matlab实现

好的，我可以为您提供MATLAB代码实现三种迭代方法。以下是示例代码：

1. 雅克比迭代

function [x, iter] = jacobi(A, b, tol, maxiter)
% Jacobi iteration to solve Ax = b
% A: coefficient matrix
% b: right-hand side vector
% tol: tolerance
% maxiter: maximum number of iterations
% x: solution vector
% iter: number of iterations

n = length(b);
x = zeros(n, 1);
iter = 0;
err = inf;

while err > tol && iter < maxiter
    xnew = zeros(n, 1);
    for i = 1:n
        xnew(i) = (b(i) - A(i, 1:i-1)*x(1:i-1) - A(i, i+1:n)*x(i+1:n)) / A(i, i);
    end
    err = norm(xnew - x);
    x = xnew;
    iter = iter + 1;
end

2. 高斯-赛德尔迭代

function [x, iter] = gauss_seidel(A, b, tol, maxiter)
% Gauss-Seidel iteration to solve Ax = b
% A: coefficient matrix
% b: right-hand side vector
% tol: tolerance
% maxiter: maximum number of iterations
% x: solution vector
% iter: number of iterations

n = length(b);
x = zeros(n, 1);
iter = 0;
err = inf;

while err > tol && iter < maxiter
    xold = x;
    for i = 1:n
        x(i) = (b(i) - A(i, 1:i-1)*x(1:i-1) - A(i, i+1:n)*xold(i+1:n)) / A(i, i);
    end
    err = norm(x - xold);
    iter = iter + 1;
end

3. 超松弛迭代

function [x, iter] = sor(A, b, omega, tol, maxiter)
% Successive over-relaxation iteration to solve Ax = b
% A: coefficient matrix
% b: right-hand side vector
% omega: relaxation parameter
% tol: tolerance
% maxiter: maximum number of iterations
% x: solution vector
% iter: number of iterations

n = length(b);
x = zeros(n, 1);
iter = 0;
err = inf;

while err > tol && iter < maxiter
    xold = x;
    for i = 1:n
        x(i) = (1 - omega)*xold(i) + omega*(b(i) - A(i, 1:i-1)*x(1:i-1) - A(i, i+1:n)*xold(i+1:n)) / A(i, i);
    end
    err = norm(x - xold);
    iter = iter + 1;
end

这些函数的使用方法如下：

% Example usage
A = [4, -1, 0, 0;
     -1, 4, -1, 0;
     0, -1, 4, -1;
     0, 0, -1, 3];
b = [1; 2; 0; 1];
tol = 1e-6;
maxiter = 1000;

% Jacobi iteration
[x, iter] = jacobi(A, b, tol, maxiter);
fprintf('Jacobi iteration: x = [%f, %f, %f, %f], iter = %d\n', x, iter);

% Gauss-Seidel iteration
[x, iter] = gauss_seidel(A, b, tol, maxiter);
fprintf('Gauss-Seidel iteration: x = [%f, %f, %f, %f], iter = %d\n', x, iter);

% Successive over-relaxation iteration
omega = 1.5;
[x, iter] = sor(A, b, omega, tol, maxiter);
fprintf('SOR iteration: x = [%f, %f, %f, %f], iter = %d\n', x, iter);

您可以根据需要修改代码中的系数矩阵 $A$、右侧向量 $b$、迭代精度 $tol$、最大迭代次数 $maxiter$ 和松弛因子 $\omega$。