二分求根法

Posted 2023-05-16

参考技术A Bisection method, 是一种方程式根的近似值求法。

二分法求方程的近似解
用实例来解答，比如求 Y^3+Y-10=0的在区间Y[0,3]之间的根,先将Y=0代入方程左边,左边=-10,将Y=3代入左边,左边=20,这样已经创造出了一正一负,在0-3之间必有解,找中点.Y=1.5代入,如果是正,就保留负的那一头,如果是负就保留正的那一头,然后重复这一过程,不断找中点,只到等式左边接近或等于零,就解得了近似根或准确根.
//例1:用二分法求方程x^3+4x-10=0在区间[1,2]内的根(精确到0.00001)
代入1得-6,代入2得6,满足二分求方程的解

方程求根——二分法

二分法求根主要应用了区间套定理，这一算法实现简单且结果也迭代的较好，但对于复杂函数其结果不理想

　　1.代码

%%二分法求根
%%f为函数表达式，interval0为初始区间，epsilon为控制精度
function RD = Roots_dichotomy(f,interval0,epsilon)
x_low = interval0(1);x_up = interval0(2);x_ave = (x_low+x_up)/2;

%%作图
t = x_low:(x_up-x_low)/1000:x_up;
T = subs(f,t);
y0 = zeros(1,max(size(t)));
h=figure;
set(h,‘color‘,‘w‘);
plot(t,T,‘r‘,t,y0,‘b‘);
grid on
legend(‘T：函数图像‘,‘y0：y = 0‘);
xlabel(‘x shaft‘);ylabel(‘y shaft‘);
title(‘函数图像‘);
syms x;

%%限定实数解及只在区间中的实数解出现在图象上
X_real = double(solve([f],[0]));
for i = 1:max(size(X_real))
    if isreal(X_real(i)) == 1
        x_real(i) = X_real(i);
    else 
        x_real(i) =0;
    end
end
x_real(x_real==0)=[];
for i = 1:max(size(x_real))
    text(x_real(i),0,[‘(‘,num2str(x_real(i)),‘,‘,num2str(0),‘)‘],‘color‘,[0.02 0.79 0.99]);
end


f_low = subs(f,x_low);f_up = subs(f,x_up);
ub = 100;e=floor(abs(log(epsilon)));
X_up(1) = x_up;X_low(1) = x_low;X_ave(1) = x_ave;
if f_low*f_up > 0
    disp(‘请修改区间！‘);
    interval0 = input(‘输入区间为：‘);
    RD = Roots_dichotomy(f,interval0,epsilon);
else 
    %%二分算法
    for i = 1:1:ub
        f_low = subs(f,x_low);
        f_up = subs(f,x_up);
        f_ave = subs(f,x_ave);
        if f_low*f_ave <0
            x_up = x_ave;
            x_ave = (x_low+x_up)/2;
        elseif f_ave*f_up < 0
            x_low = x_ave;
            x_ave = (x_low+x_up)/2;
        end
        delta = x_up -x_low;
        X_up(i+1) = x_up;X_low(i+1) = x_low;X_ave(i+1) = x_ave;
        if abs(delta) < epsilon
            break;
        end
    end
    disp(‘迭代次数为‘);
    i
    disp(‘输出结果依次是下界迭代值，中值迭代值，上界迭代值‘);
    RD = vpa([X_low;X_ave;X_up],e);
end
end

　　2.例子

clear all
clc
syms x;
f = x^3+log(x);
epsilon=1e-6;
interval0 = [0.1,1];

%%二分法
Y = Roots_dichotomy(f,interval0,epsilon)

　　结果如下

迭代次数为
i =
    20
输出结果依次是下界迭代值，中值迭代值，上界迭代值
Y =
[  0.1,  0.55,   0.55,  0.6625,   0.6625,  0.690625,  0.7046875,   0.7046875,    0.7046875,     0.7046875,      0.7046875,       0.7046875,       0.7046875,       0.7046875,       0.7046875,       0.7046875, 0.7047012329102, 0.7047080993652, 0.7047080993652, 0.7047080993652, 0.7047089576721]
[ 0.55, 0.775, 0.6625, 0.71875, 0.690625, 0.7046875, 0.71171875, 0.708203125, 0.7064453125, 0.70556640625, 0.705126953125, 0.7049072265625, 0.7047973632812, 0.7047424316406, 0.7047149658203, 0.7047012329102, 0.7047080993652, 0.7047115325928,  0.704709815979, 0.7047089576721, 0.7047093868256]
[  1.0,   1.0,  0.775,   0.775,  0.71875,   0.71875,    0.71875,  0.71171875,  0.708203125,  0.7064453125,  0.70556640625,  0.705126953125, 0.7049072265625, 0.7047973632812, 0.7047424316406, 0.7047149658203, 0.7047149658203, 0.7047149658203, 0.7047115325928,  0.704709815979,  0.704709815979]