多元函数泰勒展开证明
03 Jun 2020来自 wiki
若一元函数 f(x) 在 x = x0 点的某个邻域内具有任意阶导数，则函数 f(x) 在 x = x0 点处的泰勒展开式为:
\[f(x)=f\left(x_{0}\right)+f^{\prime}\left(x_{0}\right) \Delta x+\frac{f^{\prime \prime}\left(x_{0}\right)}{2 !} \Delta x^{2}+\cdots\]其中 △x = x - x0
同理，二元函数 f(x1, x2) 在 x0(x10, x20) 点处的泰勒展开为：
\[\begin{aligned} f\left(x_{1}, x_{2}\right)= &f\left(x_{10}, x_{20}\right)+f_{x_{1}}\left(x_{10}, x_{20}\right) \Delta x_{1}+f_{x_{2}}\left(x_{10}, x_{20}\right) \Delta x_{2}+ \\ &\frac{1}{2}\left[f_{x_{1} x_{1}}\left(x_{10}, x_{20}\right) \Delta x_{1}^{2}+2 f_{x_{1} x_{2}}\left(x_{10}, x_{20}\right) \Delta x_{1} \Delta x_{2}+f_{x_{2} x_{2}}\left(x_{10}, x_{20}\right) \Delta x_{2}^{2}\right]+\cdots \end{aligned}\]其中
\[\Delta x_{1}=x_{1}-x_{10}, \Delta x_{2}=x_{2}-x_{20}, f_{x_{1}}=\frac{\partial f}{\partial x_{1}}, f_{x_{2}}=\frac{\partial f}{\partial x_{2}} \\ \\ f_{x_{1} x_{1}}=\frac{\partial^{2} f}{\partial x_{1}^{2}}, f_{x_{2} x_{2}}=\frac{\partial^{2} f}{\partial x_{2}^{2}}, f_{x_{1} x_{2}}=\frac{\partial^{2} f}{\partial x_{1} \partial x_{2}}=\frac{\partial^{2} f}{\partial x_{2} \partial x_{1}}\]最终可以得到
\[f(x)=f\left(x_{0}\right)+ J\left(x_{0}\right)^{\mathrm{T}} \Delta x+\frac{1}{2} \Delta x^{\mathrm{T}} H\left(x_{0}\right) \Delta x+\cdots\]J, H 分别为 Jacobian 和 Hessian。