A Practical Method of Numerical Calculation of the Mapping Degree

Sunao MURASHIGE  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E89-A   No.6   pp.1813-1819
Publication Date: 2006/06/01
Online ISSN: 1745-1337
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Numerical Analysis and Optimization
Keyword: 
mapping degree,  intersection number,  numerical calculation,  interval arithmetic,  

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Summary: 
This paper proposes a simple and efficient method to numerically obtain the mapping degree deg(f, 0, B) of a C1 map f : Rn → Rn at a regular value 0 relative to a bounded open subset B ⊂ Rn. For practical application, this method adopts Aberth's algorithm which does not require computation of derivatives and determinants, and reduces the computational cost with two additional procedures, namely preconditioning using the coordinate transformation and pruning using Krawczyk's method. Numerical examples show that the proposed method gives the mapping degree with 2n+1 operations using interval arithmetic.