将阿多米安多项式一种参数化算法推广至多变量情形。
方法
非线性表达式$N(u_1,u_2,\cdots,u_m)$有$m$个变量$u_1,u_2,\cdots,u_m$,每个变量都可以展开成级数$u_i=\sum_{j=0}^{\infty}u_{i,j}$。$N(u_1,u_2,\cdots,u_m)$展开成阿多米安多项式,第$n$项为
\begin{equation}
\begin{split}
&A_n(u_{1,0},u_{1,1},\cdots,u_{1,n};u_{2,0},u_{2,1},\cdots,u_{2,n};\cdots;u_{m,0},u_{m,1},\cdots,u_{m,n})\\
&=\frac{1}{2\pi}\int_{-\pi}^{\pi}N\left(\sum_{k=0}^{n}u_{1,k}e^{ik\lambda},\sum_{k=0}^{n}u_{2,k}e^{ik\lambda},\cdots,\sum_{k=0}^{n}u_{m,k}e^{ik\lambda} \right)e^{-in\lambda}\mathrm d\lambda \\
&=N_0(u_{1,0},u_{2,0},\cdots,u_{m,0}) \mathrm{Coefficient[e^{in\lambda}]}
\end{split}
\label{mm}
\end{equation}
举例
例 $N(u_1,u_2)=\frac{u_2}{2+u_1}$
本例来自 International Mathematical Forum, 1, 2006, no. 39, 1919-1924
\begin{equation*}
\begin{split}
A_n=&\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{u_{2,0}+\sum_{k=1}^nu_{2,k}e^{ik\lambda}}{2+u_{1,0}+\sum_{k=1}^nu_{1,k}e^{ik\lambda}}e^{-in\lambda}\mathrm d\lambda \\
=& \frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{u_{2,0}+\sum_{k=1}^nu_{2,k}e^{ik\lambda}}{2+u_{1,0}}\frac{1}{1+\frac{\sum_{k=1}^nu_{1,k}e^{ik\lambda}}{2+u_{1,0}}}e^{-in\lambda}\mathrm d\lambda \\
=& \frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{u_{2,0}}{2+u_{1,0}}\left(1-\frac{\sum_{k=1}^nu_{1,k}e^{ik\lambda}}{2+u_{1,0}}+\left( \frac{\sum_{k=1}^nu_{1,k}e^{ik\lambda}}{2+u_{1,0}} \right )^2+\cdots \right)e^{-in\lambda}\mathrm d\lambda +\\
& \frac{1}{2\pi}\int_{-\pi}^{\pi}\sum_{k=1}^nu_{2,k}e^{ik\lambda}\left(1-\frac{\sum_{k=1}^nu_{1,k}e^{ik\lambda}}{2+u_{1,0}}+\left( \frac{\sum_{k=1}^nu_{1,k}e^{ik\lambda}}{2+u_{1,0}} \right )^2+\cdots \right)e^{-in\lambda}\mathrm d\lambda \\
=& \mathrm{Coefficient[e^{i\lambda}]}
\end{split}
\end{equation*}
\begin{equation*}
A_0=\frac{u_{2,0}}{2+u_{1,0}}
\end{equation*}
利用Mathematica:
| In[11]:= Subscript[u, 2, 0]/(2 + Subscript[u, 1, 0])* | Coefficient[ | Normal[Series[1/(1 + x), {x, 0, 1}]] /. | x -> ((Subscript[u, 1, 1]*Exp[I*x])/(2 + Subscript[u, 1, 0])), | Exp[I*x], 1] + | 1/(2 + Subscript[u, 1, 0])* | Coefficient[ | Subscript[u, 2, 1]*Exp[I*x]* | Normal[Series[1/(1 + t), {t, 0, 1}]] /. | t -> ((Subscript[u, 1, 1]*Exp[I*x])/(2 + Subscript[u, 1, 0])), | Exp[I*x], 1] | <p>Out[11]= -((<br /> | Subscript[u, 1, 1] Subscript[u, 2,<br /> | </p> |
|
计算得
\begin{equation*}
A_1=\frac{u_{2,1}}{u_{1,0}+2}-\frac{u_{1,1} u_{2,0}}{\left(u_{1,0}+2\right){}^2}
\end{equation*}
(可见,原文有误)
利用Mathematica:
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
