Padé approximant

Given a function f and two integers m ≥ 0 and n ≥ 1, the Padé approximant of order [m/n] is the rational function

${\displaystyle R(x)={\frac {\sum _{j=0}^{m}a_{j}x^{j}}{1+\sum _{k=1}^{n}b_{k}x^{k}}}={\frac {a_{0}+a_{1}x+a_{2}x^{2}+\dots +a_{m}x^{m}}{1+b_{1}x+b_{2}x^{2}+\dots +b_{n}x^{n}}},}$

which agrees with f(x) to the highest possible order, which amounts to

${\displaystyle {\begin{aligned}f(0)&=R(0),\\f'(0)&=R'(0),\\f''(0)&=R''(0),\\&\mathrel {\;\vdots } \\f^{(m+n)}(0)&=R^{(m+n)}(0).\end{aligned}}}$

Equivalently, if ${\displaystyle R(x)}$ is expanded in a Maclaurin series (Taylor series at 0), its first ${\displaystyle m+n}$ terms would equal the first ${\displaystyle m+n}$ terms of ${\displaystyle f(x)}$, and thus

${\displaystyle f(x)-R(x)=c_{m+n+1}x^{m+n+1}+c_{m+n+2}x^{m+n+2}+\dots }$

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

${\displaystyle [m/n]_{f}(x).}$

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

${\displaystyle T_{N}(x)=c_{0}+c_{1}x+c_{2}x^{2}+\cdots +c_{N}x^{N}}$

of the Taylor series of f, i.e., we have

${\displaystyle c_{k}={\frac {f^{(k)}(0)}{k!}}.}$

f 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 relation

${\displaystyle R(x)=P(x)/Q(x)=T_{m+n}(x){\text{ mod }}x^{m+n+1}}$

is equivalent to the existence of some factor ${\displaystyle K(x)}$ such that

${\displaystyle P(x)=Q(x)T_{m+n}(x)+K(x)x^{m+n+1},}$

which can be interpreted as the Bézout identity of one step in the computation of the extended greatest common divisor of the polynomials ${\displaystyle T_{m+n}(x)}$ and ${\displaystyle x^{m+n+1}}$.

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

${\displaystyle r_{0}=p,\;r_{1}=q,\quad r_{k-1}=q_{k}r_{k}+r_{k+1},}$

k = 1, 2, 3, ... with ${\displaystyle \deg r_{k+1}<\deg r_{k}\,}$, until ${\displaystyle r_{k+1}=0}$. For the Bézout identities of the extended greatest common divisor one computes simultaneously the two polynomial sequences

${\displaystyle u_{0}=1,\;v_{0}=0,\quad u_{1}=0,\;v_{1}=1,\quad u_{k+1}=u_{k-1}-q_{k}u_{k},\;v_{k+1}=v_{k-1}-q_{k}v_{k}}$

to obtain in each step the Bézout identity

${\displaystyle r_{k}(x)=u_{k}(x)p(x)+v_{k}(x)q(x).}$

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

${\displaystyle r_{0}=x^{m+n+1},\;r_{1}=T_{m+n}(x)}$

and stops it at the last instant that ${\displaystyle v_{k}}$ has degree n or smaller.

Then the polynomials ${\displaystyle P=r_{k},\;Q=v_{k}}$ give the [m/n] 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.

To study the resummation of a divergent series, say

${\displaystyle \sum _{z=1}^{\infty }f(z),}$

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

${\displaystyle \zeta _{R}(s)=\sum _{z=1}^{\infty }{\frac {R(z)}{z^{s}}},}$

where

${\displaystyle R(x)=[m/n]_{f}(x)\,}$

is the Padé approximation of order (m, n) of the function f(x). The zeta regularization value at s = 0 is taken to be the sum of the divergent series.

The functional equation for this Padé zeta function is

${\displaystyle \sum _{j=0}^{n}a_{j}\zeta _{R}(s-j)=\sum _{j=0}^{m}b_{j}\zeta _{0}(s-j),}$

where a_j and b_j 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 ${\displaystyle f(x)\sim |x-r|^{p}}$, one calls x = r a critical point and p the associated critical exponent of f. If sufficient terms of the series expansion of f are known, one can approximately extract the critical points and the critical exponents from respectively the poles and residues of the Padé approximants ${\displaystyle [n/n+1]_{g}(x)}$, where ${\displaystyle g=f'/f}$.

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 ${\displaystyle x=0}$, consider a case that a function ${\displaystyle f(x)}$ which is expressed by asymptotic behavior ${\displaystyle f_{0}(x)}$:

${\displaystyle f\sim f_{0}(x)+o{\big (}f_{0}(x){\big )},\quad x\to 0,}$

and at ${\displaystyle x\to \infty }$, additional asymptotic behavior ${\displaystyle f_{\infty }(x)}$:

${\displaystyle f(x)\sim f_{\infty }(x)+o{\big (}f_{\infty }(x){\big )},\quad x\to \infty .}$

By selecting the major behavior of ${\displaystyle f_{0}(x),f_{\infty }(x)}$, approximate functions ${\displaystyle F(x)}$ such that simultaneously reproduce asymptotic behavior by developing the Padé approximation can be found in various cases. As a result, at the point ${\displaystyle x\to \infty }$, where the accuracy of the approximation may be the worst in the ordinary Padé approximation, good accuracy of the 2-point Padé approximant is guaranteed. Therefore, the 2-point Padé approximant can be a method that gives a good approximation globally for ${\displaystyle x=0\sim \infty }$.

