A Family of Iterative Methods with Accelerated Convergence for Restricted Linear System of Equations

Abstract

The aim of this paper is to develop a family of iterative methods with accelerated convergence for solving restricted linear system of equations $$Ax = b$$ , $$x\in T$$ , where $$A \in \mathbb {C}^{m\times n}$$ , $$b \in \mathbb {C}^{m}$$ and T is a subspace of $$\mathbb {C}^{n}$$ . Necessary and sufficient convergence conditions along with the estimation of error bounds of derived approximations are established. The proposed family of iterative methods is applied to solve restricted linear system of equations for different subspaces T. In particular, least-squares solution of an overdetermined linear system and Krylov solution of a square singular linear system of equations are inspected. In addition, the iterative family is tested on a number of numerical examples, including randomly generated dense matrices, sparse singular matrices, and standard test matrices from the Matrix Computational Toolbox. The obtained results are compared with the existing iterative methods and with direct methods, such as the singular value decomposition technique and the QR factorization method.