Verifying Topological Indices for Higher-Order Rank Deficiencies

It has been known how to use computational fixed point theorems to verify existence and uniqueness of a true solution to a nonlinear system of equations within a small region about an approximate solution. This can be done in O n operations, where n is the number of equations and unknowns. However, these standard techniques are only valid if the Jacobi matrix for the system is nonsingular at the solution. In previous work and a dissertation (of Dian), we have shown, both theoretically and practically, that existence and multiplicity can be verified in a complex setting, and in the real setting for odd multiplicity, when the rank defect of the Jacobi matrix at an isolated solution is 1. Here, after a brief introduction, we discuss the case of higher rank defect. 1 Background Given a system of nonlinear equations, numerical methods can typically produce an approximation x̌ to a solution x∗. It is then sometimes desirable to compute bounds x = (x1,x2, . . . ,xn) = ([x1, x1], [x2, x2], . . . , [xn, xn], such that x̌ ∈ x, and such that a computational fixed point theorem can verify that there is a true solution of the nonlinear system within x. Specifically, we examine the problem Given F : x→ Rn and x ∈ IRn, rigorously verify: • there exists a unique x∗ ∈ x such that F (x∗) = 0. (1)