Padé approximant

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a Padé approximant is the "best" approximation of a function near a specific point by a <a href="/facts/Rational_function/2nJGu0Ko">rational function</a> of given order. Under this technique, the approximant's <a href="/facts/Power_series/vYh6sTps">power series</a> agrees with the power series of the function it is approximating. The technique was developed around 1890 by <a href="/facts/Henri_Pad%C3%A9/H5AH3Uhe">Henri Padé</a>, but goes back to <a href="/facts/Georg_Frobenius/4cXXQyu5">Georg Frobenius</a>, who introduced the idea and investigated the features of rational approximations of power series.
The Padé approximant often gives better approximation of the function than truncating its <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a>, and it may still work where the Taylor series does not <a href="/facts/Convergent_series/DTok170z">converge</a>. For these reasons Padé approximants are used extensively in computer <a href="/facts/Calculation/MHq42Sjw">calculations</a>. They have also been used as <a href="/facts/Auxiliary_function/QXX80Rnk">auxiliary functions</a> in <a href="/facts/Diophantine_approximation/WGVMuWF4">Diophantine approximation</a> and <a href="/facts/Transcendental_number_theory/tjyLHm5G">transcendental number theory</a>, though for sharp results ad hoc methods—in some sense inspired by the Padé theory—typically replace them. Since a Padé approximant is a rational function, an artificial singular point may occur as an approximation, but this can be avoided by Borel–Padé analysis.
The reason the Padé approximant tends to be a better approximation than a truncating Taylor series is clear from the viewpoint of the multi-point summation method. Since there are many cases in which the asymptotic expansion at infinity becomes 0 or a constant, it can be interpreted as the "incomplete two-point Padé approximation", in which the ordinary Padé approximation improves on the method of truncating a Taylor series.

Padé approximant open-in-new

Padé approximant