Making global simpler GMRES more stable

Abstract

SummaryThe global GMRES method is well known for the solution of nonsymmetric linear systems with multiple right hands. In this paper, the condition number for evaluating the stability of the global simpler GMRES method is defined. With this condition number, it is shown that Zong et al.'s global simpler method with a simple basis of the Krylov matrix subspace might lead to instability when there is a large decrease in the residual norm. An adaptive global simpler method is then presented to avoid the potential instability. For this adaptive method, theoretical results deliver the lower and upper bounds for the condition number of the basis, and when the linear system is of single right hand, the estimate is proven to be tighter than Jiránek et al.'s upper bound by numerical tests. Numerical results also show that the adaptive strategy leads to a well‐conditioned basis, and in most cases, the global simpler GMRES with normalized residuals as the basis can also be reliable.