Taylor 级数

1 Taylor级数公式

\[ y(x_{n+1}) = y(x_{n}) + \frac{h^1}{1!}y^{'}(x_n) + \frac{h^2}{2!}y^{''}(x_n) + \frac{h^3}{3!}y^{(3)}(x_n) + \dots \]

• \(y^{'}=\frac{dy}{dx}=f=f^{(0)}\)
• \(y^{''}=\frac{\partial f^{(0)}}{\partial x} + f\frac{\partial f^{(0)}}{\partial y}=f^{(1)}\)
• \(y^{(3)}=\frac{\partial f^{(1)}}{\partial x} + f\frac{\partial f^{(1)}}{\partial y}=f^{(2)}\)
• 一般地说：\(y^{j)}=\frac{\partial f^{(j-2)}}{\partial x} + f\frac{\partial f^{(j-2)}}{\partial y}=f^{(j-1)}\)

具体地：

\[ \begin{cases} y^{'} &= f\\ y^{''} &= \frac{\partial f}{\partial x} + f\frac{\partial f}{\partial y}\\ y^{(3)}&=\frac{\partial^{2} f}{\partial x^{2}}+2 f \frac{\partial^{2} f}{\partial x \partial y}+f^{2} \frac{\partial^{2} f}{\partial y^{2}}+\frac{\partial f}{\partial y}\left(\frac{\partial f}{\partial x}+f \frac{\partial f}{\partial y}\right)\\ \vdots \end{cases} \] Taylor 一阶表达式： \[ y_{n+1} = y_{n} + hy_n^{'} \]

2 截断误差

Taylor 一阶的截断误差：

\[ y(x_{n+1}) - y_{n+1} = \frac{h^{2}}{2!}y^{(2)}(\zeta) \] 其中，\(x_2 < \zeta < x_3\)

定义：如果阶段误差为 \(O(h^{p+1})\)，则称该方法有 \(p\) 阶精度。一阶 Taylor 表达式有 1 阶精度。