Computational benefit of smoothness: Parameterized bit-complexity of numerical operators on analytic functions and Gevrey's hierarchy

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The synthesis of (discrete) Complexity Theory with Recursive Analysis provides a quantitative algorithmic foundation to calculations over real numbers, sequences, and functions by approximation up to prescribable absolute error 1/2(n) (roughly corresponding to n binary digits after the radix point). In this sense Friedman and Ko have shown the seemingly simple operators of maximization and integration 'complete' for the standard complexity classes NP and #P - even when restricted to smooth (=e(infinity)) arguments. Analytic polynomial-time computable functions on the other hand are known to get mapped to polynomial-time computable functions: non-uniformly, that is, disregarding dependences other than on the output precision n. The present work investigates the uniform parameterized complexity of natural operators A on subclasses of smooth functions: evaluation, pointwise addition and multiplication, (iterated) differentiation, integration, and maximization. We identify natural integer parameters k = k(f) which, when given as enrichment to approximations to the function argument f, permit to computably produce approximations to A(f); and we explore the asymptotic worst-case running time sufficient and necessary for such computations in terms of the output precision n and said k. It turns out that Maurice Gevrey's 1918 classical hierarchy climbing from analytic to (just below) smooth functions provides for a quantitative gauge of the uniform computational complexity of maximization and integration that, non-uniformly, exhibits the phase transition from tractable (i.e. polynomial-time) to intractable (in the sense of NP-'hardness'). Our proof methods involve Hard Analysis, Approximation Theory, and an adaptation of Information-Based Complexity to the bit model. (C) 2015 The Authors. Published by Elsevier Inc.
Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
Issue Date
2015-10
Language
English
Article Type
Article
Keywords

REAL FUNCTIONS; EUCLIDEAN-SPACE; COMPUTABILITY; NUMBERS; EQUATIONS; SUBSETS; SETS

Citation

JOURNAL OF COMPLEXITY, v.31, no.5, pp.689 - 714

ISSN
0885-064X
DOI
10.1016/j.jco.2015.05.001
URI
http://hdl.handle.net/10203/212308
Appears in Collection
CS-Journal Papers(저널논문)
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