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

翻译：本文提出了一类用于逼近给定复矩阵奇异值分解（SVD）的高阶方法。据我们所知，现有文献中仅存在不超过三阶的方法。第一部分致力于在正则情形（即奇异值两两互异时）下定义并分析这类方法。该构造基于对SVD相关适定系统（SVD系统）的摄动分析。更精确地，对于给定的整数$p$，当SVD系统的初始近似满足一个取决于三个量的条件时——即初始SVD系统近似的范数、最大奇异值以及奇异值差模倒数的最大值——我们定义了一个以$p+1$阶收敛于左右奇异向量与奇异值的序列。从数值计算角度而言，这为证明和验证初始近似邻域内SVD存在性提供了高效简便的检验方法。随后我们将这些结论推广至奇异值成簇的情形。同时展示如何利用正则情形的结论检测奇异值簇，并定义SVD的消去概念。数值实验验证了理论结果。