Is there a general method to aproximate functions with ratios of polynomies?

[viraltux]

Is there a general method to approximate functions with ratios of polynomies??

Hi All,

There is a variety of methods to approximate functions (Taylor expansions, Fourier series, Splines, Interpolation, ... ) yet, when it comes to approximate some statistical distributions none of these methods seems to be as computationally efficient and accurate as using ratios of polynomials. I know of several of these ratios in the literature to approximate Gaussian distributions, but what I fail to find is a general method to adjust any arbitrary function to ratios of polynomials.

So, do you happen to know if there is a general method to find the ratio of polynomials that better fits a function?

Thank you

 

[HallsofIvy]

"Ratios of polynomials" are called "rational functions". Googling "approximation" with "rational functions" gives a number of hits including
http://en.wikipedia.org/wiki/Rational_function
http://www.measurement.sk/PAPERS/Bartkov.pdf
http://www.ams.org/journals/tran/1932-034-01/S0002-9947-1932-1501629-1/S0002-9947-1932-1501629-1.pdf
and references to books on the subject:
http://books.google.com/books/about/Approximation_With_Rational_Functions.html?id=f93ubGXa7bgC

 

[viraltux]

"Ratios of polynomials" are called "rational functions". Googling "approximation" with "rational functions" gives a number of hits including
http://en.wikipedia.org/wiki/Rational_function
http://www.measurement.sk/PAPERS/Bartkov.pdf
http://www.ams.org/journals/tran/1932-034-01/S0002-9947-1932-1501629-1/S0002-9947-1932-1501629-1.pdf
and references to books on the subject:
http://books.google.com/books/about/Approximation_With_Rational_Functions.html?id=f93ubGXa7bgC

Hi HallsofIvy,

Thank you for the links,

The first paper shows a general method but is misses the point of the procedure since it first approximates the function with a polynomial so, in cases where this approximation is not good, the cases where we would actually want to use rational functions, the rational function will not be a good approximation either.

The second paper is more theoretical and goes on in properties and no so much in general procedures to approximate any arbitrary function.

The third link, the book, I didn't buy it, but I had the chance to have a look and the lectures titles and one of them is "Rational Approximation to $e^x$" which makes me think that the state of the art of approximating arbitrary functions with rationals functions is a per case problem... Hence why I made this question.

 

[Mute]

Is this what you're looking for?:

http://en.wikipedia.org/wiki/Padé_approximant

 

[AlephZero]

The first paper shows a general method but is misses the point of the procedure since it first approximates the function with a polynomial so, in cases where this approximation is not good, the cases where we would actually want to use rational functions, the rational function will not be a good approximation either.

I haven't read the paper, but your statement is missing the point that the approximations might be of different orders.

For example, if you approximate a function by a Taylor series, you may need a large number of terms to get high accuracy, but the largest errors often occur only at the ends of the interval, and the accuracy near the center is much higher than you need. If the highest term in the approxmation in this approximation $P_n(x)$ is $x^n$, you can then approximate the $x^n$ term by a Chebyshev polynomial of order $n-1$ and get a new approximation $P_{n-1}(x)$ with a lower order. Often the maximum error of $P_{n-1}$ is no worse than $P_n$, but the errors are spread more evenly across the whole range. Sometimes you can repeat this several times to get a good approximation with a much lower order than $n$.

The same idea should apply to converting a polynomial approximation to a rational polynomial approximation.

 

[chiro]

You mentioned Fourier series, but the whole area of Fourier analysis is great if you want to find a good enough functional approximation for approximating a given function within an interval.

What you could do is to create a template rational function P(x)/Q(x) and create an orthonormal subspace and project your function or data to this new rational function.

You may have to derive this yourself using the Fourier analysis framework ideas and Gram-Schmidt process, but I recommend you check it out.

Examples of orthogonal polynomials and related include Bessel, Legendre, and Hermite just to name a few.

 

[viraltux]

Is this what you're looking for?:

http://en.wikipedia.org/wiki/Padé_approximant

Oh! WOW! it certainly looks so, first time I hear about the Padé approximant! Thank you Mute!

I haven't read the paper, but your statement is missing the point that the approximations might be of different orders.

Well, of course, I didn't mean that he paper has no application in any case, I wouldn't be surprised if there are problems where is the right approach, but definitely not the general method I was asking for.

You mentioned Fourier series, but the whole area of Fourier analysis is great if you want to find a good enough functional approximation for approximating a given function within an interval.

What you could do is to create a template rational function P(x)/Q(x) and create an orthonormal subspace and project your function or data to this new rational function.

You may have to derive this yourself using the Fourier analysis framework ideas and Gram-Schmidt process, but I recommend you check it out.

Examples of orthogonal polynomials and related include Bessel, Legendre, and Hermite just to name a few.

It seems this must be one approach since the book link offered by HallsofIvy suggests in its index the use of Hermite Polynomials, so yours was a great insight. I wonder though how it plays against the Padé approximant. Anyway, more interesting stuff to check, thanks!



Thank you all guys for the feedback, it was very helpful!