, Volume 8, Issue 4, pp 267282
First online:
On Existence and Uniqueness Verification for NonSmooth Functions
 R. Baker KearfottAffiliated withDepartment of Mathematics, University of Louisiana at Lafayette
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Given an approximate solution to a nonlinear system of equations at which the Jacobi matrix is nonsingular, and given that the Jacobi matrix is continuous in a region about this approximate solution, a small box can be constructed about the approximate solution in which interval Newton methods can verify existence and uniqueness of an actual solution. Recently, we have shown how to verify existence and uniqueness, up to multiplicity, for solutions at which the Jacobi matrix is singular. We do this by efficient computation of the topological index over a small box containing the approximate solution. Since the topological index is defined and computable when the Jacobi matrix is not even defined at the solution, one may speculate that efficient algorithms can be devised for verification in this case, too. In this note, however, we discuss, through examples, key techniques underlying our simplification of the calculations that cannot necessarily be used when the function is nonsmooth. We also present those parts of the theory that are valid in the nonsmooth case, and suggest when degree computations involving nonsmooth functions may be practical.
As a bonus, the examples lead to additional understanding of previously published work on verification involving the topological degree.
 Title
 On Existence and Uniqueness Verification for NonSmooth Functions
 Journal

Reliable Computing
Volume 8, Issue 4 , pp 267282
 Cover Date
 200208
 DOI
 10.1023/A:1016381031155
 Print ISSN
 13853139
 Online ISSN
 15731340
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Authors

 R. Baker Kearfott ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, 70504, USA