生成基于Householder变换法的QR算法matlab代码

以下是基于Householder变换法的QR算法的Matlab代码：

function [Q,R] = qr_householder(A)
% Input: A - mxn matrix
% Output: Q - mxm orthogonal matrix
%         R - mxn upper triangular matrix 

[m,n] = size(A);
Q = eye(m); % Initialize Q as mxm identity matrix
for j = 1:n
    % Compute Householder reflection vector
    v = A(j:m,j);
    v(1) = v(1) + sign(v(1))*norm(v); % Choose a sign for numerical stability
    v = v/norm(v);
    % Apply Householder transformation to A and Q
    A(j:m,j:n) = A(j:m,j:n) - 2*v*(v'*A(j:m,j:n));
    Q(j:m,:) = Q(j:m,:) - 2*v*(v'*Q(j:m,:));
end
R = A; % The upper triangular matrix is now in A
Q = Q'; % Transpose Q to get the orthogonal matrix
end

该函数接受一个 $m \times n$ 的矩阵作为输入，并返回一个 $m \times m$ 的正交矩阵 $Q$ 和一个 $m \times n$ 的上三角矩阵 $R$，满足 $A = QR$。主要思路是通过 Householder 变换将矩阵 $A$ 转化为上三角矩阵 $R$，同时构造正交矩阵 $Q$，使得 $A=QR$。