微积分笔记02：多元函数的泰勒展开式&海森矩阵

[=f(x_0,y_0) ]

[qquadqquadqquadqquadqquadqquad+Delta x cdot f'_x(x_0,y_0)+Delta ycdot f'_y(x_0,y_0) ]

[qquadqquadqquadqquadqquadqquadqquadqquadqquadquad+frac{1}{2!}cdot[(Delta x)^2 cdot f''_{xx}(x_0,y_0)+(Delta y)^2 cdot f''_{yy}(x_0,y_0)+2Delta_xDelta_ycdot f''_{xy}(x_0,y_0)] ]

[+... ]

[qquadqquadqquadqquadqquadqquadqquadqquadqquadquad+frac{1}{n!}cdot sum_{i=0}^n C_n^i (Delta x)^icdot(Delta y)^{n-i} cdot frac{alpha^n f}{alpha^i xcdot alpha^{n-i}y} Big|_{(x=x_0,y=y_0)} ]

[qquadqquadqquadqquadqquadqquadqquadqquadqquadquad+frac{1}{(n+1)!}cdot sum_{i=0}^{n+1} C_n^i (Delta x)^icdot(Delta y)^{n+1-i} cdot frac{alpha^{n+1} f}{alpha^i xcdot alpha^{n+1-i}y} Big|_{(x=x_0+theta cdot Delta x,y=y_0+theta cdot Delta y)} ]

2.2 多元函数的二阶泰勒展开式及海森矩阵

2.2.1 二元函数的二阶泰勒展开式（矩阵）

一般情况下，可直接使用多元函数的二阶泰勒展开式进行求解，由2.1.1中的n阶泰勒展开式可得：

[qquad qquad qquad f(x_0+Delta x,y_0+Delta y) ]

[=f(x_0,y_0) ]

[qquadqquadqquadqquadqquadqquad+f'_x(x_0,y_0)cdot Delta x+f'_y(x_0,y_0)cdotDelta y ]

[qquadqquadqquadqquadqquadqquadqquadqquadqquadquad+f''_{xx}(x_0,y_0)cdot (Delta x)^2+f''_{yy}cdot (Delta y)^2+2f''_{xy}(x_0,y_0)cdot Delta x Delta y ]

由梯度相关性质可得：

[nabla f(x_0,y_0)= begin{bmatrix} f'_x(x_0,y_0)\ f'_y(x_0,y_0) end{bmatrix} ]

则上式可用矩阵表示为：

[qquadqquadqquad f(x_0+Delta x,y_0+Delta y) ]

[=f(x_0,y_0) ]

[qquadqquadqquad + nabla f^T(x_0,y_0) cdot begin{bmatrix} Delta x\ Delta y end{bmatrix} ]

[qquadqquadqquadqquadqquadqquadqquadqquadquad + begin{bmatrix} Delta x & Delta y end{bmatrix} cdot begin{bmatrix} f''_{xx}(x_0,y_0)&f''_{xy}(x_0,y_0)\ f''_{xy}(x_0,y_0)&f''_{yy}(x_0,y_0) end{bmatrix} cdot begin{bmatrix} Delta x \ Delta y end{bmatrix} ]

2.2.2 多元元函数的二阶泰勒展开式（矩阵）

设存在多元函数(f(x_1,x_2,...,x_n))，若此函数满足泰勒展开式相关条件，则其二阶泰勒展开式为：

[f(x_1+Delta x_1,x_2+Delta x_2,...,x_n+Delta x_n) ]

[=f(x_1,x_2,...,x_n) ]

[+nabla f^T(x_1,x_2...,x_n) cdot begin{bmatrix} Delta x_1\ Delta x_2\ ...\ Delta x_n end{bmatrix} ]

[+ begin{bmatrix} Delta x_1 & Delta x_2 & ...& Delta x_n end{bmatrix} cdot begin{bmatrix} f''_{x1x1}(x_0,y_0)&f''_{x1x2}(x_0,y_0)&...&f''_{x1xn}(x_0,y_0)\ f''_{x2x1}(x_0,y_0)&f''_{x2x2}(x_0,y_0)&...&f''_{x2xn}(x_0,y_0)\ &......\ f''_{xnx1}(x_0,y_0)&f''_{xnx2}(x_0,y_0)&...&f''_{xnxn}(x_0,y_0) end{bmatrix} cdot begin{bmatrix} Delta x_1\ Delta x_2\ ...\ Delta x_n\ end{bmatrix} ]

[其中,矩阵 begin{bmatrix} f''_{x1x1}(x_0,y_0)&f''_{x1x2}(x_0,y_0)&...&f''_{x1xn}(x_0,y_0)\ f''_{x2x1}(x_0,y_0)&f''_{x2x2}(x_0,y_0)&...&f''_{x2xn}(x_0,y_0)\ &......\ f''_{xnx1}(x_0,y_0)&f''_{xnx2}(x_0,y_0)&...&f''_{xnxn}(x_0,y_0) end{bmatrix} 称为海森矩阵，记为H ]

则有：

[f(x_1+Delta x_1,x_2+Delta x_2,...,x_n+Delta x_n) ]

[=f(x_1,x_2,...,x_n) ]

[+nabla f^T(x_1,x_2...,x_n) cdot begin{bmatrix} Delta x_1\ Delta x_2\ ...\ Delta x_n end{bmatrix} ]

[+ begin{bmatrix} Delta x_1 & Delta x_2 & ...& Delta x_n end{bmatrix} cdot H cdot begin{bmatrix} Delta x_1\ Delta x_2\ ...\ Delta x_n\ end{bmatrix} ]