添加拉格朗日插值.

Signed-off-by: lion.chan <cy187lion@sina.com>

Algorithm/DSP/拉格朗日插值.md

@@ -1,3 +1,179 @@
 # 拉格朗日插值
 
-<https://coast.nd.edu/jjwteach/www/www/30125/pdfnotes/lecture3_6v13.pdf>
+Fit N+1 points with an $N^{th}$ degree polynomial
+
+![Fit N+1 points](img/拉格朗日插值/001.png)
+
+f(x) = exact function of which only N+1 discrete values are known and used to estab-
+lish an interpolating or approximating function g(x).
+
+g(x) = approximating or interpolating function. This function will pass through all
+specified N+1 interpolation points (also referred to as **data points** or **nodes**).
+
+The interpolation points or nodes are given as:
+
+$$\begin{gather*}
+    x_0 & f(x_0)\equiv f_0 \\
+    x_1 & f(x_1)\equiv f_1 \\
+    x_2 & f(x_2)\equiv f_2 \\
+    ... \\
+    x_N & f(x_N)\equiv f_N
+\end{gather*}$$
+
+There exists only one $N^{th}$ degree polynomial that passes through a given set of N+1 points. It’s form is (expressed as a power series):
+
+$$g(x)=a_0+a_1x+a_2x^2+a_3x^3+...+a_Nx^N$$
+
+where $a_i=\text{unknown coefficients}$, $i=0,N\text{(N+1 coefficients)}$.
+
+No matter how we derive the $N^{th}$ degree polynomial:
+
+* Fitting power series
+* Lagrange interpolating functions
+* Newton forward or backward interpolation
+
+The resulting polynomial will always be the same!
+
+## Power Series Fitting to Define Lagrange Interpolation
+
+g(x) must match f(x) at the selected data points:
+
+$$\begin{gather*}
+    g(x_0)=f_0 \to a_0+a_1x_0+a_2x_0^2+...+a_Nx_0^N=f_0 \\
+    g(x_1)=f_1 \to a_0+a_1x_1+a_2x_1^2+...+a_Nx_1^N=f_1 \\
+    ... \\
+    g(x_N)=f_N \to a_0+a_1x_N+a_2x_N^2+...+a_Nx_N^N=f_N \\
+\end{gather*}$$
+
+Solve set of simultaneous equations:
+
+$$\begin{bmatrix}
+    1 & x_0 & x_0^2 & ... & x_0^N \\
+    1 & x_1 & x_1^2 & ... & x_1^N \\
+    ... \\
+    1 & x_N & x_N^2 & ... & x_N^N
+\end{bmatrix} \begin{bmatrix}
+    a_0 \\ a_1 \\ ... \\ a_N
+\end{bmatrix} = \begin{bmatrix}
+    f_0 \\ f_1 \\ ... \\ f_N
+\end{bmatrix}$$
+
+It is relatively computationally costly to solve the coefficients of the interpolating function g(x) (i.e. you need to program a solution to these equations).
+
+## Lagrange Interpolation Using Basis Functions
+
+We note that in general:
+
+$$g(x_i)=f_i$$
+
+Let
+
+$$g(x)=\sum_{i=0}^Nf_iV_i(x)$$
+
+where $V_i(x)$ polynomial of degree associated with each node such that
+
+$$\begin{equation*}
+    V_i(x_j)\equiv \begin{cases}
+        0 & i\neq j \\
+        1 & i= j
+    \end{cases}
+\end{equation*}$$
+
+For example if we have 5 interpolation points (or nodes)
+
+$$g(x_3)=f_0V_0(x_3)+f_1V_1(x_3)+f_2V_2(x_3)+f_3V_3(x_3)+f_4V_4(x_3)$$
+
+Using the definition for $V_i(x_j)$:
+
+$$\begin{gather*}
+    V_0(x_3)=0 \\
+    V_1(x_3)=0 \\
+    V_2(x_3)=0 \\
+    V_3(x_3)=1 \\
+    V_4(x_4)=0
+\end{gather*}$$
+
+we have:
+
+$$g(x_3)=f_3$$
+
+How do we construct $V_i(x)$?
+
+* Degree N
+* Roots at $x_0,x_1,x_2,...,x_{i-1},...,x_N$ (at all nodes except $x_i$)
+* $V_i(x_i)=1$
+
+Let $W_i(x)=(x-x_0)(x-x_1)(x-x_2)...(x-x_{i+1})...(x-x_N)$
+
+* The function $W_i$ is such that we do have the required roots, i.e. it equals zeros at nodes $x_0,x_1,x_2,...,x_N$ except at node $x_i$
+* Degree of $W_i(x)$ is N
+* However $W_i(x)$ in the form presented will not equal to unity at $x_i$
+
+We normalize $W_i(x)$ and define the Lagrange basis functions $V_i(x)$:
+
+$$V_i(x)=\frac{(x-x_0)(x-x_1)(x-x_2)...(x-x_{i-1})(x-x_{i+1})...(x-x_N)}{(x_i-x_0)(x_i-x_1)(x_i-x_2)...(x_i-x_{i-1})(x_i-x_{i+1})...(x_i-x_N)}$$
+
+Now we have $V_i(x)$ such that $V_i(x_i)$ equals:
+
+$$\begin{gather*}
+V_i(x)=\frac{(x_i-x_0)(x_i-x_1)(x_i-x_2)...(x_i-x_{i-1})(1)(x_i-x_{i+1})...(x_i-x_N)}{(x_i-x_0)(x_i-x_1)(x_i-x_2)...(x_i-x_{i-1})(x_i-x_{i+1})...(x_i-x_N)} \\
+\\
+\to V_i(x_i)=1
+\end{gather*}$$
+
+We alos satisfy $V_i(x_j)=0$ for $i\neq j$
+
+e.g.
+
+$$V_1(x_2)=\frac{(x_2-x_0)(1)(x_2-x_2)(x_2-x_3)...(x_2-x_N)}{(x_1-x_0)(1)(x_1-x_2)(x_1-x_3)...(x_1-x_N)}=0$$
+
+The general form of the interpolating function g(x) with the specified form of $V_i(x)$ is:
+
+$$g(x)=\sum_{i=0}^{N}f_iV_i(x)$$
+
+* The sum of polynomials of degree N is also polynomial of degree N
+* g(x) is equivalent to fitting the power series and computing coefficients $a_0,...,a_N$
+
+## Lagrange Linear Interpolation Using Basis Functions
+
+Linear Lagrange (N=1) is the simplest form of Lagrange Interpolation:
+
+$$\begin{gather*}
+    g(x)=\sum_{i=0}^{1}f_iV_i(x) \\
+    \to g(x)=f_0V_0(x)+f_1V_1(x)
+\end{gather*}$$
+
+where
+
+$$V_0(x)=\frac{(x-x_1)}{(x_0-x_1)}=\frac{(x_1-x)}{(x_1-x_0)}$$
+
+and
+
+$$V_1(x)=\frac{(x-x_0)}{(x_1-x_0)}$$
+
+## Example
+
+Given the following data:
+
+$$\begin{gather*}
+    x_0=2 & f_0=1.5 \\
+    x_1=5 & f_1=4.0
+\end{gather*}$$
+
+Find the linear interpolating function g(x)
+
+Lagrange basis functions are:
+
+$$\begin{gather*}
+    V_0(x)=\frac{x-5}{-3} \\
+    \\
+    V_1(x)=\frac{x-2}{3}
+\end{gather*}$$
+
+Interpolating function g(x) is:
+
+$$g(x)=1.5V_0(x)+4.0V_1(x)$$
+
+## 外部参考连接
+
+1. [原文](https://coast.nd.edu/jjwteach/www/www/30125/pdfnotes/lecture3_6v13.pdf)

