Optimal interval enclosures for fractionally-linear functions, and their application to intelligent control
type: Item
instance of: scholarly article
title: Optimal interval enclosures for fractionally-linear functions, and their application to intelligent control
zbMATH Open document ID: 0862.65013
DOI: 10.1007/BF02391700
author: Robert N. Lea
author: Raul Trejo
author: Vladik Kreinovich
published in: Reliable Computing
publication date: 1997-05-25T00:00:00Z
review text: The object of the paper is the demonstration that the basic problem of interval computations is feasible for fractionally linear functions. In detail, let a fractionally linear function \(f\) be given by the tuple \((n;a_0,a_1,\dots,a_n;\) \(b_0,b_1,\dots,b_n)\) with real numbers \(a_i\), \(b_i\) \((i=0,\dots,n)\), and thus let its value for \(x=(x_1,\dots,x_n)\) be defined by \(y=f(x)=(a_0+a_1x_1+\cdots+a_nx_n)/(b_0+b_1x_1+\cdots+b_nx_n)\). A basic interval computation problem for \(f\) is denoted by the tuple \((f,X_1,\dots,X_n)\), where \(X_i=[x^-_i,x^+_i]\) are intervals, and this problem is non-degenerate if \(0\not\in b_0+b_1\cdot X_1+\cdots+b_n\cdot X_n\), where \(b_i\cdot X_i=[\min(b_ix^-_i,b_ix_i^+),\max(b_ix^-_i,b_ix_i^+)]\). Furthermore, if an algorithm computes the range \(Y=f(X_1,\dots,X_n)\) it is said to compute the optimal enclosure for the problem \((f,X_1,\dots,X_n)\). The main result of the present paper is the proof of the existence of an algorithm that computes the optimal enclosure for an arbitrary non-degenerate fractionally linear problem in quadratic time. Besides, in great detail, applications are discussed for problems of intelligent control. The bibliography contains 35 entries.
Mathematics Subject Classification ID: 65D20
Mathematics Subject Classification ID: 65G30
Mathematics Subject Classification ID: 41A20
Mathematics Subject Classification ID: 26C15
zbMATH DE Number: 915298
zbMATH Keywords: interval arithmetic
zbMATH Keywords: interval computations
zbMATH Keywords: fractionally linear functions
zbMATH Keywords: algorithm
zbMATH Keywords: intelligent control
reviewed by: M. Kracht
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 915298