Padé approximation
/ pah-DAY /
A Taylor series approximates a function by a single polynomial, but polynomials cannot bend back down or shoot up to infinity — so a Taylor approximation is poor near a pole or far from the expansion point. Padé approximation upgrades the idea: instead of one polynomial, it uses a RATIO of two polynomials, a rational function, which can capture both the gentle and the explosive behaviour a function may have.
A Padé approximant of orders (m, n) is a rational function P(x) / Q(x), with P of degree m and Q of degree n, chosen so that its own Taylor series matches the target function's Taylor series for as many terms as possible — exactly m + n + 1 terms. You find the coefficients by writing the matching condition f(x) Q(x) = P(x) up to that order and solving a small linear system. The payoff is that the denominator Q can VANISH where the function blows up, so the approximant can reproduce poles and model functions over a far wider range than the truncated Taylor series it was built from. A rational function with the same total number of coefficients usually beats the polynomial badly: a low-order Padé approximant can match a long Taylor series and remain accurate where the polynomial has long since diverged.
Padé approximation is the workhorse for turning a few series terms into a globally useful formula: it appears in summing divergent series, in control theory (approximating time delays), in evaluating special functions, and as a building block of high-order numerical schemes. The honest cautions: the coefficients come from a linear system that can be ILL-CONDITIONED, and Padé can produce SPURIOUS pole-zero pairs (a near-cancelling pole and zero, called a Froissart doublet) that signal numerical noise rather than real features. And like all approximants built from local series data, it carries no guarantee of accuracy far from where the series was taken — verify before trusting it globally.
The (1,1) Padé approximant of e^x matches the series 1 + x + x^2/2 + ... and works out to (1 + x/2) / (1 - x/2). At x = 2 it gives 4, versus the true e^2 = 7.389; the same-length Taylor polynomial 1 + x + x^2/2 gives 5. Padé is closer and, unlike the polynomial, captures the growth.
A ratio of polynomials can model what a single polynomial cannot.
Padé can fabricate SPURIOUS pole-zero pairs (Froissart doublets) from rounding noise, and its linear system can be ill-conditioned. Matching a Taylor series locally gives NO guarantee of accuracy far away — always validate the approximant's range.