Frontiers in Mathematical Sciences
University of Isfahan - January 1-3, 2020
Numerical algorithms with automatic result verification: Recent advances in Chebyshev expansions and matrix functionsSpeaker:
Behnam Hashemi, Shiraz University of Technology
Date, Time, and Venue: Friday, January 3 | 09:15-10:00 | Hall 1Abstract:
Automatic result verification is a process that enables computers to obtain rigorous inclusions for the exact solution to a mathematical problem. While it uses floating point arithmetic to be fast, the results are guaranteed to be mathematically correct, even though rounding errors are almost everywhere in floating point arithmetic. Such algorithms have been successfully used in different applications, e.g., in computer-assisted proofs of important conjectures. We start with fundamentals of machine interval arithmetic involving directed roundings as defined in IEEE standard for floating point arithmetic. We then turn our attention to three specific problems and review a wide range of numerical algorithms with automatic result verification to tackle each problem. Specifically, we consider evaluation of Chebyshev expansions and computing matrix square roots and the matrix exponential. We close this talk with a quick review of challenges and open problems in this area. Parts of the work on Chebyshev expansions are done in collaboration with Jared Aurentz (Universidad Autonoma de Madrid, Spain), while results on the matrix square root and the matrix exponential are joint work with Andreas Frommer (University of Wuppertal, Germany).