Improving Newton-Schulz Method for Approximating Matrix Generalized Inverse by Using Schemes with Memory

[EN] Some iterative schemes with memory were designed for approximating the inverse of a nonsingular square complex matrix and the Moore-Penrose inverse of a singular square matrix or an arbitrary m x n complex matrix. A Kurchatov-type scheme and Steffensen's method with memory were developed for estimating these types of inverses, improving, in the second case, the order of convergence of the Newton-Schulz scheme. The convergence and its order were studied in the four cases, and their stability was checked as discrete dynamical systems. With large matrices, some numerical examples are presented to confirm the theoretical results and to compare the results obtained with the proposed methods with those provided by other known ones.Cordero Barbero, A.; Maimo, JG.; Torregrosa Sánchez, JR.; Vassileva, MP. (2023). Improving Newton-Schulz Method for Approximating Matrix Generalized Inverse by Using Schemes with Memory. Mathematics. 11(14). https://doi.org/10.3390/math11143161111

A general class of arbitrary order iterative methods for computing generalized inverses

[EN] A family of iterative schemes for approximating the inverse and generalized inverse of a complex matrix is designed, having arbitrary order of convergence p. For each p, a class of iterative schemes appears, for which we analyze those elements able to converge with very far initial estimations. This class generalizes many known iterative methods which are obtained for particular values of the parameters. The order of convergence is stated in each case, depending on the first non-zero parameter. For different examples, the accessibility of some schemes, that is, the set of initial estimations leading to convergence, is analyzed in order to select those with wider sets. This wideness is related with the value of the first non-zero value of the parameters defining the method. Later on, some numerical examples (academic and also from signal processing) are provided to confirm the theoretical results and to show the feasibility and effectiveness of the new methods. (C) 2021 The Authors. Published by Elsevier Inc.This research was supported in part by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE) and in part by VIE from Instituto Tecnologico de Costa Rica (Research #1440037)Cordero Barbero, A.; Soto-Quiros, P.; Torregrosa Sánchez, JR. (2021). A general class of arbitrary order iterative methods for computing generalized inverses. Applied Mathematics and Computation. 409:1-18. https://doi.org/10.1016/j.amc.2021.126381S11840