Padé approximant

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

$$R(x)\; =\; \backslash frac\{\backslash sum\_\{j=0\}^m\; a\_j\; x^j\}\{1\; +\; \backslash sum\_\{k=1\}^n\; b\_k\; x^k\}\; =\; \backslash frac\{a\_0\; +\; a\_1\; x\; +\; a\_2\; x^2\; +\; \backslash dots\; +\; a\_m\; x^m\}\{1\; +\; b\_1\; x\; +\; b\_2\; x^2\; +\; \backslash dots\; +\; b\_n\; x^n\},$$

which agrees with {{math|f(x)}} to the highest possible order, which amounts to

\begin{align}

f(0) &= R(0), \\

f'(0) &= R'(0), \\

f(0) &= R(0), \\

&\mathrel{\;\vdots} \\

f^{(m+n)}(0) &= R^{(m+n)}(0).

\end{align}

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

$$f(x)\; -\; R(x)\; =\; c\_\{m+n+1\}\; x^\{m+n+1\}\; +\; c\_\{m+n+2\}\; x^\{m+n+2\}\; +\; \backslash cdots$$

When it exists, the Padé approximant is unique as a formal power series for the given m and n.{{Cite web |title=Padé Approximant |url=https://mathworld.wolfram.com/PadeApproximant.html |website=Wolfram MathWorld}}

The Padé approximant defined above is also denoted as

$$[m/n]\_f(x).$$

For given {{math|x}}, Padé approximants can be computed by Wynn's epsilon algorithmTheorem 1 in {{cite journal | title=On the Convergence and Stability of the Epsilon Algorithm| first=Peter | last=Wynn | journal=SIAM Journal on Numerical Analysis | volume=3 | date=Mar 1966 | pages=91–122| jstor=2949688| issue=1 | bibcode=1966SJNA....3...91W | doi=10.1137/0703007}} and also other sequence transformations{{cite journal | title=Extrapolation algorithms and Padé approximations | first=C. | last=Brezenski | journal=Applied Numerical Mathematics | volume=20 | year=1996 | pages=299–318 | doi=10.1016/0168-9274(95)00110-7 | issue=3 | citeseerx=10.1.1.20.9528}} from the partial sums

$$T\_N(x)\; =\; c\_0\; +\; c\_1\; x\; +\; c\_2\; x^2\; +\; \backslash cdots\; +\; c\_N\; x^N$$

of the Taylor series of {{math|f}}, i.e., we have

$$c\_k\; =\; \backslash frac\{f^\{(k)\}(0)\}\{k!\}.$$

{{math|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.{{cite book|title=Polynomial and Matrix computations - Volume 1. Fundamental Algorithms|first1=Dario | last1=Bini| first2=Victor | last2=Pan | publisher=Birkhäuser| series=Progress in Theoretical Computer Science | year=1994| isbn=978-0-8176-3786-6 | at = Problem 5.2b and Algorithm 5.2 (p. 46) }} The relation

$$R(x)\; =\; P(x)/Q(x)\; =\; T\_\{m+n\}(x)\; \backslash bmod\; x^\{m+n+1\}$$

is equivalent to the existence of some factor $K(x)$ such that

$$P(x)\; =\; Q(x)T\_\{m+n\}(x)\; +\; K(x)x^\{m+n+1\},$$

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

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

$$r\_0=p,\backslash ;r\_1=q,\backslash quad\; r\_\{k-1\}=\; q\_k\; r\_k\; +r\_\{k+1\},$$

{{math|1=k = 1, 2, 3, ...}} with , until $r\_\{k+1\}\; =\; 0$. For the Bézout identities of the extended greatest common divisor one computes simultaneously the two polynomial sequences

$$u\_0=1,\backslash ;v\_0=0,\backslash quad\; u\_1=0,\backslash ;v\_1=1,\backslash quad\; u\_\{k+1\}=u\_\{k-1\}-q\_k\; u\_k,\backslash ;v\_\{k+1\}=v\_\{k-1\}-q\_kv\_k$$

to obtain in each step the Bézout identity

$$r\_k(x)\; =\; u\_k(x)\; p(x)\; +\; v\_k(x)\; q(x).$$

For the {{math|[m/n]}} approximant, one thus carries out the extended Euclidean algorithm for

$$r\_0=x^\{m+n+1\},\backslash ;r\_1=T\_\{m+n\}(x)$$

and stops it at the last instant that $v\_k$ has degree {{math|n}} or smaller.

Then the polynomials $P=r\_k,\backslash ;Q=v\_k$ give the {{math|[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

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

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

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

where

$$R(x)\; =\; [m/n]\_f(x)$$

is the Padé approximation of order {{math|(m, n)}} of the function {{math|f(x)}}. The zeta regularization value at {{math|1=s = 0}} is taken to be the sum of the divergent series.

