High Order Numerical Methods To Approximate The Singular Value\n Decomposition

In this paper, we present a class of high order methods to approximate the\nsingular value decomposition of a given complex matrix (SVD). To the best of\nour knowledge, only methods up to order three appear in the the literature. A\nfirst part is dedicated to defline and analyse this class of method in the\nregular case, i.e., when the singular values are pairwise distinct. The\nconstruction is based on a perturbation analysis of a suitable system of\nassociated to the SVD (SVD system). More precisely, for an integer $p$ be\ngiven, we define a sequence which converges with an order $p + 1$ towards the\nleft-right singular vectors and the singular values if the initial\napproximation of the SVD system satisfies a condition which depends on three\nquantities : the norm of initial approximation of the SVD system, the greatest\nsingular value and the greatest inverse of the modulus of the difference\nbetween the singular values. From a numerical computational point of view, this\nfurnishes a very efficient simple test to prove and certifiy the existence of a\nSVD in neighborhood of the initial approximation. We generalize these result in\nthe case of clusters of singular values. We show also how to use the result of\nregular case to detect the clusters of singular values and to define a notion\nof deflation of the SVD. Moreover numerical experiments confirm the theoretical\nresults.\n