LaTeX編輯數學公式基本語法元素
LaTeX中的數學模式有兩種形式:
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inline 和 display。
- 前者是指在正文插入行間數學公式,后者獨立排列,可以有或沒有編號。
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行間公式(inline)
- 用$將公式括起來。
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塊間公式(displayed)
- 用$$將公式括起來是無編號的形式
- 還有[.....]的無編號獨立公式形式但Markdown好像不支持。
- 塊間元素默認是居中顯示的。
各類希臘字母編輯表
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上下標、根號、省略號
- 下標：x_i:\(x_i\)
- 上標：x^2: \(x^2\)
- 注意：上下標如果多於一個字母或者符號，需要用一對{}括起來 x_{i1}: \(x_{i1}\) \(x^{at}\)
- 根號: \sqrt[n]{5}: \(\sqrt[n]{5}\)
- 省略號：\cdots: \(\cdots\)
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運算符
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基本運算符+ - * ÷
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求和:
- \sum_1^n: \(\sum_1^n\)
- \sum_{x,y}: \(\sum_{x,y}\)
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積分:
- \int_1^n: \(\int_1^n\)
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極限
- lim_{x \to \infy}: \(lim\_{x \to \infty}\)
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行列式
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$$ X=\left| \begin{matrix} x_{11} & x_{12} & \cdots & x_{1d}\\ x_{21} & x_{22} & \cdots & x_{2d}\\ \vdots & \vdots & \ddots & \vdots \\ x_{11} & x_{12} & \cdots & x_{1d}\\ \end{matrix} \right| $$\[X=\left| \begin{matrix} x_{11} & x_{12} & \cdots & x_{1d}\\ x_{21} & x_{22} & \cdots & x_{2d}\\ \vdots & \vdots & \ddots & \vdots \\ x_{11} & x_{12} & \cdots & x_{1d}\\ \end{matrix} \right| \]
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矩陣
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$$ \begin{matrix} 1 & x & x^2\\ 1 & y & y^2\\ 1 & z & z^2\\ \end{matrix} $$\[\begin{matrix} 1 & x & x^2\\ 1 & y & y^2\\ 1 & z & z^2\\ \end{matrix} \]
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箭頭
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分段函數
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$$ f(n)= \begin{cases} n/2, & \text{if $n$ is even}\\ 3n+1,& \text{if $n$ is odd} \end{cases} $$\[f(n)= \begin{cases} n/2, & \text{if $n$ is even}\\ 3n+1,& \text{if $n$ is odd} \end{cases} \]
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方程組
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$$ \left\{ \begin{array}{c} a_1x+b_1y+c_1z=d_1\\ a_2x+b_2y+c_2z=d_2\\ a_3x+b_3y+c_3z=d_3 \end{array} \right. $$\[\left\{ \begin{array}{c} a_1x+b_1y+c_1z=d_1\\ a_2x+b_2y+c_2z=d_2\\ a_3x+b_3y+c_3z=d_3 \end{array} \right. \]
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常用公式
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線性模型
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$$ h(\theta) = \sum_{j=0} ^n \theta_j x_j $$\[h(\theta) = \sum_{j=0} ^n \theta_j x_j \]
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均方誤差
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$$ J(\theta) = \frac{1}{2m}\sum_{i=0}^m(y^i - h_\theta(x^i))^2 $$\[J(\theta) = \frac{1}{2m}\sum_{i=0}^m(y^i - h_\theta(x^i))^2 \]
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求積公式
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\$$ H_c=\sum_{l_1+\dots +l_p}\prod^p_{i=1} \binom{n_i}{l_i} \$$$$ H_c=\sum_{l_1+\dots +l_p}\prod^p_{i=1} \binom{n_i}{l_i} $$
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批量梯度下降
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$$ \frac{\partial J(\theta)}{\partial\theta_j} = -\frac1m\sum_{i=0}^m(y^i - h_\theta(x^i))x^i_j $$\[\frac{\partial J(\theta)}{\partial\theta_j} = -\frac1m\sum_{i=0}^m(y^i - h_\theta(x^i))x^i_j \]
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推導過程
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$$ \begin{align} \frac{\partial J(\theta)}{\partial\theta_j} & = -\frac1m\sum_{i=0}^m(y^i - h_\theta(x^i)) \frac{\partial}{\partial\theta_j}(y^i-h_\theta(x^i))\\ & = -\frac1m\sum_{i=0}^m(y^i-h_\theta(x^i)) \frac{\partial}{\partial\theta_j}(\sum_{j=0}^n\theta_j x^i_j-y^i)\\ &=-\frac1m\sum_{i=0}^m(y^i -h_\theta(x^i)) x^i_j \end{align} $$\[\begin{align} \frac{\partial J(\theta)}{\partial\theta_j} & = -\frac1m\sum_{i=0}^m(y^i - h_\theta(x^i)) \frac{\partial}{\partial\theta_j}(y^i-h_\theta(x^i))\\ & = -\frac1m\sum_{i=0}^m(y^i-h_\theta(x^i)) \frac{\partial}{\partial\theta_j}(\sum_{j=0}^n\theta_j x^i_j-y^i)\\ &=-\frac1m\sum_{i=0}^m(y^i -h_\theta(x^i)) x^i_j \end{align} \]
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字符下標
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$$ \max \limits_{a<x<b}\{f(x)\} $$\[\max \limits_{a<x<b}\{f(x)\} \]
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