| In[12]:= Subscript[u, 2, 0]/(2 + Subscript[u, 1, 0])* | Coefficient[ | Normal[Series[1/(1 + x), {x, 0, 2}]] /. | x -> ((Subscript[u, 1, 1]*Exp[I*x] + | Subscript[u, 1, 2]*Exp[2*I*x])/(2 + Subscript[u, 1, 0])), | Exp[I*x], 2] + | 1/(2 + Subscript[u, 1, 0])* | Coefficient[(Subscript[u, 2, 1]*Exp[I*x] + | Subscript[u, 2, 2]*Exp[2*I*x])* | Normal[Series[1/(1 + t), {t, 0, 2}]] /. | t -> ((Subscript[u, 1, 1]*Exp[I*x] + | Subscript[u, 1, 2]*Exp[2*I*x])/(2 + Subscript[u, 1, 0])), | Exp[I*x], 2] | <p>Out[12]= ((<br /> | !(*SubsuperscriptBox[(u), (1,<br /> | 1), (2)])/(2 + Subscript[u, 1, 0])^2 - Subscript[u, 1, 2]/(<br /> | 2 + Subscript[u, 1, 0])) Subscript[u, 2, 0])/(<br /> | 2 + Subscript[u, 1,<br /> | </p> | <p>In[13]:= Collect[%, 2 + Subscript[u, 1, 0]]</p> | <p>Out[13]= (!(<br /> | *SubsuperscriptBox[(u), (1, 1), (2)]\<br /> | *SubscriptBox[(u), (2, 0)]))/(2 + Subscript[u, 1,<br /> | </p> |
|
计算得
\begin{equation*}
A_2= \frac{u_{2,0} u_{1,1}^2}{\left(u_{1,0}+2\right){}^3}+\frac{-u_{1,2} u_{2,0}-u_{1,1} u_{2,1}}{\left(u_{1,0}+2\right){}^2}+\frac{u_{2,2}}{u_{1,0}+2}
\end{equation*}
利用Mathematica:
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 |
| In[14]:= Subscript[u, 2, 0]/(2 + Subscript[u, 1, 0])* | Coefficient[ | Normal[Series[1/(1 + x), {x, 0, 3}]] /. | x -> ((Subscript[u, 1, 1]*Exp[I*x] + | Subscript[u, 1, 2]*Exp[2*I*x] + | Subscript[u, 1, 3]*Exp[3*I*x])/(2 + Subscript[u, 1, 0])), | Exp[I*x], 3] + | 1/(2 + Subscript[u, 1, 0])* | Coefficient[(Subscript[u, 2, 1]*Exp[I*x] + | Subscript[u, 2, 2]*Exp[2*I*x] + Subscript[u, 2, 3]*Exp[3*I*x])* | Normal[Series[1/(1 + t), {t, 0, 3}]] /. | t -> ((Subscript[u, 1, 1]*Exp[I*x] + | Subscript[u, 1, 2]*Exp[2*I*x] + | Subscript[u, 1, 3]*Exp[3*I*x])/(2 + Subscript[u, 1, 0])), | Exp[I*x], 3] | <p>Out[14]= ((-(<br /> | !(*SubsuperscriptBox[(u), (1,<br /> | 1), (3)])/(2 + Subscript[u, 1, 0])^3) + (<br /> | 2 Subscript[u, 1, 1] Subscript[u, 1,<br /> | 2])/(2 + Subscript[u, 1, 0])^2 - Subscript[u, 1, 3]/(<br /> | 2 + Subscript[u, 1, 0])) Subscript[u, 2, 0])/(<br /> | 2 + Subscript[u, 1, 0]) + ((!(<br /> | *SubsuperscriptBox[(u), (1, 1), (2)]\<br /> | *SubscriptBox[(u), (2, 1)]))/(2 + Subscript[u, 1, 0])^2 - (<br /> | Subscript[u, 1, 2] Subscript[u, 2, 1])/(2 + Subscript[u, 1, 0]) - (<br /> | Subscript[u, 1, 1] Subscript[u, 2, 2])/(2 + Subscript[u, 1, 0]) +<br /> | Subscript[u, 2, 3])/(2 + Subscript[u, 1, 0])</p> | <p>In[15]:= Collect[%, 2 + Subscript[u, 1, 0]]</p> | <p>Out[15]= -((!(<br /> | *SubsuperscriptBox[(u), (1, 1), (3)]\<br /> | *SubscriptBox[(u), (2, 0)]))/(2 + Subscript[u, 1, 0])^4) + (<br /> | 2 Subscript[u, 1, 1] Subscript[u, 1, 2] Subscript[u, 2, 0] + !(<br /> | *SubsuperscriptBox[(u), (1, 1), (2)]\<br /> | *SubscriptBox[(u), (2, 1)]))/(2 + Subscript[u, 1,<br /> | </p> |
|
计算得
\begin{equation*}
A_3= -\frac{u_{2,0} u_{1,1}^3}{\left(u_{1,0}+2\right){}^4}+\frac{u_{2,1} u_{1,1}^2+2 u_{1,2} u_{2,0} u_{1,1}}{\left(u_{1,0}+2\right){}^3}+\frac{-u_{1,3} u_{2,0}-u_{1,2} u_{2,1}-u_{1,1} u_{2,2}}{\left(u_{1,0}+2\right){}^2}+\frac{u_{2,3}}{u_{1,0}+2}
\end{equation*}
类似可计算$A_4$,$A_5$,$\cdots$。