Numerical methods for computing SVD in the D-orthogonal group

In this paper, we consider the problem to compute a special kind of singular value decomposition of a square matrix A=UΣVA=UΣV, where U and V belong to the same D-orthogonal group, i.e. they are orthogonal with respect to a real diagonal orthogonal matrix D  , while ΣΣ is a real diagonal positive definite matrix. In this work, we propose an algebraic method and derive a continuous approach using the projected gradient technique. The differential systems given by the continuous approach are solved using a standard integration solver together with a projection technique obtained computing the D-orthogonal factor of the hyperbolic QR decomposition.