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{{Short description|'Best' approximation of a function by a rational function of given order}}

[[File:Henri Padé.jpeg|thumb|right|[[Henri Padé]]]] 
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 [[Convergent series|converge]]. For these reasons Padé approximants are used extensively in computer [[calculation]]s. They have also been used as [[auxiliary function]]s 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 on the method of truncating a Taylor series.

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

<!--<math display="block">R(x) = \frac{\sum_{j=0}^m p_j x^j}{1 + \sum_{k=1}^n q_k x^k} = \frac{p_0 + p_1 x + p_2 x^2 + \cdots + p_m x^m}{1 + q_1 x + q_2 x^2 + \cdots + q_n x^n}</math>-->
<math display="block">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},</math>
which agrees with {{math|''f''(''x'')}} to the highest possible order, which amounts to
<math display="block">
\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}
</math>

Equivalently, if <math>R(x)</math> is expanded in a Maclaurin series ([[Taylor series]] at 0), its first <math>m + n</math> terms would equal the first <math>m + n</math> terms of <math>f(x)</math>, and thus
<math display="block">f(x) - R(x) = c_{m+n+1} x^{m+n+1} + c_{m+n+2} x^{m+n+2} + \cdots</math>

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

The Padé approximant defined above is also denoted as
<math display="block">[m/n]_f(x).</math>

==Computation==
For given {{math|''x''}}, Padé approximants can be computed by [[Peter Wynn (mathematician)|Wynn]]'s epsilon algorithm<ref>Theorem 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}}</ref> and also other [[sequence transformation]]s<ref>{{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}}</ref> from the partial sums
<math display="block">T_N(x) = c_0 + c_1 x + c_2 x^2 + \cdots + c_N x^N </math>
of the [[Taylor series]] of {{math|''f''}}, i.e., we have
<math display="block">c_k = \frac{f^{(k)}(0)}{k!}.</math>
{{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]].<ref>{{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) }}</ref> The relation
<math display="block">R(x) = P(x)/Q(x) = T_{m+n}(x) \bmod x^{m+n+1}</math>
is equivalent to the existence of some factor <math>K(x)</math> such that
<math display="block">P(x) = Q(x)T_{m+n}(x) + K(x)x^{m+n+1},</math>
which can be interpreted as the [[Bézout's identity|Bézout identity]] of one step in the computation of the extended greatest common divisor of the polynomials <math>T_{m+n}(x)</math> and <math>x^{m+n+1}</math>.

Recall that, to compute the greatest common divisor of two polynomials ''p'' and ''q'', one computes via long division the remainder sequence
<math display="block">r_0=p,\;r_1=q,\quad r_{k-1}= q_k r_k +r_{k+1},</math>
{{math|1=''k'' = 1, 2, 3, ...}} with <math>\deg r_{k+1} < \deg r_k\,</math>, until <math>r_{k+1} = 0</math>. For the Bézout identities of the extended greatest common divisor one computes simultaneously the two polynomial sequences
<math display="block">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_kv_k</math>
to obtain in each step the Bézout identity
<math display="block">r_k(x) = u_k(x) p(x) + v_k(x) q(x).</math>

For the {{math|[''m''/''n'']}} approximant, one thus carries out the extended Euclidean algorithm for
<math display="block">r_0=x^{m+n+1},\;r_1=T_{m+n}(x)</math>
and stops it at the last instant that <math>v_k</math> has degree {{math|''n''}} or smaller.

Then the polynomials <math>P=r_k,\;Q=v_k</math> 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]].

==Riemann–Padé zeta function==
To study the resummation of a [[divergent series]], say
<math display="block"> \sum_{z=1}^\infty f(z), </math>
it can be useful to introduce the Padé or simply rational zeta function as
<math display="block"> \zeta_R(s) = \sum_{z=1}^\infty \frac{R(z)}{z^s}, </math>
where
<math display="block"> R(x) = [m/n]_f(x)</math>
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
<math display="block"> \sum_{j=0}^n a_j \zeta_R(s-j) = \sum_{j=0}^m b_j \zeta_0(s-j), </math>
where {{math|''a<sub>j</sub>''}} and {{math|''b<sub>j</sub>''}} 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]].

