PadéTransformation

The Padé transform is a method used to accelerate the convergence of sequences, particularly those that converge slowly or even diverge. It is a generalization of the Aitken's delta-squared process and is named after French mathematician Henri Padé. The core idea behind the Padé transform is to approximate a sequence by a rational function, which is a ratio of two polynomials.

Given a sequence x_0, x_1, x_2, ..., the Padé transform constructs a rational function R(n) = P(n) / Q(n), where P(n) and Q(n) are polynomials in n. The coefficients of these polynomials are determined by matching the first few terms of the sequence with the Taylor series expansion of the rational function. Specifically, if the Taylor series of R(n) around n = infinity is given by x_n = c_0 + c_1/n + c_2/n^2 + ..., the Padé approximant [m/l] is a rational function where the degree of the numerator P(n) is at most m and the degree of the denominator Q(n) is at most l. The coefficients are chosen such that the first m+l+1 terms of the Taylor series of R(n) match the first m+l+1 terms of the given sequence.

The Padé transform is particularly useful in numerical analysis for accelerating the convergence of series and sequences arising from the discretization of differential equations, or in the evaluation of mathematical functions. It can transform a slowly converging sequence into one that converges much faster, or even transform a divergent sequence into a convergent one, provided the underlying generating function is of a form that can be represented by a rational function. The choice of the order of the Padé approximant (m and l) depends on the specific problem and the desired accuracy.