Latex公式总结
CATLOG
https://zhuanlan.zhihu.com/p/261750408
一、基础
1.1、一元一次方程
$$
y = a+bx
$$
y = a + b x (1) y = a+bx \tag 1 y=a+bx(1)
1.2 、上下标
$$
y = a + b_{ij}^2
$$
y = a + b i j 2 (2) y = a + b_{ij}^2\tag 2 y=a+bij2(2)
1.3、求和
$$
\sum_{i=0}^n(x_i^2+y_j^3)
$$
∑ i = 0 n ( x i 2 + y j 3 ) (3) \sum_{i=0}^n(x_i^2+y_j^3)\tag 3 i=0∑n(xi2+yj3)(3)
1.4、分数
$$
\frac{1}{3}-\frac{x_i^2}{y_j^3}
$$
或者
$$
\cfrac{1}{3}-\cfrac{x_i^2}{y_j^3}
$$
1 3 − x i 2 y j 3 1 3 − x i 2 y j 3 (4) \frac{1}{3}-\frac{x_i^2}{y_j^3} \\ \cfrac{1}{3}-\cfrac{x_i^2}{y_j^3} \tag 4 31−yj3xi231−yj3xi2(4)
1.5、矩阵
$$
\left[
\begin{array} {cccc}
X_1&Y_1^2\\
X_2 & Y_2^2\\
\ldots \\
X_n & Y_n^2
\end{array}
\right]
$$
[ 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 ⎣ ⎡X1X2…XnY12Y22Yn2⎦ ⎤(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
| 小写 | 大写 | Latex小写命令 | Latex大写命令 |
|---|---|---|---|
| α \alpha α | A \Alpha A | \alpha | \Alpha |
| β \beta β | B \Beta B | \beta | \Beta |
| γ \gamma γ | Γ \Gamma Γ | \gamma | \Gamma |
| δ \delta δ | Δ \Delta Δ | \delta | \Delta |
| ϵ \epsilon ϵ | E \Epsilon E | \epsilon | \Epsilon |
| ζ \zeta ζ | Z \Zeta Z | \zeta | \Zeta |
| ν \nu ν | N \Nu N | \nu | \Nu |
| ξ \xi ξ | Ξ \Xi Ξ | \xi | \Xi |
| ο \omicron ο | O \Omicron O | \omicron | \Omicron |
| π \pi π | Π \Pi Π | \pi | \Pi |
| ρ \rho ρ | P \Rho P | \rho | \Rho |
| σ \sigma σ | Σ \Sigma Σ | \sigma | \Sigma |
| η \eta η | H \Eta H | \eta | \Eta |
| θ \theta θ | Θ \Theta Θ | \theta | \Theta |
| ι \iota ι | I \Iota I | \iota | \Iota |
| κ \kappa κ | K \Kappa K | \kappa | \Kappa |
| λ \lambda λ | Λ \Lambda Λ | \lambda | \Lambda |
| μ \mu μ | M \Mu M | \mu | \Mu |
| τ \tau τ | T \Tau T | \tau | \Tau |
| υ \upsilon υ | Υ \Upsilon Υ | \upsilon | \Upsilon |
| ϕ \phi ϕ | ϕ \phi ϕ | \phi | \phi |
| χ \chi χ | X \Chi X | \chi | \Chi |
| ψ \psi ψ | Ψ \Psi Ψ | \psi | \Psi |
| ω \omega ω | Ω \Omega Ω | \omega | \Omega |
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 "\\"
$$
f(x_1,x_2,\ldots,x_i) = x_1+x_2+\cdots+x_i
$$
f ( x 1 , x 2 , … , x i ) = x 1 + x 2 + ⋯ + x i (6) f(x_1,x_2,\ldots,x_i) = x_1+x_2+\cdots+x_i \tag 6 f(x1,x2,…,xi)=x1+x2+⋯+xi(6)
f ( x 1 , x 2 , … , x i ) = x 1 + x 2 + ⋯ + x i (7) f(x_1,x_2,\ldots,x_i) \\ = x_1+x_2+\cdots+x_i \tag 7 f(x1,x2,…,xi)=x1+x2+⋯+xi(7)
1.8、大括号
" \{ \}"" (注意这里直接{}打是不行的,显示不出来的)
$$
\{a+b\}
$$
{ 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
$$
\left[a+b^2+(x*r)\right]
$$
