亨利·帕德
**帕德近似**（英語：）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。

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 f(x)的[m,n]阶帕德近似为


R
(
x
)
=

∑

j
=
0

m

a

j

x

j

1
+

∑

k
=
1

n

b

k

x

k


=

a

0

+

a

1

x
+

a

2

x

2

+
⋯
+

a

m

x

m

1
+

b

1

x
+

b

2

x

2

+
⋯
+

b

n

x

n


{\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}}}}

{\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




f
(
0
)

=

R
(
0
)


f
′

(
0
)

=

R
′

(
0
)


f
″

(
0
)

=

R
″

(
0
)


⋮



f

(
m
+
n
)

(
0
)

=

R

(
m
+
n
)

(
0
)



{\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}}}

{\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m,n函数f(x)的[m,n]阶帕德近似是唯一的。

函数f(x)的帕德近似记为



[
m

/

n

]

f

(
x
)
.

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

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

例子
--

### 正弦函數



[
6

/

6

]

sin
⁡
(
x
)

=

(
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)}={\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)}={\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


x
−
(
1

/

6
)
∗

x

3

+
(
1

/

120
)
∗

x

5

−
(
1

/

5040
)
∗

x

7

+
(
1

/

362880
)
∗

x

9

−
(
1

/

39916800
)
∗

x

11

+
O
(

x

13

)

{\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})}}

{\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\sin(x)的12阶泰勒级数全同：



sin
⁡
(
x
)
≈

x
−
(
1

/

6
)
∗

x

3

+
(
1

/

120
)
∗

x

5

−
(
1

/

5040
)
∗

x

7

+
(
1

/

362880
)
∗

x

9

−
(
1

/

39916800
)
∗

x

11

+
O
(

x

13

)

{\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})}}

{\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



[
5

/

5

]

e
x
p
(
x
)

=

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


{\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}}}}

{\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


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

)

{\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})}}

{\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：


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

)

{\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})}}

{\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

f
:=

1
−
cos
⁡
(
2
∗
x

)

2

1
+
arctan
⁡
(
3
∗
x
)


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

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

[
3

/

3

]

f
(
x
)

=

(
64

/

75
)
∗

x

3

+
4
∗

x

2

1
+
(
241

/

75
)
∗
x
+
(
148

/

75
)
∗

x

2

−
(
1061

/

225
)
∗

x

3


{\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}}}}

{\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)}



−
(
9853969

/

39583665
)
∗

z

5

−
(
1493060

/

2638911
)
∗

z

3

+
z

1
+
(
968375

/

879637
)
∗

z

2

−
(
1167506

/

7916733
)
∗

z

4

+
(
867043

/

2159109
)
∗

z

6


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

{\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)}



−
(
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


{\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}}}}

{\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



(
2

/

15
)
∗
(
49140
∗
x
+
3570
∗

x

3

+
739
∗

x

5

)

(
165
∗

π

∗

x

4

+
1330
∗

π

∗

x

2

+
3276
∗

π

)


{\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 }})}}{}}

{\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 C(x)



(
1

/

135
)
∗
(
990791
∗

x

9

∗

π

4

−
147189744
∗

x

5

∗

π

2

+
8714684160
∗
x
)

(
1749
∗

π

4

∗

x

8

+
523536
∗

π

2

∗

x

4

+
64553216
)


{\displaystyle {\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)}}}

{\displaystyle {\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)}}}

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）, Scholarpedia （页面存档备份，存于）, 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,  3rd, New York: Cambridge University Press, 2007  [2015-02-24], ISBN 978-0-521-88068-8, （原始内容存档于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. Numerische Mathematik, 1966, **8** (3): 264–269, doi:10.1007/BF02162562