==DLog Padé method==
Padé approximants can be used to extract critical points and exponents of functions.<ref>{{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 }}</ref><ref>{{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 }}</ref> In thermodynamics, if a function {{math|''f''(''x'')}} behaves in a non-analytic way near a point {{math|1=''x'' = ''r''}} like <math>f(x) \sim |x - r|^p</math>, 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 <math>[n/n+1]_g(x)</math>, where <math>g = f'/f</math>.

==Generalizations==
A Padé approximant approximates a function in one variable.  An approximant in two variables is called a Chisholm approximant (after [[J. S. R. Chisholm]]),<ref>{{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}}</ref> in multiple variables a Canterbury approximant (after Graves-Morris at the University of Kent).<ref>{{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}}</ref>

==Two-points Padé approximant==
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.<ref name=":0">{{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}}</ref> At <math>x = 0</math>, consider a case that a function <math>f(x)</math> which is expressed by asymptotic behavior <math>f_0(x)</math>:
<math display="block">f \sim f_0(x) + o\big(f_0(x)\big), \quad x \to 0,</math> 
and at <math>x \to \infty</math>, additional asymptotic behavior <math>f_\infty(x)</math>:
<math display="block">f(x) \sim f_\infty(x) + o\big(f_\infty(x)\big), \quad x \to \infty.</math>

By selecting the major behavior of <math>f_0(x), f_\infty(x)</math>, approximate functions <math>F(x)</math> such that simultaneously reproduce asymptotic behavior by developing the Padé approximation can be found in various cases. As a result, at the point <math>x \to \infty</math>, 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 <math>x = 0 \sim \infty</math>.

In cases where <math>f_0(x), f_\infty(x)</math> are expressed by polynomials or series of negative powers, exponential function, logarithmic function or <math>x \ln x</math>, we can apply 2-point Padé approximant to <math>f(x)</math>. There is a method of using this to give an approximate solution of a differential equation with high accuracy.<ref name=":0" /> 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.<ref name=":0" />

== Multi-point Padé approximant ==
A further extension of the 2-point Padé approximant is the multi-point Padé approximant.<ref name=":0" /> This method treats singularity points <math>x = x_j(j = 1, 2, 3, \dots, N)</math> of a function <math>f(x)</math> which is to be approximated. Consider the cases when singularities of a function are expressed with index <math>n_j</math> by
<math display="block">f(x) \sim \frac{A_j}{(x - x_j)^{n_j}}, \quad x \to x_j.</math>

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

==Examples==
{{Unreferenced section|date=September 2018}}
;{{math|sin(''x'')}}<ref>{{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}}</ref> : <math display="block">\sin(x) \approx \frac{ \frac{12671}{4363920} x^5 - \frac{2363}{18183} x^3 + x }{ 1+ \frac{445}{12122} x^2 + \frac{601}{872784} x^4 + \frac{121}{16662240} x^6}</math>
;{{math|exp(''x'')}}<ref name="wolfram-alpha-pade-exp">{{WolframAlpha|id=PadeApproximant[Exp[x]%2C+{x%2C0%2C{5%2C5%7D%7D]|title=Padé approximant of exp(x) | accessdate=2024-01-03}}</ref> : <math display="block">\exp(x) \approx \frac{1 + \frac{1}{2} x + \frac{1}{9} x^2 + \frac{1}{72} x^3 + \frac{1}{1008} x^4 + \frac{1}{30240} x^5}{1 - \frac{1}{2} x + \frac{1}{9} x^2 - \frac{1}{72} x^3 + \frac{1}{1008} x^4 - \frac{1}{30240} x^5 }</math>
;{{math|ln(1+''x'')}}<ref name=":1">{{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}}</ref> : <math display="block">\ln(1+x) \approx \frac{x + \frac 1 2 x^2}{1 + x + \frac 1 6 x^2}</math>
;[[Jacobi elliptic functions|Jacobi]] {{math|sn(''z''{{!}}3)}}<ref>{{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}}</ref> : <math display="block">\mathrm{sn}(z|3) \approx \frac{ - \frac{9851629}{283609260} z^5 - \frac{572744}{4726821} z^3 + z }{ 1 + \frac{859490}{1575607} z^2 - \frac{5922035}{56721852} z^4 + \frac{62531591}{2977897230} z^6  }</math>
;[[Bessel function|Bessel]] {{math|''J''<sub>5</sub>(''x'')}} : <math display="block">J_5(x) \approx \frac{- \frac{107}{28416000} x^7 + \frac{1}{3840} x^5  }{ 1 + \frac{151}{5550} x^2 + \frac{1453}{3729600} x^4 + \frac{1339}{358041600} x^6 + \frac{2767}{120301977600} x^8  }</math>
;{{math|erf(''x'')}} : <math display="block">\operatorname{erf}(x) \approx \frac{2}{15\sqrt\pi} \cdot \frac{ 49140 x + 3570 x^3 + 739 x^5 }{ 165 x^4 + 1330 x^2 + 3276 }</math>
;[[Fresnel integral|Fresnel]] {{math|''C''(''x'')}} : <math display="block">C(x) \approx \frac{1}{135} \cdot \frac{ 990791\pi^4 x^9 - 147189744\pi^2 x^5 + 8714684160 x }{ 1749\pi^4 x^8 + 523536\pi^2 x^4 + 64553216 }</math>

