10 February 2023
- 10 February 2023
Range functions are an important tool for interval computations, and they can be employed for the problem of root isolation. In this talk, I will first introduce two new classes of range functions for real functions. They are based on the remainder form by Cornelius and Lohner (1984) and provide different improvements for the remainder part of this form. On the one hand, I will show how centered Taylor expansions can be used to derive a generalization of the classical Taylor form with higher than quadratic convergence. On the other hand, I will discuss a recursive interpolation procedure, in particular based on quadratic Lagrange interpolation, leading to recursive Lagrange forms with cubic and quartic convergence. These forms can be used 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, a comparison of the performance of these new range functions against the standard Taylor form will be given. Specifically, EVAL can exploit features of the recursive Lagrange forms which are not found in range functions based on Taylor expansion.
Experimentally, this yields at least a twofold speedup in EVAL.
Room B1.17