hessenberg约化matlab代码
时间: 2023-07-09 07:39:06 浏览: 46
以下是 Hessenberg 约化的 MATLAB 代码:
```matlab
function [H, Q] = hessenberg(A)
% HESSENBERG Hessenberg reduction of matrix A using Householder transformations.
% [H, Q] = HESSENBERG(A) computes the Hessenberg matrix H and the
% orthogonal matrix Q such that A = Q*H*Q', where A is the input matrix.
n = size(A, 1);
Q = eye(n);
for k = 1:n-2
% Compute the Householder reflection vector
x = A(k+1:n, k);
e1 = zeros(length(x), 1);
e1(1) = 1;
v = sign(x(1))*norm(x)*e1 + x;
v = v / norm(v);
% Apply the Householder reflection to A and Q
A(k+1:n, k:n) = A(k+1:n, k:n) - 2*v*(v'*A(k+1:n, k:n));
A(1:n, k+1:n) = A(1:n, k+1:n) - 2*(A(1:n, k+1:n)*v)*v';
Q(:, k+1:n) = Q(:, k+1:n) - 2*(Q(:, k+1:n)*v)*v';
end
H = triu(A, -1);
end
```
其中,`A` 是输入的矩阵,`H` 是经过 Hessenberg 约化后得到的上 Hessenberg 矩阵,`Q` 是正交矩阵,满足 $A=QHQ^T$。该算法使用 Householder 变换来实现 Hessenberg 约化,具体实现细节可以参考代码中的注释。