In cases where ${\displaystyle f_{0}(x),f_{\infty }(x)}$ are expressed by polynomials or series of negative powers, exponential function, logarithmic function or ${\displaystyle x\ln x}$, we can apply 2-point Padé approximant to ${\displaystyle f(x)}$. 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.[9]

A further extension of the 2-point Padé approximant is the multi-point Padé approximant.[9] This method treats singularity points ${\displaystyle x=x_{j}(j=1,2,3,\dots ,N)}$ of a function ${\displaystyle f(x)}$ which is to be approximated. Consider the cases when singularities of a function are expressed with index ${\displaystyle n_{j}}$ by

${\displaystyle f(x)\sim {\frac {A_{j}}{(x-x_{j})^{n_{j}}}},\quad x\to x_{j}.}$

Besides the 2-point Padé approximant, which includes information at ${\displaystyle x=0,x\to \infty }$, this method approximates to reduce the property of diverging at ${\displaystyle x\sim x_{j}}$. As a result, since the information of the peculiarity of the function is captured, the approximation of a function ${\displaystyle f(x)}$ can be performed with higher accuracy.

sin(x)[10]

${\displaystyle \sin(x)\approx {\frac {(12671/4363920)x^{5}-(2363/18183)x^{3}+x}{1+(445/12122)x^{2}+(601/872784)x^{4}+(121/16662240)x^{6}}}}$

exp(x)[11]

${\displaystyle \exp(x)\approx {\frac {1+(1/2)x+(1/9)x^{2}+(1/72)x^{3}+(1/1008)x^{4}+(1/30240)x^{5}}{1-(1/2)x+(1/9)x^{2}-(1/72)x^{3}+(1/1008)x^{4}-(1/30240)x^{5}}}}$

Log(1+x)[11]

${\displaystyle \log(x)\approx {\frac {x+{\frac {1}{2}}x^{2}}{1+x+{\frac {1}{6}}x^{2}}}}$

Jacobi sn(z|3)[12]

${\displaystyle \mathrm {sn} (z|3)\approx {\frac {-(9851629/283609260)z^{5}-(572744/4726821)z^{3}+z}{1+(859490/1575607)z^{2}-(5922035/56721852)z^{4}+(62531591/2977897230)z^{6}}}}$

Bessel J(5, x)

${\displaystyle J_{5}(x)\approx {\frac {-(107/28416000)x^{7}+(1/3840)x^{5}}{1+(151/5550)x^{2}+(1453/3729600)x^{4}+(1339/358041600)x^{6}+(2767/120301977600)x^{8}}}}$

erf(x)

${\displaystyle \operatorname {erf} (x)\approx {\frac {(2/15)\cdot (49140x+3570x^{3}+739x^{5})}{{\sqrt {\pi }}\cdot (165x^{4}+1330x^{2}+3276)}}}$

Fresnel C(x)

${\displaystyle C(x)\approx {\frac {(1/135)\cdot (990791x^{9}\pi ^{4}-147189744x^{5}\pi ^{2}+8714684160x)}{(1749\pi ^{4}x^{8}+523536\pi ^{2}x^{4}+64553216)}}}$

• Padé table
• Bhaskara I's sine approximation formula

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• Baker, G. A., Jr. Padé approximant, Scholarpedia, 7(6):9756.
• Brezinski, C.; Redivo Zaglia, M. Extrapolation Methods. Theory and Practice. North-Holland, 1991. ISBN 978-0444888143
• Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007), "Section 5.12 Padé Approximants", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8.
• Frobenius, G.; Ueber Relationen zwischen den Näherungsbrüchen von Potenzreihen, [Journal für die reine und angewandte Mathematik (Crelle's Journal)]. Volume 1881, Issue 90, Pages 1–17.
• Gragg, W. B.; The Pade Table and Its Relation to Certain Algorithms of Numerical Analysis [SIAM Review], Vol. 14, No. 1, 1972, pp. 1–62.
• Padé, H.; Sur la répresentation approchée d'une fonction par des fractions rationelles, Thesis, [Ann. École Nor. (3), 9, 1892, pp. 1–93 supplement.
• Wynn, P. (1966), "Upon systems of recursions which obtain among the quotients of the Padé table", Numerische Mathematik, 8 (3): 264–269, doi:10.1007/BF02162562, S2CID 123789548.

• Weisstein, Eric W. "Padé Approximant". MathWorld.
• Padé Approximants, Oleksandr Pavlyk, The Wolfram Demonstrations Project.
• Data Analysis BriefBook: Pade Approximation, Rudolf K. Bock European Laboratory for Particle Physics, CERN.
• Sinewave, Scott Dattalo, last accessed 2010-11-11.
• MATLAB function for Padé approximation of models with time delays.