An algorithm for discrete approximation by quasi-convex functions on Rm: An algorithm for discrete approximation by quasi-convex functions on \(R^m\)
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| Názov: | An algorithm for discrete approximation by quasi-convex functions on Rm: An algorithm for discrete approximation by quasi-convex functions on \(R^m\) |
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| Autori: | Vasant A. Ubhaya |
| Zdroj: | Computers & Mathematics with Applications. 47:1707-1712 |
| Informácie o vydavateľovi: | Elsevier BV, 2004. |
| Rok vydania: | 2004 |
| Predmety: | discrete approximation, computational complexity, Uniform norm, 0211 other engineering and technologies, 02 engineering and technology, quasi-convex function, Algorithm, Computational complexity, Best approximation, Chebyshev systems, Computational Mathematics, Computational Theory and Mathematics, Convex hulls, Algorithms for approximation of functions, Quasi-convex functions, Numerical aspects of computer graphics, image analysis, and computational geometry, Modelling and Simulation, 0202 electrical engineering, electronic engineering, information engineering, Best approximation, Discrete approximation, best approximation |
| Popis: | Let \(T\) be a convex subset of \(R^m\). A function \(k'\) on \(T\) is said to be quasi-convex if \[ {\;k'(\lambda s+(1-\lambda t)\leq \max(k'(s),k'(t))} \] for all \({s,t\in T}\) and all \({\lambda\in [0,1]\;}\). Let \(S\) be a finite subset of \(R^m\), \({card(S)=n}\). A function \(k\) on \(S\) is said to be quasi-convex if there exists a quasi-convex function \(k'\) on the convex hull of \(S\) such that \({k=k'}\) on \(S\). Given a real function \(f\) on \(S\), the problem is to find a best quasi-convex approximation \(g\) to \(f\) in the uniform norm. Denote by \(K_f\) the set of all quasi-convex functions \(k\) on \(S\) such that \({k\leq f}\) on \(S\). The function \[ \hat f(s)=\sup\{k(s):\;k\in K_f\}, \;\;s\in S, \] is the greatest quasi-convex minorant of \(f\). The best quasi-convex approximation to \(f\) is obtained as \({q=\hat f+\Delta}\), where \({\Delta=(1/2)\| f-\hat f\| }\). An algorithm for computing this best approximation is developed and its complexity is analyzed. In the case \({m=2}\) the worst case complexity of the algorithm is \({O(n(\log n \log r+r))}\), where \(r\) is the cardinality of the set \({\{f(s):\;s\in S\}}\). Some observations that will speed up the algorithm in the average case are given. |
| Druh dokumentu: | Article |
| Popis súboru: | application/xml |
| Jazyk: | English |
| ISSN: | 0898-1221 |
| DOI: | 10.1016/j.camwa.2004.06.023 |
| Prístupová URL adresa: | https://core.ac.uk/display/82398349 https://www.sciencedirect.com/science/article/pii/S0898122104839841 |
| Rights: | Elsevier Non-Commercial |
| Prístupové číslo: | edsair.doi.dedup.....69f2f0b5f3654adbddb465cb79c02e5c |
| Databáza: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstrakt: | Let \(T\) be a convex subset of \(R^m\). A function \(k'\) on \(T\) is said to be quasi-convex if \[ {\;k'(\lambda s+(1-\lambda t)\leq \max(k'(s),k'(t))} \] for all \({s,t\in T}\) and all \({\lambda\in [0,1]\;}\). Let \(S\) be a finite subset of \(R^m\), \({card(S)=n}\). A function \(k\) on \(S\) is said to be quasi-convex if there exists a quasi-convex function \(k'\) on the convex hull of \(S\) such that \({k=k'}\) on \(S\). Given a real function \(f\) on \(S\), the problem is to find a best quasi-convex approximation \(g\) to \(f\) in the uniform norm. Denote by \(K_f\) the set of all quasi-convex functions \(k\) on \(S\) such that \({k\leq f}\) on \(S\). The function \[ \hat f(s)=\sup\{k(s):\;k\in K_f\}, \;\;s\in S, \] is the greatest quasi-convex minorant of \(f\). The best quasi-convex approximation to \(f\) is obtained as \({q=\hat f+\Delta}\), where \({\Delta=(1/2)\| f-\hat f\| }\). An algorithm for computing this best approximation is developed and its complexity is analyzed. In the case \({m=2}\) the worst case complexity of the algorithm is \({O(n(\log n \log r+r))}\), where \(r\) is the cardinality of the set \({\{f(s):\;s\in S\}}\). Some observations that will speed up the algorithm in the average case are given. |
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| ISSN: | 08981221 |
| DOI: | 10.1016/j.camwa.2004.06.023 |