牛顿法求解动态矩阵的逆、动态矩阵Moore-Penrose逆、动态矩阵形式线性方程问题

爱被打了一巴掌 2022-12-26 00:49 193阅读 0赞

## 1. 牛顿法求解动态矩阵形式线性方程问题 ##

动态矩阵线性方程问题的定义：![A(t)x(t)=b(t)][A_t_x_t_b_t]

构造误差函数：![\\epsilon(t) = A(t)x(t)-b(t)][epsilon_t_ _ A_t_x_t_-b_t]

牛顿法的定义：

![x\_\{n+1\} = x\_n - \\frac\{y\}\{\\dot y\}][x_n_1_ _ x_n - _frac_y_dot y]

对误差函数求导：

![\\frac\{d\\epsilon(t)\}\{dx(t)\} = A(t)\\Rightarrow \\dot \\epsilon\_x(t)=A(t)][frac_d_epsilon_t_dx_t_ _ A_t_Rightarrow _dot _epsilon_x_t_A_t]

于是，根据牛顿法的迭代公式可以得到：

![x\_\{n+1\} = x\_n - A\_n^\{\\dagger\}(A\_nx\_n - b\_n)][x_n_1_ _ x_n - A_n_dagger_A_nx_n - b_n]

## 2. 牛顿法求解动态矩阵逆问题 ##

动态矩阵的逆问题的定义：![A(t)x(t)=I][A_t_x_t_I]

构造误差函数：![\\epsilon(t) = A(t)x(t)-I][epsilon_t_ _ A_t_x_t_-I]

牛顿法的定义：

![x\_\{n+1\} = x\_n - \\frac\{y\}\{\\dot y\}][x_n_1_ _ x_n - _frac_y_dot y]

对误差函数求导：

![\\frac\{d\\epsilon(t)\}\{dx(t)\} = A(t)\\Rightarrow \\dot \\epsilon\_x(t)=A(t)][frac_d_epsilon_t_dx_t_ _ A_t_Rightarrow _dot _epsilon_x_t_A_t]

于是，根据牛顿法的迭代公式可以得到：

![x\_\{n+1\} = x\_n - A\_n^\{-1\}(A\_nx\_n - I)][x_n_1_ _ x_n - A_n_-1_A_nx_n - I]

![x\_\{n+1\} = A\_n^\{-1\}][x_n_1_ _ A_n_-1]

## 3. 牛顿法求解动态矩阵Moore-Penrose逆问题 ##

动态矩阵的Moore-Penrose逆定义：

![M^\\text\{T\}(t)=M^\{\\dagger\}(t)M(t)M^\{\\text\{T\}\}(t), m<n][M_text_T_t_M_dagger_t_M_t_M_text_T_t_ m_n]

![M^\\text\{T\}(t)=M^\{\\text\{T\}\}(t)M(t)M^\{\\dagger\}(t), m>n][M_text_T_t_M_text_T_t_M_t_M_dagger_t_ m_n]

上式表示的是右Moore-Penrose逆，下式表示的是左伪逆，在这里仅讨论左伪逆的推导方式。

构造误差函数：![\\epsilon(t) = X(t)M(t)M^\{\\text\{T\}\}(t)-M^\{\\text\{T\}\}(t)][epsilon_t_ _ X_t_M_t_M_text_T_t_-M_text_T_t]

牛顿法的定义：

![x\_\{n+1\} = x\_n - \\frac\{y\}\{\\dot y\}][x_n_1_ _ x_n - _frac_y_dot y]

对误差函数求导：

![\\frac\{d\\epsilon(t)\}\{dx(t)\}=M^\{\\text\{T\}\}(t)M(t)\\Rightarrow \\dot \\epsilon\_x(t)=M^\{\\text\{T\}\}(t)M(t)][frac_d_epsilon_t_dx_t_M_text_T_t_M_t_Rightarrow _dot _epsilon_x_t_M_text_T_t_M_t]

根据牛顿法的迭代公式可以得到：

![X\_\{n+1\}=X\_n-(M^\{\\text\{T\}\}\_n M\_n)^\{-1\}(M^\{\\text\{T\}\}\_n M\_nX\_n-M^\{\\text\{T\}\}\_n)][X_n_1_X_n-_M_text_T_n M_n_-1_M_text_T_n M_nX_n-M_text_T_n]

