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 the method truncating a Taylor series.
Given a function and two integers and, the Padé approximant of order is the rational function
which agrees with to the highest possible order, which amounts to
Equivalently, if
R(x)
m+n
m+n
f(x)
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, Padé approximants can be computed by Wynn's epsilon algorithm[2] and also other sequence transformations[3] from the partial sumsof the Taylor series of, i.e., we have 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.[4] The relationis equivalent to the existence of some factor
K(x)
Tm+n(x)
xm+n+1
Recall that, to compute the greatest common divisor of two polynomials p and q, one computes via long division the remainder sequence with
\degrk+1<\degrk
rk+1=0
For the approximant, one thus carries out the extended Euclidean algorithm forand stops it at the last instant that
vk
Then the polynomials
P=rk, Q=vk
To study the resummation of a divergent series, sayit can be useful to introduce the Padé or simply rational zeta function aswhereis the Padé approximation of order of the function . The zeta regularization value at is taken to be the sum of the divergent series.
The functional equation for this Padé zeta function iswhere 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.
Padé approximants can be used to extract critical points and exponents of functions.[5] [6] In thermodynamics, if a function behaves in a non-analytic way near a point like
f(x)\sim|x-r|p
[n/n+1]g(x)
g=f'/f
A Padé approximant approximates a function in one variable. An approximant in two variables is called a Chisholm approximant (after J. S. R. Chisholm),[7] in multiple variables a Canterbury approximant (after Graves-Morris at the University of Kent).[8]
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.[9] At
x=0
f(x)
f0(x)
x\toinfty
finfty(x)
By selecting the major behavior of
f0(x),finfty(x)
F(x)
x\toinfty
x=0\siminfty
In cases where
f0(x),finfty(x)
xlnx
f(x)
A further extension of the 2-point Padé approximant is the multi-point Padé approximant. This method treats singularity points
x=xj(j=1,2,3,...,N)
f(x)
nj
Besides the 2-point Padé approximant, which includes information at
x=0,x\toinfty
x\simxj
f(x)