Title
On locating clusters of zeros of analytic functions On locating clusters of zeros of analytic functions
Author
Faculty/Department
Faculty of Social Sciences. Communication Sciences
Publication type
article
Publication
Lund ,
Subject
Computer. Automation
Source (journal)
Bit : numerical mathematics. - Lund
Volume/pages
39(1999) :4 , p. 646-682
ISSN
0006-3835
ISI
000084736100004
Carrier
E
Target language
English (eng)
Full text (Publishers DOI)
Abstract
Given an analytic function f and a Jordan curve γ that does not pass through any zero of f, we consider the problem of computing all the zeros of f that lie inside γ, together with their respective multiplicities. Our principal means of obtaining information about the location of these zeros is a certain symmetric bilinear form that can be evaluated via numerical integration along γ. If f has one or several clusters of zeros, then the mapping from the ordinary moments associated with this form to the zeros and their respective multiplicities is very ill-conditioned. We present numerical methods to calculate the centre of a cluster and its weight, i.e., the arithmetic mean of the zeros that form a certain cluster and the total number of zeros in this cluster, respectively. Our approach relies on formal orthogonal polynomials and rational interpolation at roots of unity. Numerical examples illustrate the effectiveness of our techniques.
E-info
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