黑塞矩阵
泰勒展开公式
设n是正整数。如果定义一个包含 a a a间隔函数 f f f在 a a a点处 n 1 n 1 n 1次可导，所以对于这个范围内的任何东西 x x x，都有： f ( x ) = f ( a ) f ′ ( a ) 1 ! ( x ? a ) f ( 2 ) ( a ) 2 ! ( x ? a ) 2 ? f ( n ) ( a ) n ! ( x ? a ) n R n ( x ) f(x)=f(a) \cfrac{f'(a)}{1!}(x-a) \cfrac{f^{(2)}(a)}{2!}(x-a)^2 \cdots \cfrac{f^{(n)}(a)}{n!}(x-a)^n R_n(x) f(x)=f(a) 1!f′(a)(x−a)+2!f(2)(a)(x−a)2+⋯+n!f(n)(a)(x−a)n+Rn(x) 其中的多项式称为函数在 a a a处的泰勒展开式，剩余的 R n ( x ) R_{n}(x) Rn(x)是泰勒公式的余项，是 ( x − a ) n (x-a)^n (x−a)n的高阶无穷小。
泰勒二阶展开
同理，二元函数 f ( x 1 , x 2 ) f(x_1,x_2) f(x1,x2)在 x 0 ( x 10 , x 20 ) x_0(x_{10},x_{20}) x0(x10,x20)处的泰勒展开式为 f ( x 1 , x 2 ) = f ( x 10 , x 20 ) + f x 1 ( x 10 , x 20 ) Δ x 1 + f x 2 ( x 10 , x 20 ) Δ x 2 + 1 2 [ f x 1 x 1 ( x 10 , x 20 ) Δ x 1 2 + 2 f x 1 x 2 ( x 10 , x 20 ) Δ x 1 Δ x 2 + f x 2 x 2 ( x 10 , x 20 ) Δ x 2 2 ] f(x_1,x_2)=f(x_{10},x_{20})+f_{x_1}(x_{10},x_{20})\Delta{x_1}+f_{x_2}(x_{10},x_{20})\Delta x_2+\cfrac{1}{2}[f_{x_1x_1}(x_{10},x_{20})\Delta x_1^2+2f_{x_1x_2}(x_{10},x_{20})\Delta x_1\Delta x_2+f_{x_2x_2}(x_{10},x_{20})\Delta x_2^2] f(x1,x2)=f(x10,x20)+fx1(x10,x20)Δx1+fx2(x10,x20)Δx2+21[fx1x1(x10,x20)Δx12+2fx1x2(x10,x20)Δx1Δx2+fx2x2(x10,x20)Δx22] 其中 Δ x 1 = x 1 − x 10 , Δ x 2 = x 2 − x 20 , f x 1 = ∂ f ∂ x 1 , f x 2 = ∂ f ∂ x 2 , f x 1 x 1 = ∂ 2 f ∂ x 1 2 , f x 2 x 2 = ∂ 2 f ∂ x 2 2 , f x 1 x 2 = ∂ 2 f ∂ x 1 ∂ x 2 \Delta x_1=x_1-x_{10}, \Delta x_2=x_2-x_{20}, f_{x_1}=\cfrac{\partial f}{\partial x_1}, f_{x_2}=\cfrac{\partial f}{\partial x_2}, f_{x_1x_1}=\cfrac{\partial^2f}{\partial x_1^2}, f_{x_2x_2}=\cfrac{\partial^2f}{\partial x_2^2}, f_{x_1x_2}=\cfrac{\partial^2f}{\partial x_1\partial x_2} Δx1=x1−x10,Δx2=x2−x20,fx1=∂x1∂f,fx2=∂x2∂f,fx1x1=∂x12∂2f,fx2x2=∂x22∂2f,fx1x2= 标签： fx1风向传感器