Solving linear equations on parallel distributed memory architectures by extrapolation

Extrapolation methods can be used to accelerate the convergence of vector sequences. It is shown how three di erent extrapolation algorithms, the minimal polynomial extrapolation (MPE), the reduced rank extrapolation (RRE) and the modi ed minimal polynomial extrapolation (MMPE), can be used to solve systems of linear equations. The algorithms are derived and equivalence to di erent Krylov subspace methods are established. The extrapolation algorithms are to prefer on parallel distributed memory architectures since less inter-processor communication is needed. Numerically the extrapolation methods are not as stable as the Krylov subspace methods since they require the solution of ill-conditioned overdetermined systems. Several techniques of improving convergence and stability are presented. Some of these are new to the best of the author's knowledge. The use of regularization methods and a slightly modi ed stationary method have proved to be especially useful. Error bounds and methods of estimating accuracy are given. Some aspects of implementation are discussed with emphasis on parallel distributed memory architectures. Implementations of RRE and GMRES are compared on an IBM RS/6000 Power Parallel SP. RRE has some di culties converging to solutions with errors close to the oating point relative accuracy. For slightly larger tolerances however, RRE is better than GMRES. RRE seems to be the most useful of the extrapolation algorithms, especially if it is used with the modi ed stationary method given here.