[A_t_x_t_b_t]: https://private.codecogs.com/gif.latex?A%28t%29x%28t%29%3Db%28t%29
[epsilon_t_ _ A_t_x_t_-b_t]: https://private.codecogs.com/gif.latex?%5Cepsilon%28t%29%20%3D%20A%28t%29x%28t%29-b%28t%29
[x_n_1_ _ x_n - _frac_y_dot y]: https://private.codecogs.com/gif.latex?x_%7Bn&plus;1%7D%20%3D%20x_n%20-%20%5Cfrac%7By%7D%7B%5Cdot%20y%7D
[frac_d_epsilon_t_dx_t_ _ A_t_Rightarrow _dot _epsilon_x_t_A_t]: https://private.codecogs.com/gif.latex?%5Cfrac%7Bd%5Cepsilon%28t%29%7D%7Bdx%28t%29%7D%20%3D%20A%28t%29%5CRightarrow%20%5Cdot%20%5Cepsilon_x%28t%29%3DA%28t%29
[x_n_1_ _ x_n - A_n_dagger_A_nx_n - b_n]: https://private.codecogs.com/gif.latex?x_%7Bn&plus;1%7D%20%3D%20x_n%20-%20A_n%5E%7B%5Cdagger%7D%28A_nx_n%20-%20b_n%29
[A_t_x_t_I]: https://private.codecogs.com/gif.latex?A%28t%29x%28t%29%3DI
[epsilon_t_ _ A_t_x_t_-I]: https://private.codecogs.com/gif.latex?%5Cepsilon%28t%29%20%3D%20A%28t%29x%28t%29-I
[x_n_1_ _ x_n - A_n_-1_A_nx_n - I]: https://private.codecogs.com/gif.latex?x_%7Bn&plus;1%7D%20%3D%20x_n%20-%20A_n%5E%7B-1%7D%28A_nx_n%20-%20I%29
[x_n_1_ _ A_n_-1]: https://private.codecogs.com/gif.latex?x_%7Bn&plus;1%7D%20%3D%20A_n%5E%7B-1%7D
[M_text_T_t_M_dagger_t_M_t_M_text_T_t_ m_n]: https://private.codecogs.com/gif.latex?M%5E%5Ctext%7BT%7D%28t%29%3DM%5E%7B%5Cdagger%7D%28t%29M%28t%29M%5E%7B%5Ctext%7BT%7D%7D%28t%29%2C%20m%3Cn
[M_text_T_t_M_text_T_t_M_t_M_dagger_t_ m_n]: https://private.codecogs.com/gif.latex?M%5E%5Ctext%7BT%7D%28t%29%3DM%5E%7B%5Ctext%7BT%7D%7D%28t%29M%28t%29M%5E%7B%5Cdagger%7D%28t%29%2C%20m%3En
[epsilon_t_ _ X_t_M_t_M_text_T_t_-M_text_T_t]: https://private.codecogs.com/gif.latex?%5Cepsilon%28t%29%20%3D%20X%28t%29M%28t%29M%5E%7B%5Ctext%7BT%7D%7D%28t%29-M%5E%7B%5Ctext%7BT%7D%7D%28t%29
[frac_d_epsilon_t_dx_t_M_text_T_t_M_t_Rightarrow _dot _epsilon_x_t_M_text_T_t_M_t]: https://private.codecogs.com/gif.latex?%5Cfrac%7Bd%5Cepsilon%28t%29%7D%7Bdx%28t%29%7D%3DM%5E%7B%5Ctext%7BT%7D%7D%28t%29M%28t%29%5CRightarrow%20%5Cdot%20%5Cepsilon_x%28t%29%3DM%5E%7B%5Ctext%7BT%7D%7D%28t%29M%28t%29
[X_n_1_X_n-_M_text_T_n M_n_-1_M_text_T_n M_nX_n-M_text_T_n]: https://private.codecogs.com/gif.latex?X_%7Bn&plus;1%7D%3DX_n-%28M%5E%7B%5Ctext%7BT%7D%7D_n%20M_n%29%5E%7B-1%7D%28M%5E%7B%5Ctext%7BT%7D%7D_n%20M_nX_n-M%5E%7B%5Ctext%7BT%7D%7D_n%29