Padémethode

Padé method, or Padé approximation, is a technique in numerical analysis and approximation theory for approximating a function by a rational function. The idea is to construct a ratio of two polynomials whose power series expansion matches a given function as closely as possible, often better than a corresponding truncated Taylor series.

In its common form, for a function f with a Taylor series around a point a (often a = 0), the Padé approximant [m/n] is a ratio P_m(x)/Q_n(x) where P_m has degree at most m and Q_n has degree at most n, with Q_n(a) = 1. The coefficients of P_m and Q_n are chosen so that the Taylor expansion of P_m(x)/Q_n(x) agrees with f(x) up to the order m+n. This yields a system of linear equations that can be solved to determine the coefficients. The Padé approximant is unique under these normalization conditions.

Padé approximants are valued because they can capture poles and other singularities of the target function, enabling more accurate approximation near such features and beyond the radius of convergence of the original Taylor series. They are often superior to truncating a power series, particularly for analytic continuation and for evaluating functions at larger arguments.

Common variants include diagonal and near-diagonal approximants, and the Padé table, which organizes approximants by increasing m and n. Applications span summation of divergent series, numerical evaluation of special functions, solving differential equations, control theory (for time-delay approximations), and signal processing. The method is named after Henri Padé, who introduced the concept in the late 19th century.