给我一个MINRES迭代法的Matlab精确代码

MINRES（Minimum Residual）方法是一种用于解决稀疏、不对称线性系统Ax=b的迭代求解算法，尤其适用于正定系统。以下是MATLAB中使用MINRES算法的简单精确代码：

function [x,resid] = minres(A,b,tol,maxit)
% MINRES - MINimum RESidual algorithm for solving Ax = b
% A: sparse matrix (n x n)
% b: right-hand side vector (n x 1)
% tol: desired tolerance for the residual norm
% maxit: maximum number of iterations

% Initialize variables
r0 = b - A * zeros(size(b));
z0 = r0;
H = speye(size(A)); % Identity matrix as an initial approximation to A'*A
x = zeros(size(b));
k = 0; % iteration counter
resid = [];

while k < maxit && norm(r0) > tol
    % Orthogonalize z_k with respect to previous residuals
    for i = 1:k
        q = H*z0 - resid(i);
        alpha = dot(q,r0);
        z0 = z0 - alpha*r0;
        if i > 1
            beta = alpha / beta_i(i-1);
            H = H + beta*q*q';
        end
    end
    
    % Solve the inner Krylov subspace problem for beta_k
    beta_k = dot(H*z0,r0);
    
    % Update the approximate solution and residual
    x = x + beta_k*z0;
    r1 = r0 - beta_k*H*z0;
    resid(k+1) = r1;
    r0 = r1;
    k = k + 1;
end

if k == maxit
    warning('Maximum number of iterations reached');
end

在这个代码中，`minres`函数接受稀疏矩阵`A`、右向量`b`以及停止条件（容忍度和最大迭代次数）。它返回最小残差反规范法的近似解`x`和每次迭代的残差信息。