CertiVed Numerical Root Finding

Root isolation of univariate polynomials is one of the fundamental problems in computational algebra. It aims to find disjoint regions on the real line or complex plane, each containing a single root of a given polynomial, such that the union of the regions comprises all roots. For root solving over the field of complex numbers, numerical methods are the de facto standard. They are known to be reliable, highly efficient, and apply to a broad range of different input types. Unfortunately, most of the existing solvers fail to provide guarantees on the correctness of their output. For those that do, the theoretical superiority is matched by the effort required for an efficient implementation. We strongly feel that there is no need to suffer the pain of theoretically optimal algorithms to develop an efficient library for certified algebraic root finding. As evidence, we present ARCAVOID, a highly generic and fast numerical Las Vegas-algorithm. It uses the well-established Aberth-Ehrlich simultaneous root finding iteration. Certificate on the exactness of the results are provided by rigorous application of interval arithmetics. Our implementation transparently handles the case of bitstream coefficients that can be approximated to any arbitrary precision, but cannot be represented exactly as rational numbers. While the convergence of the Aberth-Ehrlich method is not proven – and, thus, our approach cannot strictly be considered complete – we are not aware of any instance where it fails. Practically speaking, its runtime is output-sensitive in the geometric configuration of the roots, in particular the separation of close roots. In contrast, the influence of degree and complexity of the coefficients backs out, although it remains perceptible. Our algorithm requires no advanced data structures, and the implementation does not feature complicated asymptotically fast algorithms. Yet, we provide extensive benchmarks proving that ARCAVOID can compete with sophisticated state-of-the-art root solvers, whose intricacy is orders of magnitude higher. We emphasize the importance of complex root finding even for applications where it may not be considered in the first place, namely real algebraic geometry. Traditionally used real root solvers cannot deliver information about the global root structure of a polynomial. When exploiting this information, the integration of ARCAVOID into a framework for algebraic plane curve analysis yields significant benefits over existing methods.