基于单边jacobi的奇异值分解matlab

以下是基于单边Jacobi的奇异值分解的MATLAB代码示例：

function [U,S,V] = svd_jacobi(A,tol)
% SVD using Jacobi's method
% A: input matrix
% tol: tolerance for convergence
% U, S, V: output matrices

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

% Jacobi's method
converged = false;
while ~converged
    converged = true;
    for p = 1:n-1
        for q = p+1:n
            if abs(A(:,p)'*A(:,q)) > tol
                converged = false;
                % Compute Jacobi rotation matrix
                [c,s] = jacobi(A(p,p), A(q,q), A(p,q));
                J = eye(n);
                J(p,p) = c;
                J(q,q) = c;
                J(p,q) = s;
                J(q,p) = -s;
                % Update matrices
                A = J' * A * J;
                U = U * J;
                V = V * J;
            end
        end
    end
end

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

end

function [c,s] = jacobi(a,b,c)
% Computes Jacobi rotation matrix
if a == b
    c = 1;
    s = 0;
else
    tau = (b-a)/(2*c);
    t = sign(tau)/(abs(tau) + sqrt(1+tau^2));
    c = 1/sqrt(1+t^2);
    s = t*c;
end
end

这个算法的主要思想是通过Jacobi旋转来逐步将输入矩阵转化为对角矩阵，同时保持正交性。在每次旋转后，我们都会更新左右奇异向量矩阵和奇异值向量。在算法结束后，我们将奇异值向量和左右奇异向量矩阵根据奇异值的大小进行排序。