MATLAB笔记6：数值微积分与方程求解

我会带着你远行 2022-11-27 00:44 271阅读 0赞

**目录**

一、数值微分与数值积分

1.数值微分

数值微分的实现

2.数值积分

数值积分基本原理

数值积分的实现

二、线性方程组求解

1.线性方程组的直接解法

利用左除运算符的直接解法

利用矩阵分解求解线性方程组

2.线性方程组的迭代解法

雅可比迭代法

高斯—赛德尔迭代法

三、线性方程组应用组举例

1.平面绗架结构受力分析问题

2.小行星运行轨道计算问题

四、非线性方程求解与函数极值计算

1、单变量非线性方程求解

2、非线性方程组的求解

3、函数极值得计算

无约束最优化问题

有约束最优化问题

最小值问题应用实例

五、常微分方程数值求解

1、常微分方程数值求解的一般概念

2、常微分方程数值求解函数

刚性问题

六、常微分方程应用举例

--------------------

# 一、数值微分与数值积分 #

## 1.数值微分 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 1][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 2][]

### 数值微分的实现 ###

MATLAB提供了求向前差分的函数diff，其调用格式有三种：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 3][]

例子：

![20200812180847939.png][]

x=[0,sort(2*pi*rand(1,5000)),2*pi];  //rand产生随机数5000个
    >> y=sin(x);
    >> f1=diff(y)./diff(x);  //差分
    >> f2=cos(x(1:end-1));
    >> plot(x(1:end-1),f1,x(1:end-1),f2);   //绘制理论值何计算值曲线
    >> d=norm(f1-f2)                         //求范数
    
    d =
    
        0.0456

图像：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 4][]

## 2.数值积分 ##

### 数值积分基本原理 ###

![202008121815409.png][]

### 数值积分的实现 ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 5][]

例子：

分别用quad函数和quadl函数求定积分的近似值，并在相同的积分精度下，比较积函数的调用次数。

![20200812181811130.png][]

代码：

>> format long
    >> f=@(x)4./(1+x.^2);
    >> [I,n]=quad(f,0,1,1e-8)
    
    I =
    
       3.141592653733437
    
    
    n =
    
        61
    
    >> [I,n]=quadl(f,0,1,1e-8)
    
    I =
    
       3.141592653589806
    
    
    n =
    
        48
    
    >> (atan(1)-atan(0))*4
    
    ans =
    
       3.141592653589793
    
    >> format short

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 6][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 7][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 8][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 9][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 10][]

# 二、线性方程组求解 #

*  直接法
 *  迭代法

## 1.线性方程组的直接解法 ##

*  高斯（Gauss）消去法
 *  列主元消去法
 *  矩阵的三角分解法

### 利用左除运算符的直接解法 ###

Ax=b

x=A\\b

注意：如果矩阵A是奇异的或接近七一的，则MATLAB会给出警告信息。

例子：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 11][]

代码：

>> A=[2,1,-5,1;1,-5,0,7;0,2,1,-1;1,6,-1,-4];
    >> b=[13,-9,6,0]';
    >> x=A\b
    
    x =
    
      -66.5556
       25.6667
      -18.7778
       26.5556

### 利用矩阵分解求解线性方程组 ###

LU分解的基本思想：将矩阵表示为一个下三角矩阵和上三角矩阵的乘积。已经证明，只要方阵是非奇异的，LU分解总是可以进行的。

MATLAB的LU分解函数：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 12][]

用LU分解求解线性方程组：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 13][]

例子：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 14][]

代码：

>> [L,U]=lu(A);
    >> x=U\(L\b)
    
    x =
    
      -66.5556
       25.6667
      -18.7778
       26.5556

## 2.线性方程组的迭代解法 ##

所谓迭代，就是不断的用变量的原值，推导出变量的极值的过程。

*  雅可比迭代法
 *  高斯-赛德尔迭代法

### 雅可比迭代法 ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 15][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 16][]

### 高斯—赛德尔迭代法 ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 17][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 18][]

例子：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 19][]

代码：

最近时间又紧张了起来，我暂时不敲代码了，先把这个课程看完。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 20][]

例子：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 21][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 22][]

考虑收敛，考虑收敛速度怎么样。

总结：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 23][]

# 三、线性方程组应用组举例 #

## 1.平面绗架结构受力分析问题 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 24][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 25][]

代码：：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 26][]

## 2.小行星运行轨道计算问题 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 27][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 28][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 29][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 30][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 31][]

# 四、非线性方程求解与函数极值计算 #

## 1、单变量非线性方程求解 ##

函数的调用格式为：

x = fzero（filename，x0）

其中，filename是待求根方程左端的函数表达式，x0是初始值。

例子：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 32][]

图像：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 33][]

初值得选取很重要

## 2、非线性方程组的求解 ##

函数的调用格式为：

x=fsolve（filename，x0，option）

其中，x为返回的近似解，filename是待求根方程左端的函数表达式，x0是初值，option用于设置优化工具箱的优化参数，可以调用optimset函数来完成

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 34][]

例2：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 35][]

## 3、函数极值得计算 ##

### 无约束最优化问题 ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 36][]

求最小值得函数：

![20200904192837482.png][]

其中filename是定义得目标函数了第一个函数得输入标量x1、x2分别表示被研究区间得左、右边界。后两个函数的输入变量x0是一个向量，表示极值点的初值。option为优化参数，可以通过optimset函数来设置。

例子：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 37][]

### 有约束最优化问题 ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 38][]

求有约束条件下最小值的函数：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 39][]

### 最小值问题应用实例 ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 40][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 41][]

相关代码：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 42][]

例子：

![20200905091621249.png][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 43][]

# 五、常微分方程数值求解 #

## 1、常微分方程数值求解的一般概念 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 44][]

单步法    多步法

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 45][]

## 2、常微分方程数值求解函数 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 46][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 47][]

例子：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 48][]

例子：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 49][]

代码：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 50][]

## 刚性问题 ##

有一类常微分方程，其解的分量有的变化很快，有的变化很慢，且相差悬殊，这就是所谓的刚性问题（Stiff）。

例子：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 51][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 52][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 53][]

改进，使用s的函数

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 54][]

# 六、常微分方程应用举例 #

Lotka-Volterra模型

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 55][]

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[20200812180847939.png]: /images/20221124/fa88c033f49e4638ae39b90c0e1f576c.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTc1NTk2Ng_size_16_color_FFFFFF_t_70 4]: /images/20221124/63bac984994546a3848822092c73bb0a.png
[202008121815409.png]: /images/20221124/f99ecf090e794ef2b4871260f47a8fb7.png
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[20200812181811130.png]: /images/20221124/3b1af06afa15437c94a98b1ee6d911d4.png
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