用Matlab实现:基于双边Jacobi的svd算法
时间: 2023-12-18 20:05:48 浏览: 38
以下是基于双边Jacobi的SVD算法的Matlab实现:
```matlab
function [U,S,V] = bidiagonalSVD(A)
% Bidiagonal SVD using Jacobi rotations
% Input: A - m x n matrix to be decomposed
% Output: U - m x m orthogonal matrix
% S - m x n diagonal matrix with nonnegative diagonal entries
% V - n x n orthogonal matrix
[m,n] = size(A);
U = eye(m);
V = eye(n);
B = A;
for k = 1:n
% Zeroing elements below the diagonal
for i = k+2:m
G = givens(B(i-1,k),B(i,k));
B(i-1:i,k:n) = G*B(i-1:i,k:n);
U(:,i-1:i) = U(:,i-1:i)*G';
end
% Zeroing elements to the right of the super-diagonal
if k < n
for j = k+2:n
G = givens(B(k,j-1),B(k,j));
B(k:m,j-1:j) = B(k:m,j-1:j)*G;
V(:,j-1:j) = V(:,j-1:j)*G';
end
end
end
% Extracting singular values from the diagonal
S = diag(diag(B));
% Removing negative signs from the diagonal
for i = 1:n
if S(i,i) < 0
S(i,i) = -S(i,i);
U(:,i) = -U(:,i);
end
end
end
function [G] = givens(a,b)
% Compute the Givens rotation matrix for elements a and 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
G = [c -s; s c];
end
```
其中,`bidiagonalSVD`函数输入一个$m\times n$的矩阵$A$,输出其SVD分解$U,S,V$。`givens`函数计算给定两个数$a$和$b$的Givens旋转矩阵$G$。
双边Jacobi的SVD算法通过旋转矩阵来将原矩阵$A$转化为一个上三角矩阵$B$,然后再从$B$中提取出奇异值构成对角矩阵$S$。同时,记录下旋转矩阵$U$和$V$,最终得到SVD分解$A=USV^T$。
相关推荐
![zip](https://img-home.csdnimg.cn/images/20210720083736.png)
![zip](https://img-home.csdnimg.cn/images/20210720083736.png)
![-](https://csdnimg.cn/download_wenku/file_type_lunwen.png)
![-](https://csdnimg.cn/download_wenku/file_type_lunwen.png)
![-](https://csdnimg.cn/download_wenku/file_type_column_c1.png)
![-](https://csdnimg.cn/download_wenku/file_type_lunwen.png)
![-](https://csdnimg.cn/download_wenku/file_type_column_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)