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

Akitoshi Kawamura, Norbert Müller, Carsten Rösnick, Martin Ziegler

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

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 1/2n (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 (=C) 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 Λ 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 Λ(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.

Original languageEnglish
Article number1247
Pages (from-to)689-714
Number of pages26
JournalJournal of Complexity
Volume31
Issue number5
DOIs
Publication statusPublished - Oct 1 2015
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Statistics and Probability
  • Numerical Analysis
  • Control and Optimization
  • Applied Mathematics

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