Latex公式总结_latex数学公式

(by 亲持红叶)

CATLOG

基础推荐

https://zhuanlan.zhihu.com/p/261750408

一、基础

1.1、一元一次方程

$$
y = a+bx
$$
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$$y=a+bx\text{\hspace{1em}(1)}$$

1.2 、上下标

$$
y = a + b_{ij}^2
$$
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$$y=a+b_{ij}^2\text{\hspace{1em}(2)}$$

1.3、求和

$$
\sum_{i=0}^n(x_i^2+y_j^3)
$$
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$$\sum _{i=0}^n(x_i^2+y_j^3)\text{\hspace{1em}(3)}$$

1.4、分数

$$
\frac{1}{3}-\frac{x_i^2}{y_j^3}
$$

或者

$$
\cfrac{1}{3}-\cfrac{x_i^2}{y_j^3}
$$
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$$\begin{gathered} \frac{1}{3}-\frac{x_i^2}{y_j^3} \\ \frac{1}{3}-\frac{x_i^2}{y_j^3}\text{\hspace{1em}(4)} \end{gathered}$$

1.5、矩阵

$$
\left[ 
\begin{array} {cccc}
X_1&Y_1^2\\
X_2 & Y_2^2\\
\ldots \\
X_n & Y_n^2
\end{array} 
\right]
$$
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[ X 1 Y 1 2 X 2 Y 2 2 … X n Y n 2 ] (5) \left[

$$\begin{array} {cccc} X_1&Y_1^2\\ X_2 & Y_2^2\\ \ldots \\ X_n & Y_n^2 \end{array}$$ \right] \tag 5 ⎣ ⎡​X1​X2​…Xn​​Y12​Y22​Yn2​​⎦ ⎤​(5)

1.6、希腊字母

注意大小写的首字母是区分大小写的关键

1.7、省略号

-- \ldots 　与底线对齐的省略号，\cdots 与中线对齐省略号
-- 换行 "\\"

$$
f(x_1,x_2,\ldots,x_i) = x_1+x_2+\cdots+x_i
$$

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$$f(x_1,x_2,\dots ,x_i)=x_1+x_2+\cdots +x_i\text{\hspace{1em}(6)}$$

$$\begin{gathered} f(x_1,x_2,\dots ,x_i) \\ =x_1+x_2+\cdots +x_i\text{\hspace{1em}(7)} \end{gathered}$$

1.8、大括号

" \{  \}""  (注意这里直接{}打是不行的,显示不出来的)

$$
\{a+b\}
$$
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{ a + b } (8) \{a+b\} \tag 8 {a+b}(8)

1.8、大大括号

* "\left(  \right)"
* "\left[  \right]"

$$ 
\left(a+b^2+(c*r)\right) 
$$

或者

$$ 
\left[a+b^2+(x*r)\right] 
$$
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$$\begin{gathered} \left(a+b^2+(c\ast r)\right) \\ \left[a+b^2+(x\ast r)\right]\text{\hspace{1em}(9)} \end{gathered}$$

1.9、多行公式

$$
\begin{aligned}
\cos 2 \theta & = & \cos^2 \theta - \sin^2 \theta\\
&=& 2\cos^2 \theta - 1 
\end{aligned}
$$

或者

$$ 
\begin{aligned}
\cos 2 \theta & = & \cos^2 \theta - \sin^2 \theta\\
&=& 2\cos^2 \theta - 1 
\end{aligned}
$$

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cos ⁡ 2 θ = cos ⁡ 2 θ − sin ⁡ 2 θ = 2 cos ⁡ 2 θ − 1 (10)

$$\begin{aligned} \cos 2 \theta & = & \cos^2 \theta - \sin^2 \theta\\ &=& 2\cos^2 \theta - 1 \end{aligned}$$ \tag {10} cos2θ​==​cos2θ−sin2θ2cos2θ−1​(10)

cos ⁡ 2 θ = cos ⁡ 2 θ − sin ⁡ 2 θ = 2 cos ⁡ 2 θ − 1 (11)

$$\begin{aligned} \cos 2 \theta & = & \cos^2 \theta - \sin^2 \theta\\ &=& 2\cos^2 \theta - 1 \end{aligned}$$ \tag {11} cos2θ​==​cos2θ−sin2θ2cos2θ−1​(11)

