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Pipelined Krylov subspace methods (also referred to as communication-hidingmethods) have been proposed in the literature as a scalable alternative toclassic Krylov subspace algorithms for iteratively computing the solution to alarge linear system in parallel. For symmetric and positive definite systemmatrices the pipelined Conjugate Gradient method outperforms its classicConjugate Gradient counterpart on large scale distributed memory hardware byoverlapping global communication with essential computations like thematrix-vector product, thus hiding global communication. A well-known drawbackof the pipelining technique is the (possibly significant) loss of numericalstability. In this work a numerically stable variant of the pipelined ConjugateGradient algorithm is presented that avoids the propagation of local roundingerrors in the finite precision recurrence relations that construct the Krylovsubspace basis. The multi-term recurrence relation for the basis vector isreplaced by two-term recurrences, improving stability without increasing theoverall computational cost of the algorithm. The proposed modification ensuresthat the pipelined Conjugate Gradient method is able to attain a highlyaccurate solution independently of the pipeline length. Numerical experimentsdemonstrate a combination of excellent parallel performance and improvedmaximal attainable accuracy for the new pipelined Conjugate Gradient algorithm.This work thus resolves one of the major practical restrictions for theuseability of pipelined Krylov subspace methods.

Numerically Stable Recurrence Relations for the Communication Hiding Pipelined Conjugate Gradient Method