( a + b 2 + ( c ∗ r ) ) [ a + b 2 + ( x ∗ r ) ] (9) \left(a+b^2+(c*r)\right) \\ \left[a+b^2+(x*r)\right] \tag 9 (a+b2+(c∗r))[a+b2+(x∗r)](9)
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
$$
\begin{aligned}
\cos 2 \theta & = & \cos^2 \theta - \sin^2 \theta\\
&=& 2\cos^2 \theta - 1
\end{aligned}
$$
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
$$
\{ f(x) = {{{a_0}} \over 2} + \sum\limits_{n = 1}^\infty {({a_n}\cos {nx} + {b_n}\sin {nx})} \}
$$
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∑∞(ancosnx+bnsinnx){f(x)=2a0+n=1∑∞(ancosnx+bnsinnx)}(12)
2.2、高斯公式
$$
\iiint _ { \Omega } \left( \frac { \partial {P} } { \partial {x} } + \frac { \partial {Q} } { \partial {y} } + \frac { \partial {R} }{ \partial {z} } \right) \mathrm { d } V = \oint _ { \partial \Omega } ( P \cos \alpha + Q \cos \beta + R \cos \gamma ) \mathrm{ d} S
$$
∭ Ω ( ∂ P ∂ x + ∂ Q ∂ y + ∂ R ∂ z ) d V = ∮ ∂ Ω ( P cos α + Q cos β + R cos γ ) d S (13) \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 \tag {13} ∭Ω(∂x∂P+∂y∂Q+∂z∂R)dV=∮∂Ω(Pcosα+Qcosβ+Rcosγ)dS(13)
2.3、定积分
$$
\lim\limits_ { n \rightarrow + \infty } \sum _ { i = 1 } ^ { n } f \left[ a + \frac { i } { n } ( b - a ) \right] \frac { b - a } { n } = \int _ { a } ^ { b } f ( x )\mathrm {d}x
$$
lim n → + ∞ ∑ i = 1 n f [ a + i n ( b − a ) ] b − a n = ∫ a b f ( x ) d x (14) \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 \tag {14} n→+∞limi=1∑nf[a+ni(b−a)]nb−a=∫abf(x)dx(14)
2.4、和的展开式
$$
( 1 + x ) ^ { n } = 1 + \frac { n x } { 1 ! } + \frac { n ( n - 1 ) x ^ { 2 } } { 2 ! } + \cdots
$$
( 1 + x ) n = 1 + n x 1 ! + n ( n − 1 ) x 2 2 ! + ⋯ (15) ( 1 + x ) ^ { n } = 1 + \frac { n x } { 1 ! } + \frac { n ( n - 1 ) x ^ { 2 } } { 2 ! } + \cdots \tag {15} (1+x)n=1+1!nx+2!n(n−1)x2+⋯(15)
2.5、三角恒等式
$$
\sin \alpha \pm \sin \beta = 2 \sin \frac { 1 } { 2 } ( \alpha \pm \beta ) \cos \frac { 1 } { 2 } ( \alpha \mp \beta )
$$
$$
\cos \alpha + \cos \beta = 2 \cos \frac { 1 } { 2 } ( \alpha + \beta ) \cos \frac { 1 } { 2 } ( \alpha - \beta )
$$
sin α ± sin β = 2 sin 1 2 ( α ± β ) cos 1 2 ( α ∓ β ) cos α + cos β = 2 cos 1 2 ( α + β ) cos 1 2 ( α − β ) (16) \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 ) \tag {16} sinα±sinβ=2sin21(α±β)cos21(α∓β)cosα+cosβ=2cos21(α+β)cos21(α−β)(16)
2.6、欧拉公式
$$
{e^{ix}} = \cos {x} + i\sin {x}
$$
e i x = cos x + i sin x (17) {e^{ix}} = \cos {x} + i\sin {x} \tag {17} eix=cosx+isinx(17)
2.7、格林公式
$$
\int\!\!\!\int\limits_D {({{\partial Q} \over {\partial x}} - {{\partial P} \over {\partial y}})dxdy = \oint\limits_L {Pdx + Qdy} }
