多元函数泰勒展开式

实际优化问题的目标函数往往比较复杂。为了使问题简化，通常将目标函数在某点附近展开为泰勒(Taylor)多项式来逼近原函数

一元函数 $f(x)$ 在 $x_k$ 处的泰勒展开式为

f (x) = f (x_{k}) + (x - x_{k}) f^{'} (x_{k}) + \frac{1}{2!} (x - x_{k})^{2} f^{''} (x_{k}) + O^{n}

$f(x)=f(x_k)+(x-x_k)f^\prime(x_k)+\frac{1}{2!}(x-x_k)^2f^{\prime\prime}(x_k)+O^n$

二元函数 $f(x,y)$ 在 $(x_k,y_k)$ 处泰勒展开式为

f (x, y) = f (x_{k}, y_{k}) + (x - x_{k}) f_{x}^{'} (x_{k}, y_{k}) + (y - y_{k}) f_{y}^{'} (x_{k}, y_{k}) + \frac{1}{2!} (x - x_{k})^{2} f_{x x}^{''} (x_{k}, y_{k}) + \frac{1}{2!} (y - y_{k})^{2} f_{y y}^{''} (x_{k}, y_{k}) + \frac{1}{2!} (x - x_{k}) (y - y_{k}) f_{x y}^{''} (x_{k}, y_{k}) + \frac{1}{2!} (x - x_{k}) (y - y_{k}) f_{y x}^{''} (x_{k}, y_{k}) + O^{n}

$f(x,y)=f(x_k,y_k)+(x-x_k)f^\prime_x(x_k,y_k)+(y-y_k)f^\prime_y(x_k,y_k)+ \\\\ \frac{1}{2!}(x-x_k)^2f^{\prime\prime}_{xx}(x_k,y_k)+\frac{1}{2!}(y-y_k)^2f^{\prime\prime}_{yy}(x_k,y_k)+ \\\\ \frac{1}{2!}(x-x_k)(y-y_k)f^{\prime\prime}_{xy}(x_k,y_k)+\frac{1}{2!}(x-x_k)(y-y_k)f^{\prime\prime}_{yx}(x_k,y_k)+O^n$

三元函数 $f(x^1,x^2,x^3)$ 在 $\vec{x_k}=(x^1_k,x^2_k,x^3_k)$ 处泰勒展开式为，（二阶项展开共有9项）

f (x^{1}, x^{2}, x^{3}) = f (x_{k}^{1}, x_{k}^{2}, x_{k}^{3}) + \sum_{i = 1}^{3} (x^{i} - x_{k}^{i}) f_{x^{i}}^{'} (x_{k}^{1}, x_{k}^{2}, x_{k}^{3}) + \frac{1}{2!} \sum_{i = 1}^{3} \sum_{j = 1}^{3} (x^{i} - x_{k}^{i}) (x^{j} - x_{k}^{j}) f_{i j}^{''} (x_{k}^{1}, x_{k}^{2}, x_{k}^{3}) + O^{n}

$f(x^1,x^2,x^3)=f(x^1_k,x^2_k,x^3_k)+\sum_{i=1}^{3}(x^i-x^i_k)f^\prime_{x^i}(x^1_k,x^2_k,x^3_k)+\\\\ \frac{1}{2!}\sum_{i=1}^{3}\sum_{j=1}^{3}(x^i-x^i_k)(x^j-x^j_k)f^{\prime\prime}_{ij}(x^1_k,x^2_k,x^3_k)+O^n$

多元函数 $f(x^1,x^2,...,x^n)$ 在 $\vec{x_k}=(x^1_k,x^2_k,...,x^n_k)$ 处泰勒展开式为

f (x^{1}, x^{2}, . . ., x^{n}) = f (x_{k}^{1}, x_{k}^{2}, . . ., x_{k}^{n}) + \sum_{i = 1}^{n} (x^{i} - x_{k}^{i}) f_{x^{i}}^{'} (x_{k}^{1}, x_{k}^{2}, . . ., x_{k}^{n}) + \frac{1}{2!} \sum_{i = 1}^{n} \sum_{j = 1}^{n} (x^{i} - x_{k}^{i}) (x^{j} - x_{k}^{j}) f_{i j}^{''} (x_{k}^{1}, x_{k}^{2}, . . ., x_{k}^{n}) + O^{n}

$f(x^1,x^2,...,x^n)=f(x^1_k,x^2_k,...,x^n_k)+\sum_{i=1}^{n}(x^i-x^i_k)f^\prime_{x^i}(x^1_k,x^2_k,...,x^n_k)+\\\\ \frac{1}{2!}\sum_{i=1}^{n}\sum_{j=1}^{n}(x^i-x^i_k)(x^j-x^j_k)f^{\prime\prime}_{ij}(x^1_k,x^2_k,...,x^n_k)+O^n$

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