帕德近似

帕德近似（英語：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，所以多用于在计算机数学中。

亨利·帕德

例如${\displaystyle {\frac {1}{1-x}}}$的泰勒级数

${\displaystyle 1+x+x^{2}+x^{3}+\cdots }$只有在${\displaystyle -1<x<1}$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 ${\displaystyle f(x)}$ 的[m,n]阶帕德近似为

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

 

并且

$${\displaystyle {\begin{array}{rcl}f(0)&=&R(0)\\f'(0)&=&R'(0)\\f''(0)&=&R''(0)\\&\vdots &\\f^{(m+n)}(0)&=&R^{(m+n)}(0)\end{array}}}$$

 

对于给定的${\displaystyle m,n}$ 函数${\displaystyle f(x)}$ 的[m,n]阶帕德近似是唯一的。

函数${\displaystyle f(x)}$ 的帕德近似记为

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

 

例子

正弦函數

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

 

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

$${\displaystyle {x-(1/6)*x^{3}+(1/120)*x^{5}-(1/5040)*x^{7}+(1/362880)*x^{9}-(1/39916800)*x^{11}+O(x^{13})}}$$

 

和${\displaystyle \sin(x)}$ 的12阶泰勒级数全同：

$${\displaystyle \sin(x)\approx {x-(1/6)*x^{3}+(1/120)*x^{5}-(1/5040)*x^{7}+(1/362880)*x^{9}-(1/39916800)*x^{11}+O(x^{13})}}$$

 

指数函数

$${\displaystyle [5/5]_{exp(x)}={\frac {1+(1/9)*x^{2}+(1/2)*x+(1/72)*x^{3}+(1/1008)*x^{4}+(1/30240)*x^{5}}{1+(1/9)*x^{2}-(1/2)*x-(1/72)*x^{3}+(1/1008)*x^{4}-(1/30240)*x^{5}}}}$$

 

其泰勒级数为

$${\displaystyle {1+x+(1/2)*x^{2}+(1/6)*x^{3}+(1/24)*x^{4}+(1/120)*x^{5}+(1/720)*x^{6}+(1/5040)*x^{7}+(1/40320)*x^{8}+(1/362880)*x^{9}+(1/3628800)*x^{10}+(23/914457600)*x^{11}+O(x^{12})}}$$

 

与exp(x)本身的泰勒级数展开的前10阶完全等同：

$${\displaystyle {1+x+(1/2)*x^{2}+(1/6)*x^{3}+(1/24)*x^{4}+(1/120)*x^{5}+(1/720)*x^{6}+(1/5040)*x^{7}+(1/40320)*x^{8}+(1/362880)*x^{9}+(1/3628800)*x^{10}+(1/39916800)*x^{11}+O(x^{12})}}$$

 

又如

$${\displaystyle f:={\frac {1-\cos(2*x)^{2}}{1+\arctan(3*x)}}}$$

 

$${\displaystyle [3/3]_{f(x)}={\frac {(64/75)*x^{3}+4*x^{2}}{1+(241/75)*x+(148/75)*x^{2}-(1061/225)*x^{3}}}}$$

 

雅可比橢圓函數 ${\displaystyle \operatorname {sn} (x;3)}$ 

$${\displaystyle {\frac {-(9853969/39583665)*z^{5}-(1493060/2638911)*z^{3}+z}{1+(968375/879637)*z^{2}-(1167506/7916733)*z^{4}+(867043/2159109)*z^{6}}}}$$

 

第一類 5 階貝塞爾函數 ${\displaystyle J_{5}(x)}$ 

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

 

误差函数

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

 

菲涅耳積分 ${\displaystyle C(x)}$ 

$${\displaystyle {\frac {(1/135)*(990791*x^{9}*\pi ^{4}-147189744*x^{5}*\pi ^{2}[phone]*x)}{(1749*\pi ^{4}*x^{8}+523536*\pi ^{2}*x^{4}[phone])}}}$$

 

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. Padé approximant （页面存档备份，存于互联网档案馆）, Scholarpedia （页面存档备份，存于互联网档案馆）, 7(6):9756.
• Brezinski, C.; and Redivo Zaglia, M. Extrapolation Methods.= Theory and Practice. North-Holland, 1991
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP, Section 5.12 Padé Approximants, Numerical Recipes: The Art of Scientific Computing 3rd, New York: Cambridge University Press, 2007 [2015-02-24], ISBN 978-0-521-88068-8, （原始内容于2016-03-03） 
• 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. 1–62.
• Padé, H.; Sur la répresentation approchée d'une fonction par des fractions rationelles, Thesis, [Ann. \'Ecole Nor. (3), 9, 1892, pp. 1–93 supplement.
• Wynn, P., Upon systems of recursions which obtain among the quotients of the Padé table, Numerische Mathematik, 1966, 8 (3): 264–269, doi:10.1007/BF02162562