Padé approximant

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Author: Dr. George A. Baker Jr, Los Alamos National Laboratory, Group T-11, Theoretical Division, University of California, Los Alamos, NM 87545, USA

Dr. George A. Baker Jr accepted the invitation on 12 October 2009 (self-imposed deadline: 12 October 2010).

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Introduction

The relation between the Taylor series expansion and the function is given classically by the statement that if the series converges absolutely to an infinitely differentiable function, then the series defines the function uniquely and the function uniquely defines the series.

The Padé approximants are a particular type of rational approximation. The L, M Padé approximant is denoted by:

(1)
[L/M] = P_L(x)/Q_M(x)

where P_L(x) ia a polynomial of degree less than or equal to L, and Q_M(x) is a polynomial of degree less than or equal to M. Sometimes the function name f is appended as a subscript [L/M]_f where the function being approximated is not clear from the context. The formal power series

(2)
A(x) = \sum _{j=0}^\infty a_jx^j

determines the coefficients by the equation

(3)
A(x) -P_L(x)/Q_M(x)=O(x^{L+M+1})

This is the classical definition. The Baker definition adds the condition

(4)
Q_M(0)= 1.

An example is the Padé approximant to e^x.

(5)
[2/4]=\frac{360+120x+12x^2}{360-240x+72x^2-12x^3+x^4}

When it exists, the [L/M] Padé approximant is unique (Frobenius, 1881).

An encyclopedic discussion of the Padé Approximant is to be found in Baker and Graves-Morris (1996). Some of the material in this article is drawn directly from that source.


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Sample Padé Approximant Uses

One use is to compute the value of a function at a regular point. A sample case might be a(x)=[(1+2x)/(1+x)]^{1/2}. Another, more difficult, case would be an asymptotic series at an irregular point

(6)
b(x)=\int_0^\infty \frac {e^{-t}dt}{1+xt}=1-1! x +2! x^2-3! x^3 + \dots

The results of the use of the [M/M] at the point at infinity are:

[M/M] evaluated at infinity
M [(1+2x)/(1+x)]^{1/2} \int_0^\infty \frac {e^{-t}dt}{1+xt}
1 7/5 1/2
2 41/29 1/3
3 239/169 1/4
4 1393/985 1/5
5 1.414213552 1/6
Limit 1.414213552 0

The function a(z) is double valued, as the square root can be either plus or minus. The function b(z) is only unique in -i\pi < arg\ z < i\pi. The Padé approximant is single valued. The solution to this difference is to cut the complex plane in such a way that there is a single valued function. For a(z) the cut is on the negative real axis from -1 to -1/2. For b(z) the cut is the whole negative real axis. The Padé approximant simulates a branch cut in the complex plane by a line of poles and zeros.

Padé approximants can also be used to analyze the nature of singular points. As a sample, consider

(7)
f(x) = (1-x)^{-1.5}(1-\frac 12x)^{1.5}+ e^{-x}

The dominant singularity is at x=1 and shows a divergence like the 1.5 power of 1/(1-x). Then in the neighborhood of x=1 one would expect

(8)
\frac d{dx} \ln f(x) \approx \frac{-1.5}{x-1}

A sample of the Padé approximant results for this function is

Poles and residue of the [M/M]
M Poles Residues
2 0.96604 -1.25063
3 0.96526^a -1.24779^a
4 1.00471 -1.60153
5 1.00170 -1.54827
6 1.00034 -1.51563
7 1.00035^a -151605^a
8 1.00009 -1.50651
9 1.00005 -1.50443
Limit 1.00000 -1.50000

^a close pole and zero nearer to the origin.

These results show, that aside from the occasional interruption due to close poles and zeros, the procedure shows steady convergence and that the accuracy of the location is noticeably higher that that in determining the nature.

The occurrence of a close pole and zero as seen in this case is called a "defect". The defects only effect the behavior of the Padé approximant in the neighborhood where they occur and not elsewhere in the complex plane. They do however, slow the rate of convergence. As a note of history, when researchers first encountered this problem, it was supposed that there was a numerical error and the pole and zero should have canceled exactly. However further study with higher precision arithmetic showed that the phenomenon was real. There will be a more detailed discussion later in this article.

For the time being the calculation of the Padé approximant should be done separately for each approximant rather than use the recursion relations or the the determinetal method, which will be mentioned later. An appropriate method would be the solution of linear equations by the Gauss-Jordan elimination. The use of multiple precision arithmetic may sometimes be required as the defining equation can be ill-conditioned. Note that, as discussed in the Padé table section, the equation can be inconsistent.

More to follow...[ST, 19.11.09]

Suggested by: Prof. Jean Zinn-Justin, CEA, IRFU and Institut de Physique Théorique, Centre de Saclay, F-91191 Gif-sur-Yvette, France
Invited by: Dr. Riccardo Guida, Institut de Physique Théorique; CEA, IPhT; CNRS; Gif-sur-Yvette, France
Assistant editor: Mr. Sundeep Teki, Wellcome Trust Centre for Neuroimaging, University College London, UK
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