Existence Verification for Singular Zeros of Complex Nonlinear Systems

Computational fixed point theorems can be used to automatically verify existence and uniqueness of a solution to a nonlinear system of n equations in n variables ranging within a given region of n-space. Such computations succeed, however, only when the Jacobi matrix is nonsingular everywhere in this region. However, in problems such as bifurcation problems or surface intersection problems, the Jacobi matrix can be singular, or nearly so, at the solution. For n real variables, when the Jacobi matrix is singular, tiny perturbations of the problem can result in problems either with no solution in the region, or with more than one; thus no general computational technique can prove existence and uniqueness. However, for systems of n complex variables, the multiplicity of such a solution can be verified. That is the subject of this paper. Such verification is possible by computing the topological degree, but such computations heretofore have required a global search on the (n− 1)-dimensional boundary of an n-dimensional region. Here it is observed that preconditioning leads to a system of equations whose topological degree can be computed with a much lower-dimensional search. Formulas are given for this computation, and the special case of rank-defect one is studied, both theoretically and empirically. Verification is possible for certain subcases of the real case. That will be the subject of a companion paper.