Can we use Pade approximation to accurately integrate well-behaved functions?

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  • #1
Sangoku
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can we find for a well-behaved f(x) a Rational (Padé approximation) so

[tex] \int_{0}^{\infty}dx f(x) - \int_{0}^{\infty}dx Q(x) \approx 0 [/tex] ??


Where Q(x) is a rational function, the main idea is that the integral for Q(x) can be performed exactly , whereas the initial integral of f(x) not.
 
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  • #2
What do you mean by "well behaved"?
 
  • #3
smooth and differentiable function, for example sin(x) or exp(x^3) ... so it is many times differentiable.
 

FAQ: Can we use Pade approximation to accurately integrate well-behaved functions?

What is Pade integration?

Pade integration is a numerical method used to approximate the definite integral of a function over a given interval. It is based on the Pade approximation, which involves constructing a rational function that closely matches the given function.

How does Pade integration work?

In Pade integration, the interval of integration is divided into smaller subintervals, and the Pade approximation is used to approximate the integral on each subinterval. The results from each subinterval are then combined to give an overall approximation of the integral over the entire interval.

What are the advantages of using Pade integration?

Pade integration is generally more accurate than other numerical integration methods, such as the trapezoidal rule or Simpson's rule. It is also more efficient, as it requires fewer function evaluations to achieve a desired level of accuracy.

What are the limitations of Pade integration?

One limitation of Pade integration is that it may not work well for functions with singularities or very sharp peaks. In these cases, the Pade approximation may not be able to accurately capture the behavior of the function, leading to a less accurate overall result.

How can I implement Pade integration in my calculations?

There are various software packages and libraries that offer Pade integration as a built-in function. These include MATLAB, Python's SciPy library, and Mathematica. Alternatively, you can also code your own implementation using the Pade approximation formula and techniques for integrating rational functions.

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