Software/Applications/KiCAD/KiCAD_简要说明.md

@@ -1,5 +1,7 @@
 # KiCAD 简要说明
 
+TODO:
+
 1. <https://blog.csdn.net/weixin_44916154/article/details/124503031>
 2. <https://gitee.com/hooke6164/footprint_-ki-cad_-gg/>
 3. <https://blog.csdn.net/yannanxiu/article/details/52643410>

Software/Development/Language/LaTex/LaTex_数学公式.md

@@ -8,6 +8,9 @@
   - [5. 带标字母](#5-带标字母)
   - [6. 常用数集](#6-常用数集)
   - [7. 组合](#7-组合)
+    - [7.1. 基本组合](#71-基本组合)
+    - [7.2. 矩阵](#72-矩阵)
+  - [8. 排版](#8-排版)
 
 ## 1. 行内公式与行间公式
 
@@ -141,6 +144,8 @@ $$a+b$$
 
 ## 7. 组合
 
+### 7.1. 基本组合
+
 等式：
 
 $$\begin{equation}
@@ -190,3 +195,36 @@ $$\begin{split}
     F_s(s)=\int_{0}^{\infty}[\sum_{n=0}^{\infty}f(nT)\delta(t-nT)]e^{-st}dt \\
     =\sum_{n=0}^{\infty}f(nT)e^{-snT} \\
 \end{split}$$
+
+### 7.2. 矩阵
+
+$$\begin{matrix}
+    0 & 1 \\ 1 & 0 \\
+\end{matrix}$$
+
+$$\begin{pmatrix}
+    0 & 1 \\ 1 & 0 \\
+\end{pmatrix}$$
+
+$$\begin{bmatrix}
+    0 & 1 \\ 1 & 0 \\
+\end{bmatrix}$$
+
+$$\begin{Bmatrix}
+    0 & 1 \\ 1 & 0 \\
+\end{Bmatrix}$$
+
+$$\begin{vmatrix}
+    0 & 1 \\ 1 & 0 \\
+\end{vmatrix}$$
+
+$$\begin{Vmatrix}
+    0 & 1 \\ 1 & 0 \\
+\end{Vmatrix}$$
+
+## 8. 排版
+
+| LaTex | Description |
+|-------|-------------|
+| \\ | 换行 |
+| &  | 间隔，相当于 Tab |