Accurate and Efficient Expression Evaluation and Linear Algebra

Details Accurate and Efficient Expression Evaluation and Linear Algebra Accurate and Efficient Expression Evaluation and Linear Algebra

2007-12-24

arxiv.org/abs/0712.4027v1

Acta Numerica, Volume 17, May 2008, pp 87-145

Mathematics

Numerical Analysis

49 pages, 6 figures, 1 table

Scientific paper

10.1017/S0962492906350015

We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed answer has relative error less than 1, i.e., has some correct leading digits. We also address efficiency, by which we mean algorithms that run in polynomial time in the size of the input. Our results will depend strongly on the model of arithmetic: Most of our results will use the so-called Traditional Model (TM). We give a set of necessary and sufficient conditions to decide whether a high accuracy algorithm exists in the TM, and describe progress toward a decision procedure that will take any problem and provide either a high accuracy algorithm or a proof that none exists. When no accurate algorithm exists in the TM, it is natural to extend the set of available accurate operations by a library of additional operations, such as $x+y+z$, dot products, or indeed any enumerable set which could then be used to build further accurate algorithms. We show how our accurate algorithms and decision procedure for finding them extend to this case. Finally, we address other models of arithmetic, and the relationship between (im)possibility in the TM and (in)efficient algorithms operating on numbers represented as bit strings.

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