A complex singular value decomposition algorithm based on the Riemannian Newton method

In this paper, the problem of finding the singular value decomposition (SVD) of a complex matrix is formulated as an optimization problem on the product of two complex Stiefel manifolds. A new algorithm for the complex SVD is proposed on the basis of the Riemannian Newton method. This algorithm can provide the singular vectors associated with an arbitrary number of the singular values from the largest one down to a smaller one. Furthermore, once a sufficiently accurate approximate complex SVD is given, the Riemannian Newton method can improve it to be as accurate as the computer accuracy permits.