MATLAB实现Jacobi迭代法、G-S迭代法和SOR迭代法

【实验目的】

1. 编程实现Jacobi迭代法、G-S迭代法和SOR迭代法；

2. 比较Jacobi迭代法、G-S迭代法迭代次数，进而理解迭代速度与迭代矩阵谱半径的关系；

【本次实验内容】

Jacobi迭代法：

t tic;
n=length(b);
x=zeros(n,1);
x_new=zeros(n,1);
err=tolerance+1;
iter=0;
max_iter=1000;
while iter <max_iter &&err>=tolerance
    for i=1:n
         x_new(i)=(b(i)-A(i,1:i-1)*x(1:i-1)-A(i,i+1:n)*x(1+i:n))/A(i,i);
    end
    iter =iter +1;
    err=max(abs(x_new-x));
    x=x_new;
end
disp(x)
disp(iter)
toc;
err

G-S迭代法：

Ab=[4 3 0 1;3 4 -1 -5;0 -1 4 3];
toc
GaussSeidel(Ab,1e-8)
function solution = GaussSeidel(Ab,epsilon)
A=[4 3 0;3 4 -1;0 -1 4];
b=[1;-5;3];
error = 10;
n1 = 0;
n = 3;
start = zeros(3,1);
xk = start;
xknext = start;
while error > epsilon
    xk = xknext;
    xknext(1) = 1/A(1,1)*(b(1) - sum(A(1,2:n).* xk(2:n)'));
    for i = 2:n - 1
        Ssum1 = sum(A(i,1:i - 1) .* xk(1:i - 1)');
        Ssum2 = sum(A(i,i + 1:n) .* xknext(i + 1:n)');
        xknext(i) = 1/A(i,i) * (b(i) - Ssum1 - Ssum2);
    end
        xknext(n) = 1/A(n,n)*(b(n) - sum(A(n,1:n-1).* xknext(1:n-1)'));
        error = norm(xk - xknext);
        n1 = n1 + 1;
end
    fprintf('GSµü´ú´ÎÊý£º%d',n1)
    solution = xknext;
    error
    t=toc
end

tic;
A = [4 3 0; 3 4 -1; 0 -1 4];
b = [1; -5; 3];
n = length(b);
x0 = zeros(n, 1);
 
w_values = [-0.2, 0.5, 1.0, 1.24, 1.5, 2.2];
max_iterations = 20;
tolerance = 1e-6;
 
for w = w_values
    x = x0;
    for k = 1:max_iterations
        x_old = x;
        for i = 1:n
            x(i) = (1 - w) * x_old(i) + (w / A(i, i)) * (b(i) - A(i,1:i-1) * x(1:i-1) - A(i,i+1:end) * x_old(i+1:end));
        end
        if norm(x - x_old) < tolerance
            break;
        end
    end
    disp(w);
    disp(x);
    disp(k);
end   
toc;