A Theoretical Comparison of the Arnoldi and GMRES Algorithms

Two recently developed Krylov methods for solving linear systems are Arnoldi ’s method and the Generalized Minimum Residual (GMRES) method The GMRES method has been considered superior to Arnoldi’s method due in part to the fact that GMRES never breaks down in the way Arnoldi’s algorithm can. However, it is shown that there is a relationship between breakdowns in the two methods. Specifically, it is shown that GMRES does exhibit breakdowns very similar to that of Arnoldi, often referred to as the “stagnation” of GMRES. A relationship between the norms of the residuals for Arnoldi and GMRES is also given which shows exactly how much larger the residual norm for Arnoldi is than that for GMRES. In general, the results in the paper suggest that if one of the methods performs poorly on a particular problem, then so will the other.