---

二、数学公式

• 下面的公式是从wps中套用过来的

2.1、傅里叶级数

$$
f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty(a_n \cos {nx} + b_n \sin {nx}) 
$$

或者

$$
\{ f(x) = {{{a_0}} \over 2} + \sum\limits_{n = 1}^\infty  {({a_n}\cos {nx} + {b_n}\sin {nx})}  \}
$$
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f ( x ) = a 0 2 + ∑ n = 1 ∞ ( a n cos ⁡ n x + b n sin ⁡ n x ) { f ( x ) = a 0 2 + ∑ n = 1 ∞ ( a n cos ⁡ n x + b n sin ⁡ n x ) } (12) f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty(a_n \cos {nx} + b_n \sin {nx}) \\ \{ f(x) = {{{a_0}} \over 2} + \sum\limits_{n = 1}^\infty {({a_n}\cos {nx} + {b_n}\sin {nx})} \} \tag {12} f(x)=2a0​​+n=1∑∞​(an​cosnx+bn​sinnx){f(x)=2a0​​+n=1∑∞​(an​cosnx+bn​sinnx)}(12)

2.2、高斯公式

$$
\iiint _ { \Omega } \left( \frac { \partial {P} } { \partial {x} } + \frac { \partial {Q} } { \partial {y} } + \frac { \partial {R} }{ \partial {z} } \right) \mathrm { d } V = \oint _ { \partial \Omega } ( P \cos \alpha + Q \cos \beta + R \cos \gamma ) \mathrm{ d} S 
$$ 
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$${\iiint }_{\Omega }\left(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\right)\mathrm{d}V={\oint }_{\partial \Omega }(P\cos \alpha +Q\cos \beta +R\cos \gamma )\mathrm{d}S\text{\hspace{1em}(13)}$$

2.3、定积分

$$
\lim\limits_ { n \rightarrow + \infty } \sum _ { i = 1 } ^ { n } f \left[ a + \frac { i } { n } ( b - a ) \right] \frac { b - a } { n } = \int _ { a } ^ { b } f ( x )\mathrm {d}x 
$$
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$$\underset{n\to +\infty }{\lim }\sum _{i=1}^{n}f\left[a+\frac{i}{n}(b-a)\right]\frac{b-a}{n}={\int }_a^{b}f(x)\mathrm{d}x\text{\hspace{1em}(14)}$$

2.4、和的展开式

$$  
( 1 + x ) ^ { n } = 1 + \frac { n x } { 1 ! } + \frac { n ( n - 1 ) x ^ { 2 } } { 2 ! } + \cdots   
$$
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$$(1+x{)}^n=1+\frac{nx}{1!}+\frac{n(n-1)x^2}{2!}+\cdots \text{\hspace{1em}(15)}$$

2.5、三角恒等式

$$  
\sin \alpha \pm \sin \beta = 2 \sin \frac { 1 } { 2 } ( \alpha \pm \beta ) \cos \frac { 1 } { 2 } ( \alpha \mp \beta )  
$$



$$
\cos \alpha + \cos \beta = 2 \cos \frac { 1 } { 2 } ( \alpha + \beta ) \cos \frac { 1 } { 2 } ( \alpha - \beta ) 
$$
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$$\begin{gathered} \sin \alpha \pm \sin \beta =2\sin \frac{1}{2}(\alpha \pm \beta )\cos \frac{1}{2}(\alpha \mp \beta ) \\ \cos \alpha +\cos \beta =2\cos \frac{1}{2}(\alpha +\beta )\cos \frac{1}{2}(\alpha -\beta )\text{\hspace{1em}(16)} \end{gathered}$$

2.6、欧拉公式

$$ 
{e^{ix}} = \cos {x} + i\sin {x}  
$$
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$$e^{ix}=\cos x+i\sin x\text{\hspace{1em}(17)}$$

2.7、格林公式

$$  
\int\!\!\!\int\limits_D {({{\partial Q} \over {\partial x}} - {{\partial P} \over {\partial y}})dxdy = \oint\limits_L {Pdx + Qdy} }  
$$
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$$\int \underset{D}{\int }(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dxdy=\underset{L}{\oint }Pdx+Qdy\text{\hspace{1em}(18)}$$

