双边Jacobi方法求解SVD的Matlab代码示例

以下是使用双边Jacobi方法求解SVD的Matlab代码示例：

function [U,S,V] = svd_bi_jacobi(A)
% SVD using bidirectional Jacobi method
% A: input matrix
% U: left singular vectors
% S: singular values
% V: right singular vectors

% Initialization
[m,n] = size(A);
U = eye(m);
V = eye(n);
S = diag(A);
delta = 1e-10;
flag = true;

% Iteration
while flag
    flag = false;
    % Jacobi rotation on columns of A
    for i = 1:n-1
        for j = i+1:n
            p = [S(i),S(j),A(:,i)'*A(:,j)];
            [c,s,w] = jacobi(p);
            if abs(s) < delta
                continue;
            end
            flag = true;
            S(i) = c*S(i) + s*S(j);
            S(j) = c*S(j) - s*S(i);
            A(:,i) = c*A(:,i) + s*A(:,j);
            A(:,j) = c*A(:,j) - s*A(:,i);
            V(:,[i,j]) = V(:,[i,j])*[c,s;-s,c];
        end
    end
    % Jacobi rotation on rows of A
    for i = 1:m-1
        for j = i+1:m
            p = [S(i),S(j),A(i,:)*A(j,:)'];
            [c,s,w] = jacobi(p);
            if abs(s) < delta
                continue;
            end
            flag = true;
            S(i) = c*S(i) + s*S(j);
            S(j) = c*S(j) - s*S(i);
            A(i,:) = c*A(i,:) + s*A(j,:);
            A(j,:) = c*A(j,:) - s*A(i,:);
            U(:,[i,j]) = U(:,[i,j])*[c,s;-s,c];
        end
    end
end

% Sort singular values in decreasing order
[S,idx] = sort(S,'descend');
U = U(:,idx);
V = V(:,idx);

% Ensure positive singular values
signs = sign(diag(U'*A*V));
U = U*diag(signs);
V = V*diag(signs);
S = abs(S);
end

function [c,s,w] = jacobi(p)
% Jacobi rotation for 2x2 symmetric matrix
% p: vector [a,b,c] representing matrix [a,c;c,b]
% c: cos(theta), s: sin(theta), w: cot(2*theta)

if p(2) == 0
    c = 1;
    s = 0;
    w = 0;
else
    w = (p(1)-p(2))/(2*p(3));
    t = sign(w)/(abs(w) + sqrt(1+w^2));
    c = 1/sqrt(1+t^2);
    s = c*t;
end
end

这个函数接受一个矩阵作为输入，然后使用双边Jacobi方法来计算其SVD分解。返回的结果是左奇异向量矩阵U、奇异值向量S和右奇异向量矩阵V。