Frontiers in Mathematical Sciences
7th Conference University of Isfahan  January 13, 2020 
Title:
Numerical algorithms with automatic result verification:
Recent advances in Chebyshev expansions and matrix functions
Speaker:
Behnam Hashemi, Shiraz University of Technology
Date, Time, and Venue: Friday, January 3  09:1510:00  Hall 1
Abstract:
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 computerassisted 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).