The functional equation for this Padé zeta function is

$$\backslash sum\_\{j=0\}^n\; a\_j\; \backslash zeta\_R(s-j)\; =\; \backslash sum\_\{j=0\}^m\; b\_j\; \backslash zeta\_0(s-j),$$

where {{math|a_j}} and {{math|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.{{cite journal |last1=Adler |first1=Joan |title=Series expansions |journal=Computers in Physics |date=1994 |volume=8 |issue=3 |page=287 |doi=10.1063/1.168493 |bibcode=1994ComPh...8..287A |doi-access=free }}{{cite journal |last1=Baker |first1=G. A. Jr. | title=Padé approximant |journal=Scholarpedia |date=2012 |volume=7 |issue=6 |page=9756 |doi=10.4249/scholarpedia.9756 |bibcode=2012SchpJ...7.9756B |doi-access=free }} In thermodynamics, if a function {{math|f(x)}} behaves in a non-analytic way near a point {{math|1=x = r}} like $f(x)\; \backslash sim\; |x\; -\; r|^p$, one calls {{math|1=x = r}} a critical point and {{math|p}} the associated critical exponent of {{math|f}}. If sufficient terms of the series expansion of {{math|f}} are known, one can approximately extract the critical points and the critical exponents from respectively the poles and residues of the Padé approximants $[n/n+1]\_g(x)$, where $g\; =\; f\text{'}/f$.

A Padé approximant approximates a function in one variable. An approximant in two variables is called a Chisholm approximant (after J. S. R. Chisholm),{{Cite journal| last=Chisholm|first=J. S. R.| date=1973|title=Rational approximants defined from double power series| journal=Mathematics of Computation| volume=27| issue=124|pages=841–848| doi=10.1090/S0025-5718-1973-0382928-6| issn=0025-5718| doi-access=free}} in multiple variables a Canterbury approximant (after Graves-Morris at the University of Kent).{{Cite journal|last1=Graves-Morris|first1=P.R. |last2=Roberts|first2=D.E. |date=1975 |title=Calculation of Canterbury approximants |journal=Computer Physics Communications |volume=10|issue=4 |pages=234–244 |doi=10.1016/0010-4655(75)90068-5 |bibcode=1975CoPhC..10..234G}}

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.{{Cite book |last=Ueoka |first=Yoshiki |url=https://www.amazon.co.jp/dp/B08DXP73ZB/ |title=Introduction to multipoints summation method Modern applied mathematics that connects here and the infinite beyond: From Taylor expansion to application of differential equations}} At $x\; =\; 0$, consider a case that a function $f(x)$ which is expressed by asymptotic behavior $f\_0(x)$:

$$f\; \backslash sim\; f\_0(x)\; +\; o\backslash big(f\_0(x)\backslash big),\; \backslash quad\; x\; \backslash to\; 0,$$

and at $x\; \backslash to\; \backslash infty$, additional asymptotic behavior $f\_\backslash infty(x)$:

$$f(x)\; \backslash sim\; f\_\backslash infty(x)\; +\; o\backslash big(f\_\backslash infty(x)\backslash big),\; \backslash quad\; x\; \backslash to\; \backslash infty.$$

By selecting the major behavior of $f\_0(x),\; f\_\backslash infty(x)$, approximate functions $F(x)$ such that simultaneously reproduce asymptotic behavior by developing the Padé approximation can be found in various cases. As a result, at the point $x\; \backslash to\; \backslash infty$, 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 $x\; =\; 0\; \backslash sim\; \backslash infty$.

In cases where $f\_0(x),\; f\_\backslash infty(x)$ are expressed by polynomials or series of negative powers, exponential function, logarithmic function or $x\; \backslash ln\; x$, we can apply 2-point Padé approximant to $f(x)$. 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.

A further extension of the 2-point Padé approximant is the multi-point Padé approximant. This method treats singularity points $x\; =\; x\_j(j\; =\; 1,\; 2,\; 3,\; \backslash dots,\; N)$ of a function $f(x)$ which is to be approximated. Consider the cases when singularities of a function are expressed with index $n\_j$ by

$$f(x)\; \backslash sim\; \backslash frac\{A\_j\}\{(x\; -\; x\_j)^\{n\_j\}\},\; \backslash quad\; x\; \backslash to\; x\_j.$$

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

{{Unreferenced section|date=September 2018}}

;{{math|sin(x)}}{{WolframAlpha | id=PadeApproximant%5BSin%5Bx%5D%2C%7Bx%2C0%2C%7B5%2C6%7D%7D%5D|title=Padé approximant of sin(x) | accessdate=2022-01-16}} : $$\backslash sin(x)\; \backslash approx\; \backslash frac\{\; \backslash frac\{12671\}\{4363920\}\; x^5\; -\; \backslash frac\{2363\}\{18183\}\; x^3\; +\; x\; \}\{\; 1+\; \backslash frac\{445\}\{12122\}\; x^2\; +\; \backslash frac\{601\}\{872784\}\; x^4\; +\; \backslash frac\{121\}\{16662240\}\; x^6\}$$

