用matlab实现共轭梯度法求解实例,用MATLAB实现共轭梯度法求解实例

以下是使用MATLAB实现共轭梯度法求解线性方程组的实例代码：

% 定义线性方程组 A*x = b
A = [10, -1, 2; -1, 11, -1; 2, -1, 10];
b = [6; 25; -11];

% 初始解 x0
x0 = zeros(size(b));

% 定义共轭梯度法的参数
max_iter = 1000;
tol = 1e-6;

% 初始化迭代变量
r = b - A*x0;
p = r;
x = x0;
iter = 0;

% 迭代求解
while iter < max_iter
    iter = iter + 1;
    alpha = (r'*r) / (p'*A*p);
    x = x + alpha*p;
    r_new = r - alpha*A*p;
    if norm(r_new) < tol
        break;
    end
    beta = (r_new'*r_new) / (r'*r);
    p = r_new + beta*p;
    r = r_new;
end

% 输出结果
disp(['共轭梯度法迭代次数：', num2str(iter)]);
disp(['解为：']);
disp(x);

参考资料：

1. Shewchuk, J. R. (1994). An Introduction to the Conjugate Gradient Method Without the Agonizing Pain. Carnegie Mellon University, Pittsburgh, PA, USA.