What is it about?
In practical algorithms within approximation theory, one uses floating point arithmetic to evaluate a Chebyshev expansion. But, how accurate is the result of such an evaluation? This paper examines a set of efficient interval-arithmetic algorithms that can be used to compute rigorous error bounds for the result of evaluating finite Chebyshev expansions.
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Why is it important?
Chebyshev expansions are ubiquitous in approximation theory and practice. One can use our algorithms to carry out error analysis for the result of evaluating Chebyshev expansions in floating point arithmetic and the algorithms perform this automatically on a computer! Computing rigorous enclosures for Chebyshev expansions is useful also in computer-assisted proofs, ultra arithmetic, and in Chebyshev models.
Perspectives
The algorithms developed in this paper pave the way toward achieving a long-term goal. The paper explains essential tools for developing a complete software package that obtains rigorous error bounds in the paradigm of numerical computing with functions. More interval arithmetic algorithms are to be developed not only for the purpose of evaluating Chebyshev expansions efficiently and accurately, but also in the bigger context of numerical computing with functions.
Behnam Hashemi
Shiraz University of Technology
Read the Original
This page is a summary of: Enclosing Chebyshev Expansions in Linear Time, ACM Transactions on Mathematical Software, July 2019, ACM (Association for Computing Machinery),
DOI: 10.1145/3319395.
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