Padé

Padé approximants are rational approximations of a function, expressed as a ratio of two polynomials P_L(x) and Q_M(x) with degrees L and M. Given a function with a Taylor series f(x) = sum_{n=0}^\infty a_n x^n, the Padé approximant [L/M](x) is defined so that P_L(x)/Q_M(x) matches the series of f up to order L+M, i.e., f(x) - P_L(x)/Q_M(x) = O(x^{L+M+1}). Coefficients are determined by equating coefficients in the expansion of P_L(x)/Q_M(x) to those of f.

Typically one normalizes Q_M(0)=1 (by absorbing a constant into P_L). Padé approximants often converge or provide reasonable analytic continuation beyond the radius of convergence of the Taylor series. They can approximate functions with poles and other singularities more efficiently than polynomials of similar degree, though they may introduce spurious poles (defects) depending on the function.

A systematic arrangement into a Padé table lists approximants by row L and column M; diagonal or near-diagonal forms [n/n] are commonly used. Variants include multipoint Padé approximants, which interpolate or approximate a function at several points, and generalized Padé approximants that relax certain matching conditions.

Padé is named after Henri Padé, a French mathematician (1863–1953). The method is widely used in numerical analysis, physics, and engineering for tasks such as series acceleration, analytic continuation, and the numerical solution of differential equations. In control theory, Padé approximants model time delays; in physics, they aid efficient representation of special functions and propagators.