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Krylov subspace methods are widely known as efficient algebraic methods forsolving large scale linear systems. However, on massively parallel hardware theperformance of these methods is typically limited by communication latencyrather than floating point performance. With HPC hardware advancing towards theexascale regime the gap between computation and communication keeps steadilyincreasing, imposing the need for scalable alternatives to traditional Krylovsubspace methods. One such approach are the so-called pipelined Krylov subspacemethods, which reduce the number of global synchronization points and overlapglobal communication latency with local arithmetic operations, thus hiding theglobal reduction phases behind useful computations. To obtain this overlap thetraditional Krylov subspace algorithm is reformulated by introducing a numberof auxiliary vector quantities, which are computed using additional recurrencerelations. Although pipelined Krylov subspace methods are equivalent totraditional Krylov subspace methods in exact arithmetic, local rounding errorsinduced by the multi-term recurrence relations in finite precision may inpractice affect convergence significantly. This numerical stability study aimsto characterize the effect of local rounding errors on attainable accuracy invarious pipelined versions of the popular Conjugate Gradient method.Expressions for the gaps between the true and recursively computed variablesthat are used to update the search directions in the different CG variants arederived. Furthermore, it is shown how these results can be used to analyze andcorrect the effect of local rounding error propagation on the maximalattainable accuracy of pipelined CG methods. The analysis in this work issupplemented by numerical experiments that demonstrate the numerical behaviorof the pipelined CG methods.

Numerical analysis of the maximal attainable accuracy in communication hiding pipelined Conjugate Gradient methods