I Finding a Rational Function with data (Pade approximation)

[cbarker1]

TL;DR Summary

I have some points that I need to approximate a function by using a Rational function.

Dear Everybody,

I need some help understanding how to use pade approximations with a given data points (See the attachment for the data).
Here is the basic derivation of pade approximation read the Derivation of Pade Approximate.
I am confused on how to find a f(x) to the data or is there a better way to just use the values of data in order to find the rational function.
I thought about numerical differentiation in order to find f(x) at the point 0.

Thanks,
Cbarker1

 

[Mark44]

Summary:: I have some points that I need to approximate a function by using a Rational function.

Dear Everybody,

I need some help understanding how to use pade approximations with a given data points (See the attachment for the data).
Here is the basic derivation of pade approximation read the Derivation of Pade Approximate.
I am confused on how to find a f(x) to the data or is there a better way to just use the values of data in order to find the rational function.
I thought about numerical differentiation in order to find f(x) at the point 0.

Thanks,
Cbarker1

Why do you think you need to do the approximation with a rational function? You have 10 data points, so a 9th degree polynomial would go through all 10 points. As an alternative, you could use Bezier curve fitting (https://en.wikipedia.org/wiki/Bézier_curve), to find a collection of functions that fit your data points.

 

[cbarker1]

Because I am learning about pade approximation and I am presenting this method to my colleagues, I need to know where to start. I understand that I can do the polynomial interpolation.

 

[Mark44]

Summary:: I have some points that I need to approximate a function by using a Rational function.

Here is the basic derivation of pade approximation read the Derivation of Pade Approximate.

According to the article in the link, you need to have the function definition in order to approximate it as the quotient of two power series. Obviously, you don't have the function definition. In the example in the article, they find the Pade approximation for $f(x) = e^x$.

I am confused on how to find a f(x) to the data or is there a better way to just use the values of data in order to find the rational function.
I thought about numerical differentiation in order to find f(x) at the point 0.

You mean at the point (1, 20)? I don't see how that would help at all. The graph of the plotted points doesn't look to me like any rational function (other than some polynomial).

 

[cbarker1]

Polynomial interpolation would works well in this data. But pade approximate would better because the error would be smaller. The best situation is having a defined function that does not happen in many cases in application problems. It is usual given by data points.