2.8、伯努利方程

$$  
{{dy} \over {dx}} + P(x)y = Q(x){y^n}(n \ne 0,1) 
$$
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$$\frac{dy}{dx}+P(x)y=Q(x)y^n(n\ne 0,1)\text{\hspace{1em}(19)}$$

2.9 、全微分方程

$$   
du(x,y) = P(x,y)dx + Q(x,y)dy = 0  
$$
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$$du(x,y)=P(x,y)dx+Q(x,y)dy=0\text{\hspace{1em}(20)}$$

2.10 、非齐次微分方程通解

$$   
y = (\int {Q(x){e^{\int {P(x)dx} }}dx + C){e^{ - \int {P(x)dx} }}}   
$$
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$$y=(\int Q(x)e^{\int P(x)dx}dx+C)e^{-\int P(x)dx}\text{\hspace{1em}(21)}$$

2.11、柯西中值定理

 $$ 
 \frac{{f(b) - f(a)}}{{F(b) - F(a)}} = \frac{{f'(\xi )}}{{F'(\xi )}} 
 $$
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$$\frac{f(b)-f(a)}{F(b)-F(a)}=\frac{f^{\prime }(\xi )}{F^{\prime }(\xi )}\text{\hspace{1em}(22)}$$

2.12、拉格朗日中值定理

$$ 
f(b) - f(a) = f'(\xi )(b - a) 
$$
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$$f(b)-f(a)=f^{\prime }(\xi )(b-a)\text{\hspace{1em}(23)}$$

2.13、曲面面积

$$
A = \int\!\!\!\int\limits_D {\sqrt {1 + {{\left( {{{\partial z} \over {\partial x}}} \right)}^2} + {{\left( {{{\partial z} \over {\partial y}}} \right)}^2}} dxdy}
$$  
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$$A=\int \underset{D}{\int }\sqrt{1+{\left(\frac{\partial z}{\partial x}\right)}^2+{\left(\frac{\partial z}{\partial y}\right)}^2}dxdy\text{\hspace{1em}(24)}$$

2.14、重积分

 $$  
 \int\!\!\!\int\limits_D {f(x,y)dxdy}  = \int\!\!\!\int\limits_{D'} {f(r\cos \theta ,r\sin \theta )rdrd\theta }   
 $$
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$$\int \underset{D}{\int }f(x,y)dxdy=\int \underset{D^{\prime }}{\int }f(r\cos \theta ,r\sin \theta )rdrd\theta \text{\hspace{1em}(25)}$$

2.15、arcsinx 求导

$$ 
(\arcsin x)' = \frac{1}{{\sqrt {1 - {x^2}} }} 
$$
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$$(\arcsin x{)}^{\prime }=\frac{1}{\sqrt{1-x^2}}\text{\hspace{1em}(26)}$$

2.16、泰勒级数

 $$ p;ko[p';p[oooo]]
 e^x = 1 + \frac{x}{1!} +  \frac{x}{2!}+ \frac{x}{3!} + \cdots + O(x^n),     -\infty < x < \infty 
 $$
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$$e^x=1+\frac{x}{1!}+\frac{x}{2!}+\frac{x}{3!}+\cdots +O(x^n),-\infty <x<\infty \text{\hspace{1em}(27)}$$

2.17、三角函数积分

$$
\int tgx \mathrm {d} x = -\ln {|\cos x|} + c
$$
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$$\int tgx\mathrm{d}x=-\ln |\cos x|+c\text{\hspace{1em}(28)}$$

2.18、二次曲面

 $$
 \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} - \frac{{{z^2}}}{{{c^2}}} = 1 
 $$
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$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\text{\hspace{1em}(29)}$$

2.19、二阶微分

$$
\frac{d^2y}{dx^2}+P(x)\frac{dy}{dx}+Q(x)y=f(x)
$$
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$$\frac{d^{2}y}{d{x}^2}+P(x)\frac{dy}{dx}+Q(x)y=f(x)\text{\hspace{1em}(30)}$$

2.20、方向导数

$$
\frac{\partial f}{\partial l}=\frac{\partial f}{\partial x}\cos{\phi}+\frac{\partial f}{\partial y}\sin{\phi}
$$
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$$\frac{\partial f}{\partial l}=\frac{\partial f}{\partial x}\cos \varphi +\frac{\partial f}{\partial y}\sin \varphi \text{\hspace{1em}(31)}$$

