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The purpose of this paper is to derive new computable convergence bounds for GMRES. The new bounds depend on the initial guess and are thus conceptually different from standard “worst-case” bounds. Most importantly, approximations to the new bounds can be computed from information generated during the run of a certain GMRES implementation. The approximations allow predictions of how the algorithm will perform. Heuristics for such predictions are given. Numerical experiments illustrate the behavior of the new bounds as well as the use of the heuristics. [ABSTRACT FROM AUTHOR]