Research On The Simpler GMRES For Linear Systems

In this paper the simpler GMRES is studied. Firstly, a simpler block GMRES is proposed for solving linear systems with multiple right-hand sides. Then, the convergence properties of the simpler block GMRES in inexact arithmetic are discussed, with which an inexact simpler block GMRES is presented. Finally, a simpler hybrid GMRES is also formulated.Block GMRES is one of the most popular iterative algorithms for solving linear systems with multiple right-hand sides. In its usual implementation, a least-squares problem needs to be solved which involves a block upper-Hessenberg QR factorization. In Chapter 2 a simpler block GMRES is proposed. The equivalence of the simpler block GMRES and the standard block GMRES is shown. In the new algorithm, the block upper-Hessenberg least squares problem is reduced to an upper triangular least-squares problem, which doesn't need block upper-Hessenberg QR factorization. In consequence, it is easier to program and requires less cost. In Chapter 3 the convergence properties of the simpler block GMRES, when the product of the matrix and the block vector is not performed exactly, are also discussed. Theories of the inexact Krylov subspace methods are generalized to inexact simpler block GMRES. We discuss the properties of the computed residual and the approximation, the gap between the true residual and the computed residual, and the relaxation strategies.In Chapter 4, based on the equivalence of the simpler GMRES and GMRES, a simpler hybrid GMRES algorithm is proposed for solving linear systems with single right-hand side. The algorithm first runs simpler GMRES until the residual norm drops by a certain amount, then re-applies the simpler GMRES residual polynomials via a Richardson iteration.