Pade approximation to obtain numerical integration of series and integrals ?

Answer 1

Pade approximats are sometimes an alternative to a taylor approximation to a function. Imagine fitting (i.e. describing) some data with a taylor series

$D(x) = D_0 + D_1x + D_2x^2$

If this is a "correct" description of the data (i.e. higher order terms are negligible), then a Pade approximant should also describe the data. For example

$D(x) = \frac{D_0}{1-D_1x-D_2x^2}$

The reason is that the Taylor expansion of the second one is the same as the first expression plus some higher order terms that are "small". (This by the way is an explicit example of your question 2).

In many practical reasons, Pade approximants also do a nice job in describing data that have a singularity somewhere (not where you have data).

But I do not understand your questions:

1) If a series is divergent there is no way to sum it. A different thing is that you have the asymptotic expansion of a function (that can be a divergent series), and you want to reconstruct the original function (this is what Borel summation does, for example).

2) The rule is that if the taylor series is up to order $a$, then the sum of the orders of the numerator and denominator in the Pade approximant should also be $a$.

3) You can approximate the borel transform of a series by the Pade approximants, but I do not know if this would be useful...

4) Yes, they are a "good" approximation to a function, and then can be used as approximations to the integrals of the function.

Not sure if this is useful for you, in any case, If you tell us what you want to do, maybe we can help.

Answer 2

I played around wirg Pade approximates for a couple of days once, but didn't find them very useful. The basic idea is you have some sort of polynomial expansion, and you want to approximately match the first N terms, by a function which is the ratio of a Polynomial of degree N-M and a monomial of degree M. If the coefficients of the original power series are known numeric values, by multiplying both sides by the monomial, and equating coefficients of equal power, you get an (N+1)th order linear system to solve for the Pade coefficients. By adjusting M you can usually find an approximant that does somewhat better on a finite interval than the original series. But I failed to see any magic there. Perhaps someone can enlighten us as to how this method can be made useful?