The self-validated method for polynomial zeros of high efficiency
type: Item
instance of: scholarly article
title: The self-validated method for polynomial zeros of high efficiency
zbMATH Open document ID: 1204.65047
author: M. S. Petković
published in: Journal of Computational and Applied Mathematics
publication date: 2009-11-06T00:00:00Z
review text: An improved iterative method of Newton type for the simultaneous approximation of all complex zeros of a polynomial having only single roots is proposed. If \(P(z)=\prod_{j=1}^{n}(z-\zeta_{j})\) and \(z_{1},\dots,z_{n}\) are distinct approximations to the zeros, the method reads \[ \widehat{z}_{i}=z_{i}-\frac{1}{\frac{1}{u(z_{i})}-\sum^{n}_{{j=1 j\neq i}}\frac{1}{z_{i}-z_{j}} }\qquad (i\in \mathbf{I}_{n}). \] Here \(\widehat{z}_{i}\) is a new approximation to the zero \(\zeta_{i}\) and \[ u(z)=(\sum_{j=1}^{n}\frac{1}{z-\zeta_{j}})^{-1}. \] This leads to the 4th order method \[ \widehat{z}_{i}=z_{i}-\frac{1}{\frac{1}{u(z_{i})}-\sum^{n}_{{j=1 j\neq i}}\frac{1}{z_{i}-z_{j}+u(z_{j})}} \qquad (i\in \mathbf{I}_{n}). \] If we start instead of \(u(z)\) from Ostrowski's correction \[ \psi(z)=u(z)\frac{P(z-u(z))-P(z)}{2} \] we get the 6th order iterative method: \[ \widehat{z}_{i}=z_{i}-\frac{1}{\frac{1}{u(z_{i})}-\sum^{n}_{{j=1 j\neq i}}\frac{1}{z_{i}-z_{j}+\psi(z_{j})}} \qquad (i\in \mathbf{I}_{n}). \] As before \(\mathbf{I}_{n}=\{1,2,\dots,n\}.\)
Mathematics Subject Classification ID: 65H04
Mathematics Subject Classification ID: 65G20
zbMATH DE Number: 5627012
zbMATH Keywords: zeros of polynomials
zbMATH Keywords: iterative methods
zbMATH Keywords: Ostrowski's corrections
zbMATH Keywords: convergence
zbMATH Keywords: Newton method
zbMATH Keywords: self-validated method
zbMATH Keywords: inclusion methods
zbMATH Keywords: circular interval arithmetic
zbMATH Keywords: acceleration of convergence
zbMATH Keywords: computational efficiency
zbMATH Keywords: complex zeros
describes a project that uses: gmp
describes a project that uses: MPFR
describes a project that uses: Algorithm 786
MaRDI profile type: Publication
full work available at URL: j.cam.2009.09.016
OpenAlex ID: W1975241546
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DOI: 10.1016/J.CAM.2009.09.016
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description: scientific article; zbMATH DE number 5627012