Understanding Pade Approximants - Your Guide to Preparing a Talk

[Sariel]

So what's a Pade approximant? I'm supposed to give a talk on them in a few weeks, and I don't understand them. You can explain it to me, right?

 

[cristo]

So what's a Pade approximant? I'm supposed to give a talk on them in a few weeks, and I don't understand them. You can explain it to me, right?

I don't know about ElNino, but if someone asked me something in that tone, then they would be the last person I'd help!

 

[Astronuc]

So what's a Pade approximant? I'm supposed to give a talk on them in a few weeks, and I don't understand them. You can explain it to me, right?

How is it that one is supposed to give a talk on "Pade approximant," yet one does not understand them?! Perhaps one should apply some effort, do some research and then ask for assistance.

 

[awvvu]

I thought this was meant to be a sarcastic post. Why was it split off?

 

[John Creighto]

A pade approximation is similar to a taylor approximation but the approximating function is a rational function instead of a polynomial. Pade approximations provide better fits then polynomial approximations because high order polynomial approximations get very wavy. Additionally if you choose the order of the numerator equal to the order of the denominator you can get asymptotes which are constant functions. Pade approximations are often used in continuous filters to approximate time delays.

 

[kgopalak]

Hello all,

One thing that I am unable to understand about Pade approximants is when do they work? What guidelines do we have, given a truncated power series to make a best guess of the form of the rational approximant? - Thanks

 

[Count Iblis]

I recently added something on the Wiki-page on this subject. They are very useful when you have some limted number of terms of a power series of a function with a limited radius of convergence (you can roughly estimate that by talking ratios of successive coefficients of the powe series). Suppose that you need to evaluate the function way outside the radius of convergence.

If you then construct the Pade approximants, you are basically extrapolating to infinity the coeficients of the power series and summing over them. This gives rises to the poles of the rational function you construct.


I think the best way to get a feeling for this is to do some mathematical experimens with this. E.g. take the first 15 terms of the power series around x = 0 of

sin(x) + 1/(1-x)


Can you then construct a Pade approximant that gives a good approximation to this function in the neighborhood of x = 4?

You can also use Pade approximants to locate the poles and zeroes of meromorphic functions. You then construct the [n,n+1] Pade approximants to the logarthimic derivative of the function. At first sight this seems to be a worthless exercise, because if you only have some limited number of terms of the power series, the logarithmic derivative is already in an [n, n+1] form.

However, the Pade approximant based on the logarithmic derivative of the truncated power series is not the same as this. It should bew clear that the numerator can now only contain information about any poles the function has. The location of zeroes of the function can be extracted to much higher accuracy than if you simply use the truncated power series to estimate that.

In this case, there exist slightly more sophisticated methods that put in the information that the residues at the poles of the pade approximant [n, n+1] must be exactly 1 or minus 1 depending on whether the pole corresponds to a zero or pole of the original function.

You can also use the pade approximant to the logarithmic derivative to extract asymptotic power law behavior. E.g. suppose that you already know that the function you study behaves like x^p for large x. But you only have the first few terms of the series expansion and the x^p asymptotic bahavior is not visible at all from te domain of validity. If you then again cosntruct the [n,n+1] pade approximants, then the large x asymptotic behavior will show up as a pole at x = 0, the residue there will be p.

Note that the original function could have poles at some points making it impossible to see the x^p behavior directly from the power series, regardless of how many terms you use. Wit the Pade approximant, you will then effectively subtract all these singularities from the function as estimated from the terms in the power series when you estimate p.