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

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dc.contributor.authorKawamura, Akitoshiko
dc.contributor.authorMueller, Norbertko
dc.contributor.authorRoesnick, Carstenko
dc.contributor.authorZiegler, Martin A.ko
dc.date.accessioned2016-09-06T07:24:27Z-
dc.date.available2016-09-06T07:24:27Z-
dc.date.created2016-07-20-
dc.date.created2016-07-20-
dc.date.issued2015-10-
dc.identifier.citationJOURNAL OF COMPLEXITY, v.31, no.5, pp.689 - 714-
dc.identifier.issn0885-064X-
dc.identifier.urihttp://hdl.handle.net/10203/212308-
dc.description.abstractThe 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.languageEnglish-
dc.publisherACADEMIC PRESS INC ELSEVIER SCIENCE-
dc.subjectREAL FUNCTIONS-
dc.subjectEUCLIDEAN-SPACE-
dc.subjectCOMPUTABILITY-
dc.subjectNUMBERS-
dc.subjectEQUATIONS-
dc.subjectSUBSETS-
dc.subjectSETS-
dc.titleComputational benefit of smoothness: Parameterized bit-complexity of numerical operators on analytic functions and Gevrey's hierarchy-
dc.typeArticle-
dc.identifier.wosid000360250000003-
dc.identifier.scopusid2-s2.0-84938749973-
dc.type.rimsART-
dc.citation.volume31-
dc.citation.issue5-
dc.citation.beginningpage689-
dc.citation.endingpage714-
dc.citation.publicationnameJOURNAL OF COMPLEXITY-
dc.identifier.doi10.1016/j.jco.2015.05.001-
dc.contributor.localauthorZiegler, Martin A.-
dc.contributor.nonIdAuthorKawamura, Akitoshi-
dc.contributor.nonIdAuthorMueller, Norbert-
dc.contributor.nonIdAuthorRoesnick, Carsten-
dc.type.journalArticleArticle-
dc.subject.keywordAuthorComputational complexity-
dc.subject.keywordAuthorRigorous numerics-
dc.subject.keywordAuthorSmoothness-
dc.subject.keywordAuthorGevrey hierarchy-
dc.subject.keywordPlusREAL FUNCTIONS-
dc.subject.keywordPlusEUCLIDEAN-SPACE-
dc.subject.keywordPlusCOMPUTABILITY-
dc.subject.keywordPlusNUMBERS-
dc.subject.keywordPlusEQUATIONS-
dc.subject.keywordPlusSUBSETS-
dc.subject.keywordPlusSETS-
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