==See also==
{{Portal|Mathematics}}
* {{annotated link|Padé table}}
* {{annotated link|Bhaskara I's sine approximation formula}}
* {{annotated link|Approximation theory}}
* {{annotated link|Function approximation}}

==References==
<references />

==Literature==
* Baker, G. A., Jr.; and Graves-Morris, P. '' Padé Approximants''.  [[Cambridge University Press|Cambridge U.P.]], 1996.
* Baker, G. A., Jr. [http://www.scholarpedia.org/article/Pad%C3%A9_approximant Padé approximant], [http://www.scholarpedia.org/ Scholarpedia], 7(6):9756.
* Brezinski, C.; [[Michela Redivo-Zaglia|Redivo Zaglia, M.]] ''Extrapolation Methods. Theory and Practice''.  [[North-Holland]], 1991. {{ISBN|978-0444888143}}
* {{Citation |last1=Press |first1=W. H. |last2=Teukolsky |first2=S. A. |last3=Vetterling |first3=W. T. |last4=Flannery |first4=B. P. |year=2007 |title=Numerical Recipes: The Art of Scientific Computing |edition=3rd |publisher=[[Cambridge University Press]] |location=New York |isbn=978-0-521-88068-8 |chapter=Section 5.12 Padé Approximants |chapter-url=http://apps.nrbook.com/empanel/index.html?pg=245 |access-date=2011-08-09 |archive-date=2016-03-03 |archive-url=https://web.archive.org/web/20160303232840/http://apps.nrbook.com/empanel/index.html?pg=245 |url-status=dead }}.
* Frobenius, G.; {{lang|de|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.&nbsp;14, No.&nbsp;1, 1972, pp.&nbsp;1–62.
* Padé, H.; {{lang|fr|Sur la répresentation approchée d'une fonction par des fractions rationelles}}, Thesis, [Ann. École Nor. (3), 9, 1892, pp.&nbsp;1–93 supplement. 
<!--[http://www.nrbook.com/a/bookcpdf/c5-12.pdf available online]-->
* {{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 }}.

==External links==
* {{mathworld|urlname=PadeApproximant|title=Padé Approximant}}
* [http://demonstrations.wolfram.com/PadeApproximants/ Padé Approximants], Oleksandr Pavlyk, [[The Wolfram Demonstrations Project]].
* [https://web.archive.org/web/20071213000637/http://rkb.home.cern.ch/rkb/AN16pp/node203.html#SECTION0002030000000000000000 Data Analysis BriefBook: Pade Approximation], Rudolf K. Bock  [[European Laboratory for Particle Physics]], [[CERN]].
* [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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[[Category:Sequences and series]]
[[Category:Numerical analysis]]
[[Category:Rational functions]]