Pade

Padé approximants are a class of rational function approximations used to approximate a smooth function by a ratio of two polynomials. The method is named after the French mathematician Henri Padé, who introduced the construction in the late 19th century. Padé approximants are particularly valued for their ability to capture analytic structure, such as poles, that are not evident in a truncated Taylor series.

Formally, if a function f is analytic at zero with a Taylor expansion f(x) = c0 + c1 x + c2 x^2 + ..., the [m/n] Padé approximant is the rational function Pm(x)/Qn(x), where Pm has degree at most m and Qn has degree at most n with Qn(0) = 1. The coefficients are chosen so that the Taylor expansion of Pm(x)/Qn(x) agrees with f up to order m+n. Equivalently, the coefficients solve a linear system ensuring that the first m+n+1 terms of the series match.

Padé approximants come in various forms, including diagonal (m ≈ n) and near-diagonal choices, as well as multi-point Padé approximants that fit the function at several values. They can be expressed in a continued fraction form and are often organized in a Padé table. Applications span numerical analysis, physics, and engineering, where they are used to accelerate series convergence, perform analytic continuation, and approximate special functions or transfer functions in control theory.

Limitations include cases where the method is not well defined (for example, when the required linear system is singular) or where convergence to the target function is not guaranteed. Nevertheless, Padé approximants remain a central tool for rational approximation and analytic insight.

---