Padé Approximation

The Padé approximation is a rational function that can be thought of as a generalization of a Taylor polynomial. A rational function is the ratio of polynomials. Because these functions only use the elementary arithmetic operations, they are very easy to evaluate numerically. The polynomial in the denominator allows you to approximate functions that have rational singularities.

Padé approximations.

More precisely, a Padé approximation of order (n,m) to an analytic function f(x) at a regular point or pole (Subscript[x, 0]) is the rational function ((p(x))/(q(x))) where p(x) is a polynomial of degree n, q(x) is a polynomial of degree m, and the formal power series of f(x)q(x)-p(x) about the point (Subscript[x, 0]) begins with the term (((x-(Subscript[x, 0])))^(n+m+1)). If m is equal to n, the approximation is called a diagonal Padé approximation of order n.

Here is the Padé approximation of order (2,4) to cos(x) at x==0:

PadeApproximant[Cos[x],{x,0,{2,4}}]

This gives another Padé approximation of the same order:

pd=PadeApproximant[(E^x),{x,1,{2,4}}]

The initial terms of this series vanish. This is the property that characterizes the Padé approximation:

Series[(E^x) Denominator[pd]-Numerator[pd],{x,1,8}]

This plots the difference between the approximation and the true function. Notice that the approximation is very good near the center of expansion, but the error increases rapidly as you move away:

Plot[pd-(E^x),{x,0,2}]

In the Wolfram Language (RefLink[PadeApproximant, paclet:ref/PadeApproximant]) is generalized to allow expansion about branch points.

This gives the diagonal Padé approximation of order 1 to a generalized rational function at x==0:

PadeApproximant[((Sqrt[x])/((1+Sqrt[x])^3)),{x,0,1}]

This gives the diagonal Padé approximation of order 5 to the logarithm of a rational function at the branch point x==0:

PadeApproximant[Log[(x/(1+x))],{x,0,5}]

The series expansion of the function agrees with the diagonal Padé approximation up to order 10:

Series[%-Log[(x/(1+x))],{x,0,11}]