求解线性方程的数值方法
求解线性方程的数值方法主要包括以下几种：
- 二分法
- 牛顿法
- 简化牛顿法
- 弦截法
实现代码见博客最后
算法实现
- 用二分法、牛顿法、简化牛顿法、弦截法分别计算115的平方根，精确到小数点后六位
设计思路
- 首先罗列出方程y = x^2-115，而要求的平方根就在于跟x轴的交点，也就是零点处
- 具体迭代的方法见代码
- 迭代结束的标志是前后两次迭代的结果，相差小于小数点后六位，也就是log10（a-b）< -6
数值实验
二分法迭代步数跟收敛精度的关系图
牛顿法迭代步数跟收敛精度的关系图
简化牛顿法迭代步数跟收敛精度的关系图
弦截法迭代步数跟收敛精度的关系图
四种方法的迭代时间对比
结果分析
- 从迭代步数上看，相同的求解的结果如下：
| 求解方法 | 迭代步数 |
|---|---|
| 二分法 | 20 |
| 牛顿法 | 4 |
| 简化牛顿法 | 6 |
| 弦截法 | 4 |
- 从时间上看，二分法的迭代时间明显高于其他三种方法，其他三种方法的迭代时间不分上下。总体而言，在本次求解中，牛顿法可能速度稍微快一点，简化牛顿法由于迭代步数较多，迭代时间会稍微长一点。
实现代码
- 二分法
clear;
format long;
a = 10;
b = 11;
res = binary(a,b);
x = res(1,:);
y = res(2,:);
y = -y;
plot(x,y);
function result = binary(a, b)
k = 0;
acc = -6;
now = 0.0;
x = 0.0;
while(now >= acc)
x = (a+b)/2;
n = x^2-115;
m = a^2-115;
if n == 0
return;
elseif n*m < 0
b = x;
else
a = x;
end
k = k+1;
now = log10(b-a);
result(1,k) = k;
result(2,k) = now;
end
return;
end- 牛顿法
clear;
format long;
a = 10;
b = 11;
res = newton(10);
x = res(1,:);
y = res(2,:);
y = -y;
plot(x,y);
function result = newton(a)
x0 = 0.0;
x1 = a;
acc = -6;
k = 0;
now = 0.0;
while(now >= acc)
x0 = x1;
w = x0^2-115;
p = 2*x0;
x1 = x0 - w/p;
k = k + 1;
result(1,k) = k;
now = log10(abs(x1-x0));
result(2,k) = now;
end
end- 简化牛顿法
clear;
format long;
a = 10;
b = 11;
res = newton(10);
x = res(1,:);
y = res(2,:);
y = -y;
plot(x,y);
function result = newton(a)
x0 = 0.0;
x1 = a;
acc = -6;
k = 0;
now = 0.0;
p = 2*a;
while(now >= acc)
x0 = x1;
w = x0^2-115;
x1 = x0 - w/p;
k = k + 1;
result(1,k) = k;
now = log10(abs(x1-x0));
result(2,k) = now;
end
end- 弦截法
clear;
format long;
a = 10;
b = 11;
res = secant(a, b);
x = res(1,:);
y = res(2,:);
y = -y;
plot(x, y);
function result = secant(a, b)
x0 = a;
x1 = b;
x2 = 0.0;
w = 0.0;
p = 0.0;
k = 0;
acc = -6;
now = 0.0;
while(now >= acc)
w = x1^2-115;
p = ((x1^2-115)-(x0^2-115))/(x1-x0);
x2 = x1 - w/p;
x0 = x1;
x1 = x2;
k = k+1;
now = log10(abs(x1-x0));
result(1,k) = k;
result(2,k) = now;
end
end- 时间对比代码
clear;
format long;
a = 10;
b = 11;
x = [1,2,3,4];
y(1) = binary(a, b);
y(2) = newton(a);
y(3) = better_newton(a);
y(4) = secant(a, b);
bar(x,y);
function time = binary(a, b)
k = 0;
acc = -6;
now = 0.0;
x = 0.0;
time = 0.0;
while(now >= acc)
tic;
x = (a+b)/2;
n = x^2-115;
m = a^2-115;
if n == 0
return;
elseif n*m < 0
b = x;
else
a = x;
end
k = k+1;
now = log10(b-a);
toc;
time = time+toc;
result(1,k) = time;
result(2,k) = now;
end
return;
end
function time = newton(a)
x0 = 0.0;
x1 = a;
acc = -6;
k = 0;
now = 0.0;
time = 0.0;
while(now >= acc)
x0 = x1;
tic;
w = x0^2-115;
p = 2*x0;
x1 = x0 - w/p;
k = k + 1;
toc;
time = time + toc;
result(1,k) = time;
now = log10(abs(x1-x0));
result(2,k) = now;
end
end
function time = better_newton(a)
x0 = 0.0;
x1 = a;
acc = -6;
k = 0;
now = 0.0;
time = 0.0;
p = 2*a;
while(now >= acc)
x0 = x1;
tic;
w = x0^2-115;
x1 = x0 - w/p;
k = k + 1;
toc;
time = time + toc;
result(1,k) = time;
now = log10(abs(x1-x0));
result(2,k) = now;
end
end
function time = secant(a, b)
x0 = a;
x1 = b;
x2 = 0.0;
w = 0.0;
p = 0.0;
k = 0;
acc = -6;
now = 0.0;
time = 0.0;
while(now >= acc)
tic;
w = x1^2-115;
p = ((x1^2-115)-(x0^2-115))/(x1-x0);
x2 = x1 - w/p;
x0 = x1;
x1 = x2;
toc;
time = time + toc;
k = k+1;
now = log10(abs(x1-x0));
result(1,k) = time;
result(2,k) = now;
end
end