Computing zeros of analytic mappings: A logarithmic residue approach
type: Item
instance of: scholarly article
title: Computing zeros of analytic mappings: A logarithmic residue approach
zbMATH Open document ID: 0916.65053
DOI: 10.1007/BF02510261
author: Ronald Cools
author: Peter Kravanja
author: Ann Haegemans
published in: BIT
publication date: 1999-07-06T00:00:00Z
review text: Let \(D\) be a polydisk in \(\mathbb{C}^n\) and \(f:\overline D\to \mathbb{C}^n\) a mapping that is analytic in \(\overline D\) and has no zero on the boundary of \(D\). Then \(f\) has only a finite number of zeros in \(D\) and these zeros are all isolated. Consider the problem of computing these zeros. A multidimensional generalization of the classical logarithmic residue formula from the theory of functions of one complex variable is obtained by means of information about the location of these zeros. This integral formula involves the integral of a differential form, which is transformed into a sum of \(n\) Riemann integrals of dimension \(2n-1\). The authors show how the zeros and their multiplicities can be computed from these integrals by solving a generalized eigenvalue problem that has Hankel structure, and \(n\) Vandermonde systems. Numerical examples are included.
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 65E05
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 1228556
zbMATH Keywords: zeros of analytic mappings
zbMATH Keywords: logarithmic residue integrals
zbMATH Keywords: Hankel matrix techniques
zbMATH Keywords: numerical examples
Wikidata QID: Q57778989
reviewed by: Nai-Yang Deng
describes a project that uses: CLAPACK
describes a project that uses: RFSFNS
describes a project that uses: DCUHRE
describes a project that uses: LAPACK
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1228556