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Wikipedia / 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.

Definition

Given a function and two integers and , the Padé approximant of order is the rational function

which agrees with

is expanded in a Maclaurin series (Taylor series

When it exists, the Padé approximant is unique as a formal power series for the given m and n.

Computation

For given , Padé approximants can be computed by Wynn's epsilon algorithm and also other sequence transformations of the Taylor series of can also be a formal power series, and, hence, Padé approximants can also be applied to the summation of divergent series.

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.

Recall that, to compute the greatest common divisor of two polynomials p and q

For the has degree or smaller.

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

Riemann–Padé zeta function

To study the resummation of a divergent series is the Padé approximation of order of the function . The zeta regularization value at is taken to be the sum of the divergent series.

where and 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.

DLog Padé method

Padé approximants can be used to extract critical points and exponents of functions. In thermodynamics, if a function behaves in a non-analytic way near a point , one calls a critical point and the associated critical exponent of . If sufficient terms of the series expansion of .

Generalizations

A Padé approximant approximates a function in one variable. An approximant in two variables is called a Chisholm approximant (after J. S. R. Chisholm), in multiple variables a Canterbury approximant (after Graves-Morris at the University of Kent).

Two-points Padé approximant

The conventional Padé approximation is determined to reproduce the Maclaurin expansion up to a given order. Therefore, the approximation at the value apart from the expansion point may be poor. This is avoided by the 2-point Padé approximation, which is a type of multipoint summation method.

.

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

Multi-point Padé approximant

A further extension of the 2-point Padé approximant is the multi-point Padé approximant.

can be performed with higher accuracy.

Examples

  1. Jacobi

  2. Bessel

  3. Error

  4. Fresnel

See also

References

Literature