The number of centered forms for a polynomial
type: Item
instance of: scholarly article
title: The number of centered forms for a polynomial
zbMATH Open document ID: 0661.65046
DOI: 10.1007/BF01954905
author: J. Rokne
author: Paul Bao
published in: BIT
publication date: 1988-01-01T00:00:00Z
review text: If f: \(R^ n\to R\) and \(c\in R^ n\) are given, every representation of f as expression \(f(x)=f(c)+g(x,c)\cdot (x-c)\) is called a centered form of f at c. In this definition, g: \(R^{2n}\to R\) is called the kernel of the centered form and the dot denotes the standard inner product in \(R^ n\). The concept of centered forms is due to \textit{R. E. Moore} [Interval analysis (1966; Zbl 0176.133)]. The interval extension of a centered form over an n-dimensional interval X is denoted by \(f(X)=f(c)+g(X,c)\cdot (X-c).\) Interval extensions of centered forms are an excellent means to approximate the range of f over X from without, i.e., f(X)\(\supseteq \{f(x):\) \(x\in X\}\). The kernel g underlying the centered form idea is not uniquely defined and, furthermore, different kernels will, in general, provide different inclusions f(X) of the range. The search for kernels which lead to optimum inclusions of the range is an old and interesting problem. In connection which such considerations the authors develop formulas for the number of different centered forms of a given multi-variate polynomial at a point c. Methodically, the number of different centered forms of such a polynomial is reduced to the number of different arrangements of multi-variate variables. Conversely, an algorithm is presented that generates a centered form at c if an arrangement of the variables is given. The algorithm generates different centered forms when the arrangements are different. Hence it suffices to develop recursive formulas for the number of different arrangements.
Mathematics Subject Classification ID: 65G30
zbMATH DE Number: 4080769
zbMATH Keywords: centered form
zbMATH Keywords: kernel
zbMATH Keywords: interval extension
zbMATH Keywords: optimum inclusions
zbMATH Keywords: algorithm
zbMATH Keywords: recursive formulas
reviewed by: H. Ratschek
MaRDI profile type: Publication
cites work: Q5577946
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full work available at URL: https://doi.org/10.1007/bf01954905
OpenAlex ID: W2001164886
description: scientific article; zbMATH DE number 4080769