DC Field | Value | Language |
---|---|---|
dc.contributor.author | Kawamura, Akitoshi | ko |
dc.contributor.author | Mueller, Norbert | ko |
dc.contributor.author | Roesnick, Carsten | ko |
dc.contributor.author | Ziegler, Martin A. | ko |
dc.date.accessioned | 2016-09-06T07:24:27Z | - |
dc.date.available | 2016-09-06T07:24:27Z | - |
dc.date.created | 2016-07-20 | - |
dc.date.created | 2016-07-20 | - |
dc.date.created | 2016-07-20 | - |
dc.date.issued | 2015-10 | - |
dc.identifier.citation | JOURNAL OF COMPLEXITY, v.31, no.5, pp.689 - 714 | - |
dc.identifier.issn | 0885-064X | - |
dc.identifier.uri | http://hdl.handle.net/10203/212308 | - |
dc.description.abstract | 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. | - |
dc.language | English | - |
dc.publisher | ACADEMIC PRESS INC ELSEVIER SCIENCE | - |
dc.subject | REAL FUNCTIONS | - |
dc.subject | EUCLIDEAN-SPACE | - |
dc.subject | COMPUTABILITY | - |
dc.subject | NUMBERS | - |
dc.subject | EQUATIONS | - |
dc.subject | SUBSETS | - |
dc.subject | SETS | - |
dc.title | Computational benefit of smoothness: Parameterized bit-complexity of numerical operators on analytic functions and Gevrey's hierarchy | - |
dc.type | Article | - |
dc.identifier.wosid | 000360250000003 | - |
dc.identifier.scopusid | 2-s2.0-84938749973 | - |
dc.type.rims | ART | - |
dc.citation.volume | 31 | - |
dc.citation.issue | 5 | - |
dc.citation.beginningpage | 689 | - |
dc.citation.endingpage | 714 | - |
dc.citation.publicationname | JOURNAL OF COMPLEXITY | - |
dc.identifier.doi | 10.1016/j.jco.2015.05.001 | - |
dc.contributor.localauthor | Ziegler, Martin A. | - |
dc.contributor.nonIdAuthor | Kawamura, Akitoshi | - |
dc.contributor.nonIdAuthor | Mueller, Norbert | - |
dc.contributor.nonIdAuthor | Roesnick, Carsten | - |
dc.type.journalArticle | Article | - |
dc.subject.keywordAuthor | Computational complexity | - |
dc.subject.keywordAuthor | Rigorous numerics | - |
dc.subject.keywordAuthor | Smoothness | - |
dc.subject.keywordAuthor | Gevrey hierarchy | - |
dc.subject.keywordPlus | REAL FUNCTIONS | - |
dc.subject.keywordPlus | EUCLIDEAN-SPACE | - |
dc.subject.keywordPlus | COMPUTABILITY | - |
dc.subject.keywordPlus | NUMBERS | - |
dc.subject.keywordPlus | EQUATIONS | - |
dc.subject.keywordPlus | SUBSETS | - |
dc.subject.keywordPlus | SETS | - |
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