Simpler Hybrid GMRES

Abstract

Hybrid GMRES algorithms are effective for solving large nonsymmetric linear systems. GMRES is \u00a0employed \u00a0at \u00a0the \u00a0first \u00a0phase \u00a0to \u00a0produce \u00a0iterative \u00a0polynomials, \u00a0which \u00a0will \u00a0be \u00a0used \u00a0at \u00a0the \u00a0second \u00a0phase \u00a0to implement the Richardson iteration. In the process of GMRES, a least squares problem needs to be solved which involves an upper Hessenberg factorization. Instead of using GMRES, we may use simpler GMRES. Correspondingly, \u00a0simpler \u00a0hybrid \u00a0GMRES \u00a0algorithms \u00a0are \u00a0formulated. \u00a0It \u00a0is \u00a0described \u00a0how \u00a0to \u00a0construct \u00a0the iterative polynomials from simpler GMRES. The new algorithms avoid the upper Hessenberg factorization so that \u00a0they \u00a0are \u00a0easier \u00a0to \u00a0program \u00a0and \u00a0require \u00a0a \u00a0less \u00a0amount \u00a0of \u00a0work. \u00a0Numerical \u00a0examples \u00a0are \u00a0conducted \u00a0to illustrate the good performance of the new algorithms.