$$
∫ ∫ D ( ∂ Q ∂ x − ∂ P ∂ y ) d x d y = ∮ L P d x + Q d y (18) \int\!\!\!\int\limits_D {({{\partial Q} \over {\partial x}} - {{\partial P} \over {\partial y}})dxdy = \oint\limits_L {Pdx + Qdy} } \tag {18} ∫D∫(∂x∂Q−∂y∂P)dxdy=L∮Pdx+Qdy(18)
2.8、伯努利方程
$$
{{dy} \over {dx}} + P(x)y = Q(x){y^n}(n \ne 0,1)
$$
d y d x + P ( x ) y = Q ( x ) y n ( n ≠ 0 , 1 ) (19) {{dy} \over {dx}} + P(x)y = Q(x){y^n}(n \ne 0,1) \tag {19} dxdy+P(x)y=Q(x)yn(n=0,1)(19)
2.9 、全微分方程
$$
du(x,y) = P(x,y)dx + Q(x,y)dy = 0
$$
d u ( x , y ) = P ( x , y ) d x + Q ( x , y ) d y = 0 (20) du(x,y) = P(x,y)dx + Q(x,y)dy = 0 \tag {20} du(x,y)=P(x,y)dx+Q(x,y)dy=0(20)
2.10 、非齐次微分方程通解
$$
y = (\int {Q(x){e^{\int {P(x)dx} }}dx + C){e^{ - \int {P(x)dx} }}}
$$
y = ( ∫ Q ( x ) e ∫ P ( x ) d x d x + C ) e − ∫ P ( x ) d x (21) y = (\int {Q(x){e^{\int {P(x)dx} }}dx + C){e^{ - \int {P(x)dx} }}} \tag {21} y=(∫Q(x)e∫P(x)dxdx+C)e−∫P(x)dx(21)
2.11、柯西中值定理
$$
\frac{{f(b) - f(a)}}{{F(b) - F(a)}} = \frac{{f'(\xi )}}{{F'(\xi )}}
$$
f ( b ) − f ( a ) F ( b ) − F ( a ) = f ′ ( ξ ) F ′ ( ξ ) (22) \frac{{f(b) - f(a)}}{{F(b) - F(a)}} = \frac{{f'(\xi )}}{{F'(\xi )}} \tag {22} F(b)−F(a)f(b)−f(a)=F′(ξ)f′(ξ)(22)
2.12、拉格朗日中值定理
$$
f(b) - f(a) = f'(\xi )(b - a)
$$
f ( b ) − f ( a ) = f ′ ( ξ ) ( b − a ) (23) f(b) - f(a) = f'(\xi )(b - a) \tag{23} f(b)−f(a)=f′(ξ)(b−a)(23)
2.13、曲面面积
$$
A = \int\!\!\!\int\limits_D {\sqrt {1 + {{\left( {{{\partial z} \over {\partial x}}} \right)}^2} + {{\left( {{{\partial z} \over {\partial y}}} \right)}^2}} dxdy}
$$
A = ∫ ∫ D 1 + ( ∂ z ∂ x ) 2 + ( ∂ z ∂ y ) 2 d x d y (24) A = \int\!\!\!\int\limits_D {\sqrt {1 + {{\left( {{{\partial z} \over {\partial x}}} \right)}^2} + {{\left( {{{\partial z} \over {\partial y}}} \right)}^2}} dxdy} \tag{24} A=∫D∫1+(∂x∂z)2+(∂y∂z)2dxdy(24)
2.14、重积分
$$
\int\!\!\!\int\limits_D {f(x,y)dxdy} = \int\!\!\!\int\limits_{D'} {f(r\cos \theta ,r\sin \theta )rdrd\theta }
$$
∫ ∫ D f ( x , y ) d x d y = ∫ ∫ D ′ f ( r cos θ , r sin θ ) r d r d θ (25) \int\!\!\!\int\limits_D {f(x,y)dxdy} = \int\!\!\!\int\limits_{D'} {f(r\cos \theta ,r\sin \theta )rdrd\theta } \tag{25} ∫D∫f(x,y)dxdy=∫D′∫f(rcosθ,rsinθ)rdrdθ(25)
2.15、arcsinx 求导
$$
(\arcsin x)' = \frac{1}{{\sqrt {1 - {x^2}} }}
$$
( arcsin x ) ′ = 1 1 − x 2 (26) (\arcsin x)' = \frac{1}{{\sqrt {1 - {x^2}} }} \tag{26} (arcsinx)′=1−x21(26)
2.16、泰勒级数
$$ p;ko[p';p[oooo]]
e^x = 1 + \frac{x}{1!} + \frac{x}{2!}+ \frac{x}{3!} + \cdots + O(x^n), -\infty < x < \infty
