一元和二元的泰勒展开式

一元函数的泰勒展开式

由高等数学知识可知，对于一元函数$f(x)$ 在$k$点，即$x=x^{(k)}$的泰勒展开式为：
$$\begin{gathered} f(x)=f(x^{(k)})+f^{\prime }(x^{(k)})(x-x^{(k)})+\frac{1}{2!}{f}^{{}^{\prime \prime }}(x^{(k)})(x-x^{(k)}{)}^2 \\ +\cdots +\frac{1}{n!}{f}^{(n)}(x^{(k)})(x-x^{(k)}{)}^n+R_n \end{gathered}$$
式子中的余项$R_n$为
$$R_n=\frac{1}{(n+1)!}{f}^{(n+1)}(\xi )(x-x^{(k)}{)}^{(n+1)}$$
其中$\xi$在$x$和$x^{(k)}$之间。

二元函数的泰勒展开式

定理

设 $z=f(x,y)$在点$(x_0,y_0)$的某一个领域内连续且有直到$n+1$阶的连续偏导数，$(x_0+h,y_0+h)$为此领域内任一点，则有
$$\begin{gathered} f(x_0+h,y_0+h)=f(x_0,y_0)+(h\frac{\partial }{\partial x}+k\frac{\partial }{\partial y})f(x_0,y_0)+\frac{1}{2!}(h\frac{\partial }{\partial x}+k\frac{\partial }{\partial y}{)}^{2}f(x_0,y_0) \\ +\cdots +\frac{1}{n!}(h\frac{\partial }{\partial x}+k\frac{\partial }{\partial y}{)}^{n}f(x_0,y_0)+\frac{1}{(n+1)!}(h\frac{\partial }{\partial x}+k\frac{\partial }{\partial y}{)}^{(n+1)}f(x_0+\theta h,y_0+\theta k),(0<\theta <1) \end{gathered}$$

举个例子
函数$f(X)=f(x_1,x_2)$在点$X^{(k)}=(x_1^{(k)},x_2^{(k)})$附近的泰勒展开，若只去到二次项可写成
$$\begin{gathered} f(x)\approx f(X^{(k)})+\frac{\partial f}{\partial x_1}{|}_{X=X^{(k)}}(x_1-x_1^{(k)})+\frac{\partial f}{\partial x_2}{|}_{X=X^{(k)}}(x_2-x_2^{(k)}) \\ +\frac{1}{2!}\left[\frac{{\partial }^{2}f}{\partial x_1^2}{|}_{X=X^{(k)}}(x_1-x_1^{(k)}{)}^2+\frac{{\partial }^{2}f}{\partial x_1\partial x_2}{|}_{X=X^{(k)}}(x_1-x_1^{(k)})(x_2-x_2^{(k)})+\frac{{\partial }^{2}f}{\partial x_2^2}{|}_{X=X^{(k)}}(x_2-x_2^{(k)}{)}^2\right] \end{gathered}$$
将这个式子写成矩阵形式，即
$$f(X)\approx f(X^{(k)})+\left(\begin{array}{cc}\frac{\partial f}{\partial x_1}& \frac{\partial f}{\partial x_2}\end{array}\right)\left(\begin{array}{c}{x}_1-x_1^{(k)}\\ x_2-x_2^{(k)}\end{array}\right)+\frac{1}{2}\left(\begin{array}{cc}{x}_1-x_1^{(k)}& x_2-x_2^{(k)}\end{array}\right)\left(\begin{array}{cc}\frac{{\partial }^{2}f}{\partial x_1^2}& \frac{{\partial }^{2}f}{\partial x_1\partial x_2}\\ \frac{{\partial }^{2}f}{\partial x_1\partial x_2}& \frac{{\partial }^{2}f}{\partial x_2^2}\end{array}\right)\left(\begin{array}{c}{x}_1-x_1^{(k)}\\ x_2-x_2^{(k)}\end{array}\right)$$
式子中，我们令
$$\begin{gathered} \nabla f=\left(\begin{array}{cc}\frac{\partial f}{\partial x_1}& \frac{\partial f}{\partial x_2}\end{array}\right) \\ X-X^{(k)}=\left(\begin{array}{c}{x}_1-x_1^{(k)}\\ x_2-x_2^{(k)}\end{array}\right) \\ {\nabla }^{2}f=\left(\begin{array}{cc}\frac{{\partial }^{2}f}{\partial x_1^2}& \frac{{\partial }^{2}f}{\partial x_1\partial x_2}\\ \frac{{\partial }^{2}f}{\partial x_1\partial x_2}& \frac{{\partial }^{2}f}{\partial x_2^2}\end{array}\right) \end{gathered}$$
其中${\nabla }^{2}f$是函数$f(X)$在$X(k)$点的二阶偏导数矩阵，称为海森矩阵，可以用$H(X)$表示，它是一个对称矩阵。引用上述符号后，二元函数泰勒展开式可以简写为：
$$f(X)\approx f(X^{(k)})+\nabla f(X^{(k)})(X-X^{(k)})+\frac{1}{2}(X-X^{(k)}{)}^{T}H(X^{(k)})(X-X^{(k)})$$