Advances in Mathematical Physics

Volume 2018, Article ID 1369707, 12 pages

https://doi.org/10.1155/2018/1369707

## Residual-Based Simpler Block GMRES for Nonsymmetric Linear Systems with Multiple Right-Hand Sides

^{1}School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China^{2}College of Science, Hunan University of Science and Engineering, Yongzhou 425100, China^{3}Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

Correspondence should be addressed to Qinghua Wu; moc.361@hqwkcaj

Received 16 November 2017; Accepted 12 March 2018; Published 16 April 2018

Academic Editor: Soheil Salahshour

Copyright © 2018 Qinghua Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose in this paper a residual-based simpler block GMRES method for solving a system of linear algebraic equations with multiple right-hand sides. We show that this method is mathematically equivalent to the block GMRES method and thus equivalent to the simpler block GMRES method. Moreover, it is shown that the residual-based method is numerically more stable than the simpler block GMRES method. Based on the deflation strategy proposed by Calandra et al. (2013), we derive a deflation strategy to detect the possible linear dependence of the residuals and a near rank deficiency occurring in the block Arnoldi procedure. Numerical experiments are conducted to illustrate the performance of the new method.

#### 1. Introduction

In this paper, we consider iterative methods for solving a system of linear algebraic equations:where is a nonsingular matrix of order and and are rectangular matrices of dimension with . For solving such systems, the block GMRES [1] and its variants are very popular. Block GMRES is based on the block Arnoldi process and is formally fully analogous to the ordinary GMRES algorithm by Saad and Schultz [2].

The following notation is used throughout the paper. Subscripts denote the iteration index and superscripts distinguish between individual columns in a block. We denote by the Euclidean vector norm and the induced matrix norm and by the Frobenius norm. Moreover, for of rank , is the spectral condition number, where are the extremal singular values of .

Given an initial approximation to the solution of (1), letand then in analogy to the unblocked case, we build a sequence of iterates such thatwhere . Equation (3) is equivalent to minimizing every column of , that is,and also to the orthogonality conditionwhere is the orthogonality relation induced by the Euclidean inner product. Assume that is of the form , where is a basis of . Then, we obtain the th residual matrix

Central to the usual implementations of block GMRES is the block Arnoldi process [1], which can be used to construct the orthonormal basis of . In practice, the possible linear dependence of the residuals of the systems requires an explicit reduction of the number of right-hand sides. In [3], this was called deflation. If the block residual is nearly rank deficient, block GMRES should be implemented with deflation and there are various sophisticated rank-revealing QR factorizations. For details, see [3] and the references therein. We can write the nondeflated block Arnoldi process as shown in Algorithm 1.