$$
e x = 1 + x 1 ! + x 2 ! + x 3 ! + ⋯ + O ( x n ) , − ∞ < x < ∞ (27) e^x = 1 + \frac{x}{1!} + \frac{x}{2!}+ \frac{x}{3!} + \cdots + O(x^n), -\infty < x < \infty \tag{27} ex=1+1!x+2!x+3!x+⋯+O(xn),−∞<x<∞(27)
2.17、三角函数积分
$$
\int tgx \mathrm {d} x = -\ln {|\cos x|} + c
$$
∫ t g x d x = − ln ∣ cos x ∣ + c (28) \int tgx \mathrm {d} x = -\ln {|\cos x|} + c \tag {28} ∫tgxdx=−ln∣cosx∣+c(28)
2.18、二次曲面
$$
\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} - \frac{{{z^2}}}{{{c^2}}} = 1
$$
x 2 a 2 + y 2 b 2 − z 2 c 2 = 1 (29) \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} - \frac{{{z^2}}}{{{c^2}}} = 1 \tag{29} a2x2+b2y2−c2z2=1(29)
2.19、二阶微分
$$
\frac{d^2y}{dx^2}+P(x)\frac{dy}{dx}+Q(x)y=f(x)
$$
d 2 y d x 2 + P ( x ) d y d x + Q ( x ) y = f ( x ) (30) \frac{d^2y}{dx^2}+P(x)\frac{dy}{dx}+Q(x)y=f(x) \tag{30} dx2d2y+P(x)dxdy+Q(x)y=f(x)(30)
2.20、方向导数
$$
\frac{\partial f}{\partial l}=\frac{\partial f}{\partial x}\cos{\phi}+\frac{\partial f}{\partial y}\sin{\phi}
$$
∂ f ∂ l = ∂ f ∂ x cos ϕ + ∂ f ∂ y sin ϕ (31) \frac{\partial f}{\partial l}=\frac{\partial f}{\partial x}\cos{\phi}+\frac{\partial f}{\partial y}\sin{\phi} \tag{31} ∂l∂f=∂x∂fcosϕ+∂y∂fsinϕ(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
$$
\begin{aligned}
\min {f_1} &= \frac{\sqrt{\frac{1}{n-1}\sum(W_j-\overline{W}^2)}}{\overline{W}}
\\
\min{f_2} &= \max\limits_{1<j<n}{T_j}
\end{aligned}
\\
\operatorname{ s.t. }
\left\{
\begin{array}{}
x_{ij} &= 1, i = j \\
\sum\limits_{j=1}^n x_{ij}&=1, i \neq j \\
\frac{S_{ij}}{V} &\leq 3\\
x_{ij} & \in \{0,1\} \\
\overline{W} &= \frac{1}{n}\sum\limits_{j=1}^n W_j\\
&(i = 1,2,3,\cdots,m;j=1,2,\cdots,n)
\end{array}
\right.
$$
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} minf1minf2=Wn−11∑(Wj−W2)=1<j<nmaxTjs.t.⎩ ⎨ ⎧xijj=1∑nxijVSijxijW=1,i=j=1,i=j≤3∈{0,1}=n1j=1∑nWj(i=1,2,3,⋯,m;j=1,2,⋯,n)(32)
3.2、练习二
$$
\begin{aligned}
\int_{k-1}^k \frac{dx_0^{(1)}(t)}{dt}\mathrm{dt} & \approx - \hat{a}\int_{k-1}^kx_0^{(1)}dt+\int_{k-1}^k\hat b dt
\\
& = \int_{k-1}^k \left[-\hat a x_0 ^ {(1)} (t) + \hat b \right]dt
\end{aligned}
$$
∫ 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−1kdtdx0(1)(t)dt≈−a^∫k−1kx0(1)dt+∫k−1kb^dt=∫k−1k[−a^x0(1)(t)+b^]dt(33)
3.3、练习三
$$
\left\{
\begin{aligned}
z_1 & = l_{11}x_1 + l_{12}x_2 + \cdots + l_{1p}x_p
\\
z_2 & = l_{21}x_1 + l_{22}x_2 + \cdots + l_{2p}x_p
\\
& \vdots
\\
z_m & = l_{m1}x_1 + l_{m2}x_2 + \cdots + l_{mp}x_p
\end{aligned}
\right.
