【Matlab】牛顿迭代法实现

短命女 2022-11-20 03:55 338阅读 0赞

### 文章目录 ###

*   *  题目：牛顿迭代法
     *  程序1：牛顿迭代法通用函数
     *  程序2：求最大Delta
     *  程序3：观察结果

## 题目：牛顿迭代法 ##

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## 程序1：牛顿迭代法通用函数 ##

function [x] = newton(x0,epsilon,f,print_flag)
    digits(10)  % 控制牛顿迭代法的运算精度，精度太大迭代速度过慢
    syms x
    f(x) = f;
    df(x) = diff(f);
    count = 0;
    e = 1;
        while abs(e) > epsilon
            x1 = vpa(x0 - f(x0)/df(x0));
            e = vpa(x1 -x0);
            x0 = x1;
            count = count + 1;
            if count>100
                fprintf('牛顿迭代发散。\n')
                break
            end
            if print_flag
                fprintf('已迭代 %d 次,', count)
                fprintf('x为：%f,', x0)
                fprintf('误差为：%f\n', e)
            end
        end
        if print_flag
            fprintf('Newton迭代的近似解 x = %f\n',x1)
            fprintf('迭代次数count = %d\n',count)
        end
    x = x0;
    end

## 程序2：求最大Delta ##

clc;clear;close all
    
    %% 定义方程
    syms x
    f(x) = x^3 / 3 - x
    
    %% 寻找delta范围
    delta = 0;             % delta的绝对值：|delta|
    step = 0.1;            % delta递增的步长，初始步长为0.1
    epsilon = 10^(-6);     % 允许的误差
    while step > 10^(-7)
        a = 0; 
        b = 0;
        while abs(a) < epsilon & abs(b) < epsilon
            a = newton(delta,epsilon,f,0);
            b = newton(-delta,epsilon,f,0);
            delta = delta + step;
        end
            % 当牛顿迭代发散时，修改步长大小
            delta = delta - 2 * step;
            step = step * 0.1;
            delta = delta + step;
            step
            
        fprintf('delta:%f\n', delta)
    end
    fprintf('Newton迭代序列收敛于根x2时, delta应小于：%f\n',delta)

## 程序3：观察结果 ##

clc;clear;close all
    
    %% 定义方程
    syms x
    f(x) = x^3 / 3 - x
    
    %% 寻找delta范围
    delta = 0;             % delta的绝对值：|delta|
    step = 0.1;            % delta递增的步长，初始步长为0.1
    epsilon = 10^(-6);     % 允许的误差
    
    
    %% A
    x0 = -200;
    x = newton(x0,epsilon,f,0);
    fprintf('x0 = %f\n',x0)
    fprintf('迭代结果x = %f\n',x)
    
    x0 = -50;
    x = newton(x0,epsilon,f,0);
    fprintf('x0 = %f\n',x0)
    fprintf('迭代结果x = %f\n',x)
    
    x0 = -2;
    x = newton(x0,epsilon,f,0);
    fprintf('x0 = %f\n',x0)
    fprintf('迭代结果x = %f\n',x)
    
    x0 = -1.1;
    x = newton(x0,epsilon,f,0);
    fprintf('x0 = %f\n',x0)
    fprintf('迭代结果x = %f\n',x)
    fprintf('\n')
    
    %% B
    x0 = -0.9;
    x = newton(x0,epsilon,f,0);
    fprintf('x0 = %f\n',x0)
    fprintf('迭代结果x = %f\n',x)
    
    x0 = -0.85;
    x = newton(x0,epsilon,f,0);
    fprintf('x0 = %f\n',x0)
    fprintf('迭代结果x = %f\n',x)
    
    x0 = -0.8;
    x = newton(x0,epsilon,f,0);
    fprintf('x0 = %f\n',x0)
    fprintf('迭代结果x = %f\n',x)
    
    x0 = -0.78;
    x = newton(x0,epsilon,f,0);
    fprintf('x0 = %f\n',x0)
    fprintf('迭代结果x = %f\n',x)
    fprintf('\n')
    %% D
    x0 = 0.78;
    x = newton(x0,epsilon,f,0);
    fprintf('x0 = %f\n',x0)
    fprintf('迭代结果x = %f\n',x)
    
    x0 = 0.8;
    x = newton(x0,epsilon,f,0);
    fprintf('x0 = %f\n',x0)
    fprintf('迭代结果x = %f\n',x)
    
    x0 = 0.85;
    x = newton(x0,epsilon,f,0);
    fprintf('x0 = %f\n',x0)
    fprintf('迭代结果x = %f\n',x)
    
    x0 = 0.9;
    x = newton(x0,epsilon,f,0);
    fprintf('x0 = %f\n',x0)
    fprintf('迭代结果x = %f\n',x)
    fprintf('\n')
    %% E
    x0 = 1.1;
    x = newton(x0,epsilon,f,0);
    fprintf('x0 = %f\n',x0)
    fprintf('迭代结果x = %f\n',x)
    
    x0 = 2;
    x = newton(x0,epsilon,f,0);
    fprintf('x0 = %f\n',x0)
    fprintf('迭代结果x = %f\n',x)
    
    x0 = 50;
    x = newton(x0,epsilon,f,0);
    fprintf('x0 = %f\n',x0)
    fprintf('迭代结果x = %f\n',x)
    
    x0 = 200;
    x = newton(x0,epsilon,f,0);
    fprintf('x0 = %f\n',x0)
    fprintf('迭代结果x = %f\n',x)

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