---

三、公式

• 下面的公式来自清风数学建模中有一张教latex语法的时候给出的4个例子

3.1、练习一

$$
\begin{aligned}
\min {f_1} &= \frac{\sqrt{\frac{1}{n-1}\sum(W_j-\overline{W}^2)}}{\overline{W}}
\\
\min{f_2} &= \max\limits_{1<j<n}{T_j}
\end{aligned}
\\
\operatorname{ s.t. }
\left\{
\begin{array}{}
x_{ij} &= 1, i = j \\
\sum\limits_{j=1}^n x_{ij}&=1, i \neq j \\
\frac{S_{ij}}{V} &\leq 3\\
x_{ij} & \in \{0,1\} \\
\overline{W} &= \frac{1}{n}\sum\limits_{j=1}^n W_j\\
&(i = 1,2,3,\cdots,m;j=1,2,\cdots,n)
\end{array}
\right.
$$
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min ⁡ f 1 = 1 n − 1 ∑ ( W j − W ‾ 2 ) W ‾ min ⁡ f 2 = max ⁡ 1 < j < n T j s.t. ⁡ { x i j = 1 , i = j ∑ j = 1 n x i j = 1 , i ≠ j S i j V ≤ 3 x i j ∈ { 0 , 1 } W ‾ = 1 n ∑ j = 1 n W j ( i = 1 , 2 , 3 , ⋯   , m ; j = 1 , 2 , ⋯   , n ) (32)

$$\begin{aligned} \min {f_1} &= \frac{\sqrt{\frac{1}{n-1}\sum(W_j-\overline{W}^2)}}{\overline{W}} \\ \min{f_2} &= \max\limits_{1<j<n}{T_j} \end{aligned}$$ \\ \operatorname{ s.t. } \left\{ $$\begin{array}{} x_{ij} &= 1, i = j \\ \sum\limits_{j=1}^n x_{ij}&=1, i \neq j \\ \frac{S_{ij}}{V} &\leq 3\\ x_{ij} & \in \{0,1\} \\ \overline{W} &= \frac{1}{n}\sum\limits_{j=1}^n W_j\\ &(i = 1,2,3,\cdots,m;j=1,2,\cdots,n) \end{array}$$ \right. \tag{32} minf1​minf2​​=Wn−11​∑(Wj​−W2) ​​=1<j<nmax​Tj​​s.t.⎩ ⎨ ⎧​xij​j=1∑n​xij​VSij​​xij​W​=1,i=j=1,i=j≤3∈{0,1}=n1​j=1∑n​Wj​(i=1,2,3,⋯,m;j=1,2,⋯,n)​(32)

3.2、练习二

$$
\begin{aligned}
\int_{k-1}^k \frac{dx_0^{(1)}(t)}{dt}\mathrm{dt} & \approx  - \hat{a}\int_{k-1}^kx_0^{(1)}dt+\int_{k-1}^k\hat b dt 
\\
& =  \int_{k-1}^k \left[-\hat a x_0 ^ {(1)} (t) + \hat b \right]dt
\end{aligned}
$$
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∫ k − 1 k d x 0 ( 1 ) ( t ) d t d t ≈ − a ^ ∫ k − 1 k x 0 ( 1 ) d t + ∫ k − 1 k b ^ d t = ∫ k − 1 k [ − a ^ x 0 ( 1 ) ( t ) + b ^ ] d t (33)

$$\begin{aligned} \int_{k-1}^k \frac{dx_0^{(1)}(t)}{dt}\mathrm{dt} & \approx - \hat{a}\int_{k-1}^kx_0^{(1)}dt+\int_{k-1}^k\hat b dt \\ & = \int_{k-1}^k \left[-\hat a x_0 ^ {(1)} (t) + \hat b \right]dt \end{aligned}$$ \tag{33} ∫k−1k​dtdx0(1)​(t)​dt​≈−a^∫k−1k​x0(1)​dt+∫k−1k​b^dt=∫k−1k​[−a^x0(1)​(t)+b^]dt​(33)