$$
{ 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} ⎩ ⎨ ⎧z1z2zm=l11x1+l12x2+⋯+l1pxp=l21x1+l22x2+⋯+l2pxp⋮=lm1x1+lm2x2+⋯+lmpxp(34)
3.4、练习四
$$
x =
x = \left[
\begin{aligned}
x_{11} & x_{12} & \cdots & x_{1p} \\
x_{21} & x_{22} & \cdots & x_{2p} \\
\vdots & \vdots & \ddots & \vdots \\
x_{n1} & x_{n2} & \cdots & x_{np} \\
\end{aligned} \right]
=(x_1,x_2,\cdots,x_p)
$$
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=⎣ ⎡x11x21⋮xn1x12x22⋮xn2⋯⋯⋱⋯x1px2p⋮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=⎣ ⎡x1x2⋮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⎠ ⎞
ω ^ i = ∏ j = 1 n a i j n ∑ j = 1 n ∏ j = 1 n a i j n , i ∈ [ 1 , m ] (m为指标数量) \hat{\omega}_i = \frac{\sqrt[n]{\prod\limits_{j=1}^na_{ij}}}{\sum\limits_{j=1}^n\sqrt[n]{\prod\limits_{j=1}^na_{ij}}} , i \in [1,m]\tag{m为指标数量} ω^i=j=1∑nnj=1∏naijnj=1∏naij,i∈[1,m](m为指标数量)
C R = C I R I = λ m a x − n R I ( n − 1 ) CR = \frac{CI}{RI} = \frac{\lambda_{max} - n}{RI(n-1)} CR=RICI=RI(n−1)λmax−n
B j = − ∑ i = 1 m p i j ln p i j ln m B_j = -\sum\limits_{i=1}^m p_{ij}\frac{\ln{p_{ij}}}{\ln{m}} Bj=−i=1∑mpijlnmlnpij
w j = 1 − B j ∑ j = 1 n ( 1 − B j ) w_{j} = \frac{1-B_j}{\sum\limits_{j=1}^n(1-B_j)} wj=j=1∑n(1−Bj)1−Bj
∑ j = 1 n w j = 1 , j ∈ [ 1 , n ] \sum_{j=1}^n w_j = 1,j \in [1,n] j=1∑nwj=1,j∈[1,n]
ω ‾ = { W 1 w 1 ∑ j = 1 n W j w j , W 2 w 2 ∑ j = 1 n W j w j , ⋯ , W n w n ∑ j = 1 n W j w j } = ( ω 1 , ω 2 , ⋯ , ω 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) ω={∑j=1nWjwjW1w1,∑j=1nWjwjW2w2,⋯,∑j=1nWjwjWnwn}=(ω1,ω2,⋯,ωn)
s.t. ∑ j = 1 n ω j = 1 ; ω j > 0 \text { s.t. } \sum_{j=1}^{n} \omega_{j}=1 ; \omega_{j}>0 s.t. j=1∑nωj=1;ωj>0
w j ( j = 1 , 2 , ⋯ , n ) w_j(j=1,2,\cdots,n) wj(j=1,2,⋯,n)
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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=⎣
⎡x11x21⋮xn1x12x22⋮xn2⋯⋯⋱⋯x1mx2m⋮xnm⎦
⎤
那么，对其标准化的矩阵记为Z，
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z_{i j}=x_{i j} / \sqrt{\sum_{i=1}^{n} x_{i j}^{2}}
zij=xij/i=1∑nxij2
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\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 + = ∑ j = 1 m ( Z j + − z i j ) 2 D_i^+ = \sqrt{\sum\limits_{j=1}^m(Z_j^+ - z_{ij})^2} Di+=j=1∑m(Zj+−zij)2
S i = D i − D i + + D i − S i ∈ [ 0 , 1 ] S_i = \frac{D_i^-}{D_i^+ + D_i^-} S_i \in [0,1] Si=Di++Di−Di−Si∈[0,1]
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\tilde{S_i} = \frac{S_i}{\sum\limits_{i=1}^n \tilde{S_i}}
Si~=i=1∑nSi~Si
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min f 1 = 1 n − 1 ∑ j = 1 n ( W j − W ‾ ) 2 W ‾ \min {f_1} = \frac{\sqrt{\frac{1}{n-1}\sum \limits _{j=1}^n (W_j - \overline {W})^2}}{\overline {W}} minf1=Wn−11j=1∑n(Wj−W)2
min f 2 = max 1 < j < n T j \min{f_2} = \max\limits _ {1<j<n}{T_j} minf2=1<j<nmaxTj
∫ 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−1kdtdx0(1)(t)dt≈−a^∫k−1kx0(1)