Existence and Uniqueness Verification of Singular Zeros of Nonlinear Systems
by R. Baker Kearfott

If the system F(X) = 0 of n nonlinear equations in n unknowns has a solution X* at which the Jacobi matrix of F is nonsingular, and a small box B in n-space is constructed that contains X*, then interval Newton methods can be used to have a computer prove that there exists a unique solution of F(X)=0 within the box. Such interval Newton methods are based on a linear model for the function F, and fail if B contains a point at which the Jacobi matrix of F is singular. However, with a quadratic model of F, computation of the topological degree of F over B can be made practical, even for large systems of equations resulting from discretization of PDE's, and holds promise for providing tools for analysis of bifurcation phenomena.

In this talk, numerical verification with traditional interval Newton methods will be briefly reviewed. Time permitting, the topological degree techniques will be presented. The latter is work in progress, in collaboration with Arnold Neumaier at the University of Vienna.