;{{math|exp(x)}}{{WolframAlpha|id=PadeApproximant[Exp[x]%2C+{x%2C0%2C{5%2C5%7D%7D]|title=Padé approximant of exp(x) | accessdate=2024-01-03}} : $$\backslash exp(x)\; \backslash approx\; \backslash frac\{1\; +\; \backslash frac\{1\}\{2\}\; x\; +\; \backslash frac\{1\}\{9\}\; x^2\; +\; \backslash frac\{1\}\{72\}\; x^3\; +\; \backslash frac\{1\}\{1008\}\; x^4\; +\; \backslash frac\{1\}\{30240\}\; x^5\}\{1\; -\; \backslash frac\{1\}\{2\}\; x\; +\; \backslash frac\{1\}\{9\}\; x^2\; -\; \backslash frac\{1\}\{72\}\; x^3\; +\; \backslash frac\{1\}\{1008\}\; x^4\; -\; \backslash frac\{1\}\{30240\}\; x^5\; \}$$

;{{math|ln(1+x)}}{{WolframAlpha|id=PadeApproximant%5BLog%5B1%2Bx%5D%2C%7Bx%2C0%2C%7B2%2C2%7D%7D%5D|title=Padé approximant of log(1+x)|accessdate=2023-09-16}} : $$\backslash ln(1+x)\; \backslash approx\; \backslash frac\{x\; +\; \backslash frac\; 1\; 2\; x^2\}\{1\; +\; x\; +\; \backslash frac\; 1\; 6\; x^2\}$$

;Jacobi {{math|sn(z{{!}}3)}}{{WolframAlpha |id=PadeApproximant%5BJacobiSN%5Bz%2C3%5D%2C%7Bz%2C0%2C%7B5%2C6%7D%7D%5D|title=Padé approximant of sn(x{{!}}3) | accessdate=2022-01-16}} : $$\backslash mathrm\{sn\}(z|3)\; \backslash approx\; \backslash frac\{\; -\; \backslash frac\{9851629\}\{283609260\}\; z^5\; -\; \backslash frac\{572744\}\{4726821\}\; z^3\; +\; z\; \}\{\; 1\; +\; \backslash frac\{859490\}\{1575607\}\; z^2\; -\; \backslash frac\{5922035\}\{56721852\}\; z^4\; +\; \backslash frac\{62531591\}\{2977897230\}\; z^6\; \}$$

;Bessel {{math|J₅(x)}} : $$J\_5(x)\; \backslash approx\; \backslash frac\{-\; \backslash frac\{107\}\{28416000\}\; x^7\; +\; \backslash frac\{1\}\{3840\}\; x^5\; \}\{\; 1\; +\; \backslash frac\{151\}\{5550\}\; x^2\; +\; \backslash frac\{1453\}\{3729600\}\; x^4\; +\; \backslash frac\{1339\}\{358041600\}\; x^6\; +\; \backslash frac\{2767\}\{120301977600\}\; x^8\; \}$$

;{{math|erf(x)}} : $$\backslash operatorname\{erf\}(x)\; \backslash approx\; \backslash frac\{2\}\{15\backslash sqrt\backslash pi\}\; \backslash cdot\; \backslash frac\{\; 49140\; x\; +\; 3570\; x^3\; +\; 739\; x^5\; \}\{\; 165\; x^4\; +\; 1330\; x^2\; +\; 3276\; \}$$

;Fresnel {{math|C(x)}} : $$C(x)\; \backslash approx\; \backslash frac\{1\}\{135\}\; \backslash cdot\; \backslash frac\{\; 990791\backslash pi^4\; x^9\; -\; 147189744\backslash pi^2\; x^5\; +\; 8714684160\; x\; \}\{\; 1749\backslash pi^4\; x^8\; +\; 523536\backslash pi^2\; x^4\; +\; 64553216\; \}$$

{{Portal|Mathematics}}

• {{annotated link|Padé table}}
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• {{Citation |first1=P. |last1=Wynn |authorlink=Peter Wynn (mathematician) |journal=Numerische Mathematik |title=Upon systems of recursions which obtain among the quotients of the Padé table |volume=8 |issue=3 |pages=264–269 |doi=10.1007/BF02162562 |year=1966 |s2cid=123789548 }}.

• {{mathworld|urlname=PadeApproximant|title=Padé Approximant}}
• [http://demonstrations.wolfram.com/PadeApproximants/ Padé Approximants], Oleksandr Pavlyk, The Wolfram Demonstrations Project.
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• [https://web.archive.org/web/20100220110347/http://www.dattalo.com/technical/theory/sinewave.html Sinewave], Scott Dattalo, last accessed 2010-11-11.
• [http://www.mathworks.com/help/toolbox/control/ref/pade.html MATLAB function] for Padé approximation of models with time delays.

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