用Matlab实现：Jacobi SVD

Jacobi SVD 是一种基于 Jacobi 旋转的奇异值分解方法。它的基本思路是通过 Jacobi 旋转将矩阵逐步转化为对角矩阵，然后对角线上的元素就是矩阵的奇异值。下面是用 Matlab 实现 Jacobi SVD 的代码：

function [U,S,V] = jacobiSVD(A)
% Jacobi SVD algorithm
% A: input matrix
% U: left singular vectors
% S: singular values
% V: right singular vectors

[m,n] = size(A);
U = eye(m);
V = eye(n);
S = diag(A);

max_iter = 100;
eps = 1e-6;

for i = 1:max_iter
    % Find maximum off-diagonal element
    [p,q] = findMaxOffDiag(S);
    if abs(S(p,q)) < eps
        break;
    end
    
    % Compute Jacobi rotation matrix
    theta = atan2(S(q,q) - S(p,p), 2*S(p,q));
    c = cos(theta);
    s = sin(theta);
    J = eye(m);
    J(p,p) = c;
    J(q,q) = c;
    J(p,q) = -s;
    J(q,p) = s;
    
    % Update matrices
    S = J'*S*J;
    U = U*J;
    V = V*J;
end

% Sort singular values and vectors
[S,idx] = sort(S,'descend');
U = U(:,idx);
V = V(:,idx);

end

function [p,q] = findMaxOffDiag(A)
% Find maximum off-diagonal element of matrix A
n = size(A,1);
max_val = 0;
for i = 1:n-1
    for j = i+1:n
        if abs(A(i,j)) > max_val
            max_val = abs(A(i,j));
            p = i;
            q = j;
        end
    end
end
end

使用方法：

A = [1 2 3; 4 5 6; 7 8 9];
[U,S,V] = jacobiSVD(A);

其中，输入参数 A 是待分解的矩阵，输出参数 U、S、V 分别是左奇异向量、奇异值和右奇异向量。