Padéteknikker

Padé approximanter are rational function approximations to a given function. A Padé approximant is constructed by matching the Taylor series expansion of the function at a given point with the Taylor series expansion of a rational function. For a function f(x) and a point x0, the Padé approximant of order [m/n] is a rational function P(x)/Q(x) where P(x) is a polynomial of degree m and Q(x) is a polynomial of degree n, such that the Taylor series expansion of f(x) around x0 agrees with the Taylor series expansion of P(x)/Q(x) around x0 up to the term of degree m+n.

The construction of a Padé approximant involves solving a system of linear equations. If f(x) = c0 + c1*x + c2*x^2 + ..., then the Padé approximant [m/n] is given by P(x)/Q(x) where P(x) = a0 + a1*x + ... + am*x^m and Q(x) = b0 + b1*x + ... + bn*x^n (with b0=1 conventionally), and (a0 + a1*x + ... + am*x^m) = (b0 + b1*x + ... + bn*x^n) * (c0 + c1*x + c2*x^2 + ...). The coefficients ai and bj are determined by equating the coefficients of x^k for k from 0 to m+n.

Padé approximants are useful for approximating functions where Taylor series expansions may converge slowly or not at all. They can provide good approximations in regions where the function has singularities or poles. Applications of Padé approximants are found in various fields of science and engineering, including numerical analysis, quantum mechanics, and control theory, particularly for problems involving differential equations and asymptotic expansions.