Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Accurate and Efficient Expression Evaluation and Linear Algebra   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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The categories of this paper are: nlin
13
The categories of this paper are: quant-ph
22
The categories of this paper are: chao-dyn
14
The categories of this paper are: math-ph
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
28
The categories of this paper are: alg-geom
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The categories of this paper are: q-bio
34
The categories of this paper are: chem-ph
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The categories of this paper are: plasm-ph
35
The categories of this paper are: mtrl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-lat
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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc