Some efficient methods for enclosing simple zeros of nonlinear equations
type: Item
instance of: scholarly article
title: Some efficient methods for enclosing simple zeros of nonlinear equations
zbMATH Open document ID: 0756.65070
DOI: 10.1007/BF01994885
author: Florian A. Potra
author: Gotz Alefeld
published in: BIT
publication date: 1993-01-16T00:00:00Z
review text: Three methods for solving univariate equations are presented. The methods are derivative free and render nested sequences of intervals converging to a zero of the equation. The common feature of the methods are so- called double length secant steps, which aim to improve enclosing intervals that have been obtained by preceding bracketing phases. The interval widths of the first algorithm converge \(Q\)-quadratically to zero. Two function evaluations each iteration are needed finally (that is, from a certain stage of the computation). The second algorithm uses quadratic interpolation polynomials, converges with \(Q\)-order 4, and needs three function evaluations finally each iteration. The third algorithm arises from the second one by replacing a secant step with a simple bisection. In this case, convergence is of \(R\)-order \((3+\sqrt {13})/2\), and three function evaluations each step are needed.
reviewed by: H. Ratschek
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 65G30
zbMATH DE Number: 89627
zbMATH Keywords: interval arithmetic
zbMATH Keywords: enclosing intervals
zbMATH Keywords: numerical experiments
zbMATH Keywords: iteration method
zbMATH Keywords: high order of convergence
zbMATH Keywords: double length secant steps
describes a project that uses: BRENT
MaRDI profile type: Publication
cites work: Q3345690
cites work: On Two Higher Order Enclosing Methods of J. W. Schmidt
cites work: A new class of interval methods with higher order of convergence
cites work: Q4071019
cites work: Q5595535
cites work: Methods without secant steps for finding a bracketed root
cites work: Q4060241
cites work: On Q-order and R-order of convergence
cites work: Eingrenzung von Lösungen nichtlinearer Gleichungen durch Verfahren mit höherer Konvergenzgeschwindigkeit. (Inclusion of solutions of nonlinear equations by methods with higher convergence rate).
full work available at URL: https://doi.org/10.1007/bf01994885
OpenAlex ID: W2039120927
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description: scientific article; zbMATH DE number 89627