3.3、练习三

$$
\left\{
\begin{aligned}
z_1 & =  l_{11}x_1 + l_{12}x_2 + \cdots  +  l_{1p}x_p 
\\
z_2 & =  l_{21}x_1 + l_{22}x_2 + \cdots  +  l_{2p}x_p 
\\
& \vdots 
\\
z_m & =  l_{m1}x_1 + l_{m2}x_2 + \cdots  +  l_{mp}x_p 
\end{aligned}
\right.
$$
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{ z 1 = l 11 x 1 + l 12 x 2 + ⋯ + l 1 p x p z 2 = l 21 x 1 + l 22 x 2 + ⋯ + l 2 p x p ⋮ z m = l m 1 x 1 + l m 2 x 2 + ⋯ + l m p x p (34) \left\{

$$\begin{aligned} z_1 & = l_{11}x_1 + l_{12}x_2 + \cdots + l_{1p}x_p \\ z_2 & = l_{21}x_1 + l_{22}x_2 + \cdots + l_{2p}x_p \\ & \vdots \\ z_m & = l_{m1}x_1 + l_{m2}x_2 + \cdots + l_{mp}x_p \end{aligned}$$ \right. \tag{34} ⎩ ⎨ ⎧​z1​z2​zm​​=l11​x1​+l12​x2​+⋯+l1p​xp​=l21​x1​+l22​x2​+⋯+l2p​xp​⋮=lm1​x1​+lm2​x2​+⋯+lmp​xp​​(34)

3.4、练习四

$$
x = 
x =  \left[ 
\begin{aligned} 
x_{11} & x_{12} & \cdots & x_{1p} \\ 
x_{21} & x_{22} & \cdots & x_{2p} \\ 
\vdots & \vdots & \ddots & \vdots \\ 
x_{n1} & x_{n2} & \cdots & x_{np} \\ 
\end{aligned} \right] 
=(x_1,x_2,\cdots,x_p)
$$
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x = x = [ x 11 x 12 ⋯ x 1 p x 21 x 22 ⋯ x 2 p ⋮ ⋮ ⋱ ⋮ x n 1 x n 2 ⋯ x n p ] = ( x 1 , x 2 , ⋯   , x p ) (35) x = x = \left[

$$\begin{aligned} x_{11} & x_{12} & \cdots & x_{1p} \\ x_{21} & x_{22} & \cdots & x_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ x_{n1} & x_{n2} & \cdots & x_{np} \\ \end{aligned}$$ \right] =(x_1,x_2,\cdots,x_p) \tag{35} x=x=⎣ ⎡​x11​x21​⋮xn1​​x12​x22​⋮xn2​​⋯⋯⋱⋯​x1p​x2p​⋮xnp​​⎦ ⎤​=(x1​,x2​,⋯,xp​)(35)

四、其他的一些例子

• 博主平时学习、工作中用的，顺着写的就不一一整理了

y = f ( x ) = W T ⋅ x + b = [ x 1 x 2 ⋮ x n ] + b y = f(x)=W^T\cdot x + b = \left[

$$\begin{array}{} x_1 \\ x_2 \\ \vdots \\ x_n \end{array}$$ \right] + b y=f(x)=WT⋅x+b=⎣ ⎡​x1​x2​⋮xn​​⎦ ⎤​+b

χ A = { 1 ,  A occurs  , 0 ,  A doesn’t occur.  \chi_{A}=\left\{

$$\begin{array}{lc} 1, & \text { A occurs }, \\ 0, & \text { A doesn't occur. } \end{array}$$ \right. χA​={1,0,​ A occurs , A doesn’t occur. ​

A = ( a 11 ⋯ a 1 n ⋮ ⋱ ⋮ a m 1 ⋯ a m n ) A = \left(

$$\begin{array}{ccc} a_{11}&\cdots&a_{1n}\\ \vdots&\ddots & \vdots \\ a_{m1} &\cdots & a_{mn} \end{array}$$ \right) A=⎝ ⎛​a11​⋮am1​​⋯⋱⋯​a1n​⋮amn​​⎠ ⎞​

$${\hat{\omega }}_i=\frac{\sqrt[n]{\prod _{j=1}^n{a}_{ij}}}{\sum _{j=1}^n\sqrt[n]{\prod _{j=1}^n{a}_{ij}}},i\in [1,m]\text{\hspace{1em}(m为指标数量)}$$

