Existence verification for singular and nonsmooth zeros of real nonlinear systems

Abstract:

Traditional computational fixed point theorems, such as the Kantorovich theorem (made rigorous with directed roundings), Krawczyk’s meth- od, or interval Newton methods use a computer’s floating-point hardware computations to mathematically prove existence and uniqueness of a solution to a nonlinear system of equations within a given region of $n$-space. Such computations require the Jacobi matrix of the system to be nonsingular in a neighborhood of a solution. However, in previous work we showed how we could mathematically verify existence of singular solutions in a small region of complex $n$-space containing an approximate real solution. We verified existence of such singular solutions by verifying that the topological degree of a small region is nonzero; a nonzero topological degree implies existence of a solution in the interior of the region. Here, we show that, when the actual topological degree in complex space is odd and the rank defect of the Jacobi matrix is one, the topological degree of a small region containing the singular solution can be verified to be plus or minus one in real space. The algorithm for verification in real space is significantly simpler and more efficient. We demonstrate this efficiency with numerical experiments. Since our verification procedure uses only values on the surfaces of a bounding box that contains the solution, the method can also be applied to cases where the system is nonsmooth at the solution.