Newton's method with deflation for isolated singularities of polynomial systems
Theoretical Computer Science
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Finding approximate solutions to systems of n nonlinear equations in n real variables is a much studied problem in numerical analysis. Somewhat more recently, researchers have developed numerical methods to provide mathematically rigorous error bounds on such solutions. (We say that we "verify" existence of the solution within those bounds on the variables.) However, when the Jacobi matrix is singular at the solution, no computational techniques to verify existence can handle the general case. Nonetheless, computational verification that one or more solutions exists within a region in complex space containing the real bounds is possible by computing the topological degree. In a previous paper, we presented theory and algorithms for the simplest case, when the rank-defect of the Jacobian matrix at the solution is 1 and the topological index is 2. Here, we generalize that result to arbitrary topological index $d\ge2$: We present theory, algorithms, and experimental results. We also present a heuristic for determining the degree, obtaining a value that we can subsequently verify with our algorithms. Although execution times are slow compared to corresponding bound verification processes for nonsingular systems, the order with respect to system size is still cubic.