On Existence and Uniqueness Verification for Non-Smooth Functions

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 in underlying our simplification of the calculations that cannot necessarily be used when the function is non-smooth...

• Type: Journal
• Year: 2002
• Where: RC
• Authors: R. Baker Kearfott
• Keywords: Approximate Solution, Jacobi Matrix, RC 2002, Small Box