Padéapproximáció

Padéapproximáció is a method for approximating a function by a rational function. A rational function is a ratio of two polynomials. The Padé approximant of a function f(x) at x=0 is a rational function R(x) = P(x)/Q(x) where P(x) and Q(x) are polynomials, such that the Taylor series expansion of R(x) at x=0 matches the Taylor series expansion of f(x) up to the highest possible order.

The order of the Padé approximant is determined by the degrees of the numerator and denominator polynomials. For example, a [m/n] Padé approximant has a numerator polynomial of degree m and a denominator polynomial of degree n. The goal is to choose the coefficients of these polynomials such that the Taylor series of the rational function aligns with the function being approximated. This alignment is achieved by setting the coefficients of the first m+n+1 terms of the Taylor series of R(x) equal to the corresponding terms of the Taylor series of f(x).

Padé approximants are particularly useful for approximating functions that have singularities or for extending the radius of convergence of a Taylor series. They can often provide a more accurate approximation than a Taylor polynomial of the same total degree (m+n) over a wider range of x values, especially near singularities. The method has applications in various fields, including numerical analysis, applied mathematics, and physics.