Viktor Levandovskyy

Jonathan D. Hauenstein, Viktor Levandovskyy

Smale's alpha-theory certifies that Newton iterations will converge quadratically to a solution of a square system of analytic functions based on the Newton residual and all higher order derivatives at the given point. Shub and Smale presented a bound for the higher order derivatives of a system of polynomial equations based in part on the degrees of the equations. For a given system of polynomial-exponential equations, we consider a related system of polynomial-exponential equations and provide a bound on the higher order derivatives of this related system. This bound yields a complete algorithm for certifying solutions to polynomial-exponential systems, which is implemented in alphaCertified. Examples are presented to demonstrate this certification algorithm.

Albert Heinle, Viktor Levandovskyy, Andreas Nareike

In this paper we will present SDeval, a software project that contains tools for creating and running benchmarks with a focus on problems in computer algebra. It is built on top of the Symbolic Data project, able to translate problems in the database into executable code for various computer algebra systems. The included tools are designed to be very flexible to use and to extend, such that they can be utilized even in contexts of other communities. With the presentation of SDEval, we will also address particularities of benchmarking in the field of computer algebra. Furthermore, with SDEval, we provide a feasible and automatizable way of reproducing benchmarks published in current research works, which appears to be a difficult task in general due to the customizability of the available programs. We will simultaneously present the current developments in the Symbolic Data project.