Existence and Uniqueness Verification of Singular Zeros of
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
and a small box B in n-space is constructed that contains X*,
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.