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SEQUENTIALLY OPTIMAL ALGORITHM FOR NUMERICAL-INTEGRATION

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1979年第3期28卷 363-373页

作者： SUKHAREV, AG MV LOMONOSOV STATE UNIV DEPT COMPUTAT MATH & CYBERNETMOSCOW 117234USSR

For the class of functions of one variable, satisfying the Lipschitz condition with a fixed constant, an optimal passive algorithm for numerical integration (an optimal quadrature formula) has been found by Nikol'skii. In this paper, a sequentially optimal algorithm is constructed; i.e., the algorithm on each step makes use in an optimal way of all relevant information which was accumulated on previous steps. Using the algorithm, it is necessary to solve an integer program at each step. An effective algorithm for solving these problems is given.

关键词： minimax problems Numerical integration quadrature formulas worst-case algorithms

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OPTIMAL algorithms FOR ITERATED NUMERICAL-INTEGRATION

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1979年第3期28卷 375-390页

作者： SUKHAREV, AG MV LOMONOSOV STATE UNIV DEPT COMPUTAT MATH & CYBERNETMOSCOW 117234USSR

Iterated (repeated, successive) integration is used for integrating functions satisfying the Lipschitz condition. To construct an optimal (minimax) algorithm, it is necessary to integrate optimally functions evaluated with indefinite errors. Such a problem is solved by generalization of an algorithm from Ref. 1 which is sequentially optimal for one-dimensional problems.

关键词： Numerical integration quadrature formulas worst-case algorithms minimax problems analytic complexity optimal algorithms

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Functional Data Stuctures and algorithms

Functional Data Stuctures and Algorithms

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作者： Straka, Milan Charles University of Prague

Title: Functional Data Structures and algorithms Author: Milan Straka Institute: Computer Science Institute of Charles University Supervisor of the doctoral thesis: doc. Mgr. Zdeněk Dvořák, Ph.D, Computer Science Institute of Charles University Abstract: Functional programming is a well established programming paradigm and is becoming increasingly popular, even in industrial and commercial appli- cations. Data structures used in functional languages are principally persistent, that is, they preserve previous versions of themselves when modified. The goal of this work is to broaden the theory of persistent data structures and devise efficient implementations of data structures to be used in functional languages. Arrays are without any question the most frequently used data structure. Despite being conceptually very simple, no persistent array with constant time access operation exists. We describe a simplified implementation of a fully per- sistent array with asymptotically optimal amortized complexity Θ(log log n) and especially a nearly optimal worst-case implementation. Additionally, we show how to effectively perform a garbage collection on a persistent array. The most efficient data structures are not necessarily based on asymptotically best structures. On that account, we also focus on data structure...

关键词： persistent data structures persistent arrays worst-case algorithms purely functional data structures Haskell

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