diff options
Diffstat (limited to 'doc/html/formula.repository')
| -rw-r--r-- | doc/html/formula.repository | 56 |
1 files changed, 12 insertions, 44 deletions
diff --git a/doc/html/formula.repository b/doc/html/formula.repository index bfa1d28..7bee4b5 100644 --- a/doc/html/formula.repository +++ b/doc/html/formula.repository @@ -19,47 +19,15 @@ \form#18:$ M $ \form#19:\[ \begin{aligned} & (1 - p_0^N)^M \leq(1 - P) \\ \Rightarrow & M \log(1 - p_0^N) \leq \log(1 - P) \\ \Rightarrow & M \geq \frac{\log(1 - p)}{\log(1 - p_0^N)},~~ \because (1-p_0^N<1 \Rightarrow \log(1-p_0^N)<0) \end{aligned} \] \form#20:$ M = \lceil \frac{\log(1 - P)}{\log(1 - p_0^N)} \rceil $ -\form#21:$ F: \mathbb{R} ^N \mapsto \mathbb{R}^M $ -\form#22:$ v $ -\form#23:$ F(v)^T F(v) = 0$ -\form#24:$ \epsilon $ -\form#25:$ F(v)^T F(v) < \epsilon $ -\form#26:$ v_0 $ -\form#27:$ v_1, v_2, v_3, v_4... $ -\form#28:$ v_k $ -\form#29:$ F(v_k)^TF(v_k)<\epsilon $ -\form#30:\[ v_{i+1} = v_i + (J(v_i)^TJ(v_i)+\lambda I_{N\times N})^{-1} J(v_i)^T F(v_i) \] -\form#31:$ J(v) $ -\form#32:\[ J(v) = \frac{d}{dv}F(v) = \left[ \begin{array}{ccccc} \frac{\partial F_1(v)}{\partial v_1} & \frac{\partial F_1(v)}{\partial v_2} & \frac{\partial F_1(v)}{\partial v_3} & ... & \frac{\partial F_1(v)}{\partial v_N} \\ \frac{\partial F_2(v)}{\partial v_1} & \frac{\partial F_2(v)}{\partial v_2} & \frac{\partial F_2(v)}{\partial v_3} & ... & \frac{\partial F_2(v)}{\partial v_N} \\ \frac{\partial F_3(v)}{\partial v_1} & \frac{\partial F_3(v)}{\partial v_2} & \frac{\partial F_3(v)}{\partial v_3} & ... & \frac{\partial F_3(v)}{\partial v_N} \\ . & . & . & & . \\ . & . & . & & . \\ . & . & . & & . \\ \frac{\partial F_M(v)}{\partial v_1} & \frac{\partial F_M(v)}{\partial v_2} & \frac{\partial F_M(v)}{\partial v_3} & ... & \frac{\partial F_M(v)}{\partial v_N} \\ \end{array} \right] \] -\form#33:$ \lambda $ -\form#34:$ F $ -\form#35:$ J $ -\form#36:$ \lambda I_{N \times N} $ -\form#37:\[ S_{top}(v) = \begin{cases} true & if~F(v)<\epsilon \\ false & else \end{cases} \] -\form#38:$ R $ -\form#39:\[ \left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ . \\ . \\ . \\ x_N \\ \end{array} \right] \stackrel{transformate}{\rightarrow} \left[ \begin{array}{c} \frac{x_1 \times R}{L} \\ \frac{x_2 \times R}{L} \\ \frac{x_3 \times R}{L} \\ . \\ . \\ . \\ \frac{x_N \times R}{L} \\ \end{array} \right] \\ \] -\form#40:$ L=\sqrt{x_1^2 + x_2^2 + x_3^2 + ... + x_N^2 } $ -\form#41:$ L $ -\form#42:$ f $ -\form#43:\[ \left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ . \\ . \\ . \\ x_N \\ \end{array} \right] \stackrel{transformate}{\rightarrow} \left[ \begin{array}{c} \frac{-x_1 \times f}{x_N} \\ \frac{-x_2 \times f}{x_N} \\ \frac{-x_3 \times f}{x_N} \\ . \\ . \\ . \\ -f \\ \end{array} \right] \\ \] -\form#44:$ x_N = -f $ -\form#45:$ L=\sqrt{x_1^2+x_2^2+...+x_N^2} $ -\form#46:\[ \frac{R}{L^3} \times \left[ \begin{array}{ccccc} L^2-x_1^2 & -x_1x_2 & -x_1x_3 & ... & -x_1x_N \\ -x_2x_1 & L^2-x_2^2 & -x_2x_3 & ... & -x_2x_N \\ -x_3x_1 & -x_3x_2 & L^2-x_3^2 & ... & -x_3x_N \\ . & . & . & & . \\ . & . & . & & . \\ . & . & . & & . \\ -x_Nx_1 & -x_Nx_2 & -x_Nx_3 & ... & L^2-x_N^2 \\ \end{array} \right] \] -\form#47:\[ R \times \left[ \begin{array}{c} \frac{x_1}{L} \\ \frac{x_2}{L} \\ \frac{x_3}{L} \\ . \\ . \\ . \\ \frac{x_N}{L} \\ \end{array} \right] \] -\form#48:\[ f \times \left[ \begin{array}{ccccc} \frac{-1}{x_N} & 0 & 0 & ... & \frac{1}{x_N^2} \\ 0 & \frac{-1}{x_N} & 0 & ... & \frac{1}{x_N^2} \\ 0 & 0 & \frac{-1}{x_N} & ... & \frac{1}{x_N^2} \\ . & . & . & & . \\ . & . & . & & . \\ . & . & . & & . \\ 0 & 0 & 0 & ... & 0 \\ \end{array} \right] \] -\form#49:\[ f \times \left[ \begin{array}{c} \frac{-x_1}{x_N} \\ \frac{-x_2}{x_N} \\ \frac{-x_3}{x_N} \\ . \\ . \\ . \\ -1 \\ \end{array} \right] \] -\form#50:\[ v_{output} = H(h_1, G(g_1, g_2, g_3, F(f_1, v_{input}))) \] -\form#51:$ v_{input}(x,y,z), v_{output} $ -\form#52:$ y $ -\form#53:\[ m_{jacobian} = \frac{\partial H(h_1, G(g_1, g_2, g_3, F(f_1, v_{input})))} {\partial G(g_1, g_2, g_3, F(f_1, v_{input})) } \frac{\partial G(g_1, g_2, g_3, F(f_1, v_{input}))} {\partial F(f_1, v_{input}) } \frac{\partial F(f_1, v_{input})} {\partial v_{input} } \frac{\partial v_{input}} {\partial y} \] -\form#54:\[ \frac{\partial v_{input}}{\partial y} = \left[ \begin{array}{c} 0 \\ 1 \\ 0 \\ \end{array} \right] \] -\form#55:\[ v_{output} = H(h_1, h_2, G(g_1, g_2, g_3, F(f_1, v_{input}))) \] -\form#56:$ f_1, g_1, g_2, g_3, h_1, h_2 $ -\form#57:$ F, G, H $ -\form#58:\[ M_{jacobian} = \frac{\partial H(h_1, h_2, G(g_1, g_2, g_3, F(f_1, v_{input})))} {\partial G(g_1, g_2, g_3, F(f_1, v_{input})) } \frac{\partial G(g_1, g_2, g_3, F(f_1, v_{input})) } {\partial F(f_1, v_{input}) } \frac{\partial F(f_1, v_{input}) } {\partial v_{input} } \frac{\partial v_{input} } {\partial y } \] -\form#59:\[ v_{output} = I(i_1,i_2, H(h_1,h_2, G(g_1,g_2,g_3, F(f_1, v_{input})))) \] -\form#60:$ f_1, g_1,g_2,g_3, h_1,h_2, i_1,i_2 $ -\form#61:$ F, G, H, I $ -\form#62:$ g_2 $ -\form#63:\[ M_{jacobian} = \frac{\partial I(i_1,i_2, H(h_1,h_2, G(g_1,g_2,g_3, F(f_1, v_{input}))))} {\partial H(h_1,h_2, G(g_1,g_2,g_3, F(f_1, v_{input}))) } \frac{\partial H(h_1,h_2, G(g_1,g_2,g_3, F(f_1, v_{input}))) } {\partial G(g_1,g_2,g_3, F(f_1, v_{input})) } \frac{\partial G(g_1,g_2,g_3, F(f_1, v_{input})) } {\partial g_2 } \] -\form#64:$ f_1, g_1, g_2, g_3, h_1, h_2, i_1, i_2 $ +\form#21:$ R $ +\form#22:\[ \left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ . \\ . \\ . \\ x_N \\ \end{array} \right] \stackrel{transformate}{\rightarrow} \left[ \begin{array}{c} \frac{x_1 \times R}{L} \\ \frac{x_2 \times R}{L} \\ \frac{x_3 \times R}{L} \\ . \\ . \\ . \\ \frac{x_N \times R}{L} \\ \end{array} \right] \\ \] +\form#23:$ L=\sqrt{x_1^2 + x_2^2 + x_3^2 + ... + x_N^2 } $ +\form#24:$ L $ +\form#25:$ f $ +\form#26:\[ \left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ . \\ . \\ . \\ x_N \\ \end{array} \right] \stackrel{transformate}{\rightarrow} \left[ \begin{array}{c} \frac{-x_1 \times f}{x_N} \\ \frac{-x_2 \times f}{x_N} \\ \frac{-x_3 \times f}{x_N} \\ . \\ . \\ . \\ -f \\ \end{array} \right] \\ \] +\form#27:$ x_N = -f $ +\form#28:$ L=\sqrt{x_1^2+x_2^2+...+x_N^2} $ +\form#29:\[ \frac{R}{L^3} \times \left[ \begin{array}{ccccc} L^2-x_1^2 & -x_1x_2 & -x_1x_3 & ... & -x_1x_N \\ -x_2x_1 & L^2-x_2^2 & -x_2x_3 & ... & -x_2x_N \\ -x_3x_1 & -x_3x_2 & L^2-x_3^2 & ... & -x_3x_N \\ . & . & . & & . \\ . & . & . & & . \\ . & . & . & & . \\ -x_Nx_1 & -x_Nx_2 & -x_Nx_3 & ... & L^2-x_N^2 \\ \end{array} \right] \] +\form#30:\[ R \times \left[ \begin{array}{c} \frac{x_1}{L} \\ \frac{x_2}{L} \\ \frac{x_3}{L} \\ . \\ . \\ . \\ \frac{x_N}{L} \\ \end{array} \right] \] +\form#31:\[ f \times \left[ \begin{array}{ccccc} \frac{-1}{x_N} & 0 & 0 & ... & \frac{1}{x_N^2} \\ 0 & \frac{-1}{x_N} & 0 & ... & \frac{1}{x_N^2} \\ 0 & 0 & \frac{-1}{x_N} & ... & \frac{1}{x_N^2} \\ . & . & . & & . \\ . & . & . & & . \\ . & . & . & & . \\ 0 & 0 & 0 & ... & 0 \\ \end{array} \right] \] +\form#32:\[ \left[ \begin{array}{c} \frac{-x_1}{x_N} \\ \frac{-x_2}{x_N} \\ \frac{-x_3}{x_N} \\ . \\ . \\ . \\ -1 \\ \end{array} \right] \] |