On Existence and Uniqueness Verification for Non-Smooth Functions

It is known that interval Newton methods can verify existence and uniqueness of solutions of a nonlinear system of equations near points where the Jacobi matrix of the system is not ill-conditioned. 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. Algorithmically, our techniques mimic the non-singular case (both in algorithmic steps and computational complexity), and can be considered as incomplete Gauss–Seidel sweeps. 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 talk, we discuss, through examples, key techniques underlying our simplification of the calculations that cannot necessarily be used when the function is non-smooth. We also suggest when degree computations involving non-smooth functions may be practical. Our examples also shed light on our published work on verification involving the topological degree.