New iterative methods for dense linear systems

Lizi Yin,A. Caggiano,Ke Wang,Jinmei Wang

doi:10.1051/e3sconf/202129302013

Abstract

Solving dense linear systems of equations is quite time consuming and requires an efficient parallel implementation on powerful supercomputers. Du, Zheng and Wang presented some new iterative methods for linear systems [Journal of Applied Analysis and Computation, 2011, 1(3): 351-360]. This paper shows that their methods are suitable for solving dense linear system of equations, compared with the classical Jacobi and Gauss-Seidel iterative methods.

Highlights

Linear system plays an important role of applications in engineering and scientific computing such as boundary element methods, quantum mechanical problems and large least squares problems [1, 2, 7, 12, 16, 17]
For an n × n ( ) dense linear system (1), if the coefficient matrix A is diagonally dominant and all entries are positive, it can be numerically solved by the method (2)
Two new iterative methods are discussed for solving dense linear system, which is easy to establish and meet the convergence conditions

Summary

Introduction

Linear system plays an important role of applications in engineering and scientific computing such as boundary element methods, quantum mechanical problems and large least squares problems [1, 2, 7, 12, 16, 17]. Large sparse linear systems can be solved efficiently by iterative methods, especially those based on a Krylov subspace. For large dense linear systems, it is hard to develop good numerical methods. The case where A is dense can be solved numerically by direct methods, iterative methods and parallel methods [11]. Zheng and Wang [3] suggested some new iterative methods for solving linear systems, and they showed that these methods, compared with the classical Jacobi and Gauss-Seidel methods, can be applied to more systems and have faster convergence. The new methods presented by Du, Zheng and Wang [3] are discussed for solving dense linear systems.

New iterative methods

Discussion for dense linear system

Numerical experiments

Conclusion