$$CR=\frac{CI}{RI}=\frac{{\lambda }_{max}-n}{RI(n-1)}$$

$$B_j=-\sum _{i=1}^m{p}_{ij}\frac{\ln p_{ij}}{\ln m}$$

$$w_j=\frac{1-B_j}{\sum _{j=1}^n(1-B_j)}$$

$$\sum _{j=1}^n{w}_j=1,j\in [1,n]$$

$$\overline{\omega }=\left\{\frac{W_1{w}_1}{{\sum }_{j=1}^n{W}_j{w}_j},\frac{W_2{w}_2}{{\sum }_{j=1}^n{W}_j{w}_j},\cdots ,\frac{W_n{w}_n}{{\sum }_{j=1}^n{W}_j{w}_j}\right\}=\left({\omega }_1,{\omega }_2,\cdots ,{\omega }_n\right)$$

$$\text{ s.t. }\sum _{j=1}^n{\omega }_j=1;{\omega }_j>0$$

$$w_j(j=1,2,\cdots ,n)$$

X = [ x 11 x 12 ⋯ x 1 m x 21 x 22 ⋯ x 2 m ⋮ ⋮ ⋱ ⋮ x n 1 x n 2 ⋯ x n m ] X=\left[

$$\begin{array}{cccc} x_{11} & x_{12} & \cdots & x_{1 m} \\ x_{21} & x_{22} & \cdots & x_{2 m} \\ \vdots & \vdots & \ddots & \vdots \\ x_{n 1} & x_{n 2} & \cdots & x_{n m} \end{array}$$ \right] X=⎣ ⎡​x11​x21​⋮xn1​​x12​x22​⋮xn2​​⋯⋯⋱⋯​x1m​x2m​⋮xnm​​⎦ ⎤​
那么，对其标准化的矩阵记为Z， $Z$ 中的每一个元素:
$$z_{ij}=x_{ij}/\sqrt{\sum _{i=1}^n{x}_{ij}^2}$$
Z − = ( Z 1 − , Z 2 − , ⋯   , Z m − ) = ( min ⁡ { z 11 , z 21 , ⋯   , z n 1 } , min ⁡ { z 12 , z 22 , ⋯   , z n 2 } , ⋯   , min ⁡ { z 1 m , z 2 m , ⋯   , z n m } ) $$\begin{aligned} Z^{-} &=\left(Z_{1}^{-}, Z_{2}^{-}, \cdots, Z_{m}^{-}\right) \\ &=\left(\min \left\{z_{11}, z_{21}, \cdots, z_{n 1}\right\}, \min \left\{z_{12}, z_{22}, \cdots, z_{n 2}\right\}, \cdots, \min \left\{z_{1 m}, z_{2 m}, \cdots, z_{n m}\right\}\right) \end{aligned}$$ Z−​=(Z1−​,Z2−​,⋯,Zm−​)=(min{z11​,z21​,⋯,zn1​},min{z12​,z22​,⋯,zn2​},⋯,min{z1m​,z2m​,⋯,znm​})​

$$D_i^{+}=\sqrt{\sum _{j=1}^m(Z_j^{+}-z_{ij}{)}^2}$$

$$S_i=\frac{D_i^{-}}{D_i^{+}+D_i^{-}}{S}_i\in [0,1]$$

$$\widetilde{S_i}=\frac{S_i}{\sum _{i=1}^n\widetilde{S_i}}$$
$$\sum _{i=1}^n\widetilde{S_i}=1$$

$$\min f_1=\frac{\sqrt{\frac{1}{n-1}\sum _{j=1}^n(W_j-\overline{W}{)}^2}}{\overline{W}}$$

$$\min f_2=\underset{1<j<n}{\max }{T}_j$$

∫ k − 1 k d x 0 ( 1 ) ( t ) d t d t ≈ − a ^ ∫ k − 1 k x 0 ( 1 )

$$\begin{aligned} \int_{k-1}^k \frac{dx_0^{(1)}(t)}{dt} dt & \approx & - \hat{a}\int_{k-1}^kx_0^{(1)} \end{aligned}$$ ∫k−1k​dtdx0(1)​(t)​dt​≈​−a^∫k−1k​x0(1)​​