FIx ch03

docs/chapter3/chapter3_2_1.md

@@ -30,12 +30,12 @@ $$
 
 $$
 \begin{gathered}
-f(x, y)=f\left(x_k, y_k\right)+\left(x-x_k\right) f_x^{\prime}\left(x_k, y_k\right)+\left(y-y_k\right) f_y^{\prime}\left(x_k, y_k\right) \\
-+\frac{1}{2 !}\left(x-x_k\right)^2 f_{x x}^{\prime \prime}\left(x_k, y_k\right)+\frac{1}{2 !}\left(x-x_k\right)\left(y-y_k\right) f_{x y}^{\prime \prime}\left(x_k, y_k\right) \\
-+\frac{1}{2 !}\left(x-x_k\right)\left(y-y_k\right) f_{y x}^{\prime \prime}\left(x_k, y_k\right)+\frac{1}{2 !}\left(y-y_k\right)^2 f_{y y}^{\prime \prime}\left(x_k, y_k\right) +\ldots
-\end{gathered} \tag{5}
+f\left(x^1, x^2, \ldots, x^n\right)=f\left(x_k^1, x_k^2, \ldots, x_k^n\right)+\sum_{i=1}^n\left(x^i-x_k^i\right) f_{x^i}^{\prime}\left(x_k^1, x_k^2, \ldots, x_k^n\right) \\
++\frac{1}{2!} \sum_{i, j=1}^n\left(x^i-x_k^i\right)\left(x^j-x_k^j\right) f_{i j}^{\prime \prime}\left(x_k^1, x_k^2, \ldots, x_k^n\right)  +o^n \tag{5}
+\end{gathered}
 $$
 
+
   推广到矩阵形式，可表示为：
 
 $$

docs/chapter3/chapter3_2_2.md

@@ -32,10 +32,9 @@ $$
 
 $$
 \begin{gathered}
-f(x, y)=f\left(x_k, y_k\right)+\left(x-x_k\right) f_x^{\prime}\left(x_k, y_k\right)+\left(y-y_k\right) f_y^{\prime}\left(x_k, y_k\right) \\
-+\frac{1}{2 !}\left(x-x_k\right)^2 f_{x x}^{\prime \prime}\left(x_k, y_k\right)+\frac{1}{2 !}\left(x-x_k\right)\left(y-y_k\right) f_{x y}^{\prime \prime}\left(x_k, y_k\right) \\
-+\frac{1}{2 !}\left(x-x_k\right)\left(y-y_k\right) f_{y x}^{\prime \prime}\left(x_k, y_k\right)+\frac{1}{2 !}\left(y-y_k\right)^2 f_{y y}^{\prime \prime}\left(x_k, y_k\right) +\ldots
-\end{gathered} \tag{5}
+f\left(x^1, x^2, \ldots, x^n\right)=f\left(x_k^1, x_k^2, \ldots, x_k^n\right)+\sum_{i=1}^n\left(x^i-x_k^i\right) f_{x^i}^{\prime}\left(x_k^1, x_k^2, \ldots, x_k^n\right) \\
++\frac{1}{2!} \sum_{i, j=1}^n\left(x^i-x_k^i\right)\left(x^j-x_k^j\right) f_{i j}^{\prime \prime}\left(x_k^1, x_k^2, \ldots, x_k^n\right)  +o^n \tag{5}
+\end{gathered}
 $$
 
   推广到矩阵形式，可表示为：
@@ -213,6 +212,7 @@ $$
 H_{m+1}^{-1}=H_m^{-1}-\frac{H_m^{-1} \cdot X^{[m+1]} \cdot X^{[m+1] T} \cdot H_m^{-1}}{P+X^{[m+1] T} \cdot H_m^{-1} \cdot X^{[m+1]}} \tag{30}
 $$
 其中，$H_0^{-1}=\alpha^{-1} I$，$H_P^{-1}=H^{-1}$,$10^{-8}\leq \alpha \leq 10^{-4}$($\alpha$是一个常量，使得${H}_0^{-1}$有意义)，只要知道第一项，$H^{-1}$就可以一步步计算出来。
+
   推广到多个输出，可表示为：
 $$
 \begin{array}{r}