matlab 吉文斯(givens)变换求qr分解的代码
时间: 2024-02-03 19:02:08 浏览: 9
以下是使用吉文斯变换求 QR 分解的 Matlab 代码:
```matlab
function [Q, R] = QR_givens(A)
% A: m x n matrix
% Q: m x m orthogonal matrix
% R: m x n upper triangular matrix
[m, n] = size(A);
Q = eye(m);
R = A;
for j = 1:n
for i = m:-1:j+1
% Compute the Givens rotation matrix G(i-1, i) to zero out R(i, j)
[c, s] = givens(R(i-1, j), R(i, j));
G = [c s; -s c];
% Apply the Givens rotation matrix to R and Q
R([i-1, i], j:n) = G * R([i-1, i], j:n);
Q(:, [i-1, i]) = Q(:, [i-1, i]) * G';
end
end
function [c, s] = givens(a, b)
% Compute the Givens rotation matrix G(i-1, i) to zero out b
if b == 0
c = 1;
s = 0;
else
if abs(b) > abs(a)
r = a / b;
s = 1 / sqrt(1 + r^2);
c = s * r;
else
r = b / a;
c = 1 / sqrt(1 + r^2);
s = c * r;
end
end
```
其中,`givens` 函数用于计算吉文斯旋转矩阵,主函数 `QR_givens` 则通过循环逐步将矩阵 $A$ 转化为上三角矩阵 $R$,并构造正交矩阵 $Q$,最终返回 $Q$ 和 $R$。
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