Application of the Faber polynomials in calculation of eigenvalues
type: Item
instance of: scholarly article
title: Application of the Faber polynomials in calculation of eigenvalues
zbMATH Open document ID: 0802.65046
author: N. A. Farkova
published in: Russian Mathematics
publication date: 1994-10-05T00:00:00Z
review text: Let \(A\) be a complex matrix of order \(n\times n\) with eigenvalues \(\lambda_ 1,\dots,\lambda_ n\); \(K\) be a bounded continuum on the plane \(\overline{\mathbb{C}}\). Let the domain \(D= \overline{\mathbb{C}}\backslash K\) be simply connected and let there be known that \(\lambda_ 1\in D\), \(\lambda_ j\in K\) \((j=2,\dots,n)\). The construction of such a continuum \(K\) for a given matrix \(A\) can be done, in number of cases, by means of the well-known technique of the localization of eigenvalues. If \(K\) is a disk, then, in order to calculate \(\lambda_ 1\) and the corresponding eigenvector \(\bar g_ 1\), \(A\bar g_ 1= \lambda_ 1\bar g_ 1\), one can use the power method. But if \(K\) is a segment, then \(\lambda_ 1\) and \(\bar g_ 1\) can be calculated by the Chebyshev method. In this article we show that if there exists a technique for the calculation of the Faber polynomials for the continuum \(K\), then the mentioned problem can be solved by Faber's integrative method. This method generalizes the classical power method and the Chebyshev method, having higher rate of convergence in certain cases.
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 640170
zbMATH Keywords: localization of eigenvalues
zbMATH Keywords: power method
zbMATH Keywords: Chebyshev method
zbMATH Keywords: Faber polynomials
zbMATH Keywords: Faber's integrative method
zbMATH Keywords: convergence
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 640170