Novel Range Functions via Taylor Expansions and Recursive Lagrange Interpolation with Application to Real Root Isolation

Range functions are an important tool for interval computations, and they can be employed for the problem of root isolation. In this paper, we first introduce two new classes of range functions for real functions. They are based on the remainder form by Cornelius and Lohner [7] and provide different improvements for the remainder part of this form. On the one hand, we use centered Taylor expansions to derive a generalization of the classical Taylor form with higher than quadratic convergence. On the other hand, we propose a recursive interpolation procedure, in particular based on quadratic Lagrange interpolation, leading to recursive Lagrange forms with cubic and quartic convergence. We then use these forms for isolating the real roots of square-free polynomials with the algorithm Eval, a relatively recent algorithm that has been shown to be effective and practical. Finally, we compare the performance of our new range functions against the standard Taylor form. Range functions are often compared in isolation; in contrast, our holistic comparison is based on their performance in an application. Specifically, Eval can exploit features of our recursive Lagrange forms which are not found in range functions based on Taylor expansion. Experimentally, this yields at least a twofold speedup in Eval.