查看“︁帕德近似”︁的源代码

{{NoteTA
|G1=Math
}}
[[File:Henri Padé.jpeg|thumb|亨利·帕德]]
'''帕德近似'''（{{lang-en|Padé approximant}}）是[[法国]][[数学家]][[亨利·帕德]]发明的有理多项式近似法。帕德近似往往比截断的[[泰勒級數]]准确，而且当泰勒级数不收敛时，帕德近似往往仍可行，所以多用于在[[电子计算机|计算机]]数学中。

例如<math>\frac{1}{1-x}</math>的泰勒级数

<math>1+x+x^2+x^3+\cdots</math>只有在<math>-1<x<1</math>时收敛，不如原函数广泛。

==定义==
给定[[自然数]]m和正整数n, 函数 <math>f(x)</math>的[m,n]阶帕德近似为

<!--:<math>R(x)= \frac{\sum_{j=0}^{m}p_j x^j}{1+\sum_{k=1}^{n}q_k x^k}=\frac{p_0+p_1x+p_2x^2+\cdots+p_mx^m}{1+q_1 x+q_2x^2+\cdots+q_nx^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_1x+a_2x^2+\cdots+a_mx^m}{1+b_1 x+b_2x^2+\cdots+b_nx^n}</math>

并且

<math display="block">\begin{array}{rcl}
        f(0)&=&R(0)\\
       f'(0)&=&R'(0)\\
      f''(0)&=&R''(0)\\
            &\vdots& \\
f^{(m+n)}(0)&=&R^{(m+n)}(0)\end{array}</math>

对于给定的<math>m,n</math>函数<math>f(x)</math>的[m,n]阶帕德近似是唯一的。

函数<math>f(x)</math>的帕德近似记为

<math display="block">[m/n]_f(x). \,</math>

==例子==
===[[正弦]]函數===
<math display="block">[6/6]_{\sin(x)}=\frac{ (12671/4363920)*x^5-(2363/18183)*x^3+x  }{ 1+(445/12122)*x^2+(601/872784)*x^4+(121/16662240)*x^6    }</math>

<math>[6/6]_{\sin(x)}</math>的6+6=12阶泰勒级数展开为

<math display="block">{x-(1/6)*x^3+(1/120)*x^5-(1/5040)*x^7+(1/362880)*x^9-(1/39916800)*x^{11}+O(x^{13})}</math>
和<math>\sin(x)</math>的12阶泰勒级数全同：

<math display="block">\sin(x)\approx {x-(1/6)*x^3+(1/120)*x^5-(1/5040)*x^7+(1/362880)*x^9-(1/39916800)*x^{11}+O(x^{13})}</math>

===[[指数函数]]===
<math display="block">[5/5]_{exp(x)}=\frac{1+(1/9)*x^2+(1/2)*x+(1/72)*x^3+(1/1008)*x^4+(1/30240)*x^5}{1+(1/9)*x^2-(1/2)*x-(1/72)*x^3+(1/1008)*x^4-(1/30240)*x^5   }</math>
其泰勒级数为
<math display="block">{1+x+(1/2)*x^2+(1/6)*x^3+(1/24)*x^4+(1/120)*x^5+(1/720)*x^6+(1/5040)*x^7+(1/40320)*x^8+(1/362880)*x^9+(1/3628800)*x^{10}+(23/914457600)*x^{11}+O(x^{12})}</math>

与exp(x)本身的泰勒级数展开的前10阶完全等同：
<math display="block">{1+x+(1/2)*x^2+(1/6)*x^3+(1/24)*x^4+(1/120)*x^5+(1/720)*x^6+(1/5040)*x^7+(1/40320)*x^8+(1/362880)*x^9+(1/3628800)*x^{10}+(1/39916800)*x^{11}+O(x^{12})}</math>

:又如
<math display="block">f := \frac{1-\cos(2*x)^2}{1+\arctan(3*x)}</math>

<math display="block">[3/3]_{f(x)}=\frac{(64/75)*x^3+4*x^2}{ 1+(241/75)*x+(148/75)*x^2-(1061/225)*x^3  }</math>


===[[雅可比橢圓函數]] <math>\operatorname{sn}(x; 3)</math>===
<math display="block">\frac{ -(9853969/39583665)*z^5-(1493060/2638911)*z^3+z }{ 1+(968375/879637)*z^2-(1167506/7916733)*z^4+(867043/2159109)*z^6  }</math>

===[[贝塞尔函数#第一类贝塞尔函数|第一類 5 階貝塞爾函數]] <math>J_5(x)</math>===
<math display="block">\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  }</math>

===[[误差函数]]===
<math display="block">\frac{  (2/15)*(49140*x+3570*x^3+739*x^5)}{(165*\sqrt\pi*x^4+1330*\sqrt\pi*x^2+3276*\sqrt\pi)}{ }</math>

===[[菲涅耳積分]] <math>C(x)</math>===
<math display="block">\frac{ (1/135)*(990791*x^9*\pi^4-147189744*x^5*\pi^2+8714684160*x)}{(1749*\pi^4*x^8+523536*\pi^2*x^4+64553216) }</math>

==Maple计算==
Maple中

pade(f(x),x,[m,n]);

其中 m,n 分别表示 分子、分母的级数；

==参考文献==
* Baker, G. A., Jr.; and Graves-Morris, P. '' Padé Approximants''.  Cambridge U.P., 1996
* Baker, G. A., Jr. [http://www.scholarpedia.org/article/Pad%C3%A9_approximant Padé approximant] {{Wayback|url=http://www.scholarpedia.org/article/Pad%C3%A9_approximant |date=20210121023401 }}, [http://www.scholarpedia.org/ Scholarpedia] {{Wayback|url=http://www.scholarpedia.org/ |date=20111125192650 }}, 7(6):9756.
* Brezinski, C.; and Redivo Zaglia, M. ''Extrapolation Methods.= Theory and Practice''.  North-Holland, 1991
* {{Citation | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=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 | accessdate=2015-02-24 | archive-date=2016-03-03 | archive-url=https://web.archive.org/web/20160303232840/http://apps.nrbook.com/empanel/index.html?pg=245 | dead-url=no }}
* Frobenius, G.; ''Ueber Relationen zwischem 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.&nbsp;1–62.
* Padé, H.; ''Sur la répresentation approchée d'une fonction par des fractions rationelles'', Thesis, [Ann. \'Ecole 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}}

[[Category:数学近似]]
[[Category:数值分析]]