Padé approximant

In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order.

Under this technique, the approximant's power series agrees with the power series of the function it is approximating.

The technique was developed around 1890 by Henri Padé, but goes back to Georg Frobenius, 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 Taylor series, and it may still work where the Taylor series does not converge.

For these reasons Padé approximants are used extensively in computer calculations.

They have also been used as auxiliary functions in Diophantine approximation and transcendental number theory, 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.

When it exists, the Padé approximant is unique as a formal power series for the given m and n.[1] The Padé approximant defined above is also denoted as

For given x, Padé approximants can be computed by Wynn's epsilon algorithm[2] and also other sequence transformations[3] from the partial sums

One way to compute a Padé approximant is via the extended Euclidean algorithm for the polynomial greatest common divisor.

which can be interpreted as the Bézout identity of one step in the computation of the extended greatest common divisor of the polynomials

Recall that, to compute the greatest common divisor of two polynomials p and q, one computes via long division the remainder sequence

For the Bézout identities of the extended greatest common divisor one computes simultaneously the two polynomial sequences

For the [m/n] approximant, one thus carries out the extended Euclidean algorithm for

If one were to compute all steps of the extended greatest common divisor computation, one would obtain an anti-diagonal of the Padé table.

it can be useful to introduce the Padé or simply rational zeta function as

The zeta regularization value at s = 0 is taken to be the sum of the divergent series.

where aj and bj are the coefficients in the Padé approximation.

The subscript '0' means that the Padé is of order [0/0] and hence, we have the Riemann zeta function.

Padé approximants can be used to extract critical points and exponents of functions.

[5][6] In thermodynamics, if a function f(x) behaves in a non-analytic way near a point x = r like

[8] The conventional Padé approximation is determined to reproduce the Maclaurin expansion up to a given order.

This is avoided by the 2-point Padé approximation, which is a type of multipoint summation method.

such that simultaneously reproduce asymptotic behavior by developing the Padé approximation can be found in various cases.

There is a method of using this to give an approximate solution of a differential equation with high accuracy.

[9] Also, for the nontrivial zeros of the Riemann zeta function, the first nontrivial zero can be estimated with some accuracy from the asymptotic behavior on the real axis.

Consider the cases when singularities of a function are expressed with index

, this method approximates to reduce the property of diverging at