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Computational Science, Engineering & Technology Series
ISSN 1759-3158
Edited by: P. Iványi and B.H.V. Topping
Chapter 8

Finite Precision Computation for Linear Fixed Point Methods of Parallel Asynchronous Iterations

P. Spiteri

Institut National Polytechnique and Institut de Recherche en Informatique, Toulouse, France

Full Bibliographic Reference for this chapter
P. Spiteri, "Finite Precision Computation for Linear Fixed Point Methods of Parallel Asynchronous Iterations", in P. Iványi and B.H.V. Topping, (Editor), "Techniques for Parallel, Distributed and Cloud Computing in Engineering", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 8, pp 163-196, 2015. doi:10.4203/csets.36.8
Keywords: fixed point methods, parallel asynchronous iterations, roundoff errors, approximate contraction, stopping criteria, backward and forward errors.

This paper addresses the difficult question of defining reliable stopping criteria for a parallel asynchronous iterative method applied on an affine fixed point iteration. Such a method allows the solving of a linear algebraic system. The study takes into account the fact that the computations are performed in finite precision. Several types of norms are considered for the estimations of the errors and the corresponding error bounds are proved. More precisely, in the finite dimensional case, this study relies on a topological framework involving the use of a uniform Perron-Frobenius weighted norm and also equivalent norms. New algorithmic aspects based on sliding macro iteration are considered in order to obtain effective stopping criteria.

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