Abstract
In approximation theory, it is standard to approximate functions by polynomials expressed in the Chebyshev basis. Evaluating a polynomial f of degree n given in the Chebyshev basis can be done in O(n) arithmetic operations using the Clenshaw algorithm. Unfortunately, the evaluation of f on an interval I using the Clenshaw algorithm with interval arithmetic returns an interval of width exponential in n. We describe a variant of the Clenshaw algorithm based on ball arithmetic that returns an interval of width quadratic in n for an interval of small enough width. As an application, our variant of the Clenshaw algorithm can be used to design an efficient root finding algorithm .
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Ledoux, V., Moroz, G. (2020). Evaluation of Chebyshev Polynomials on Intervals and Application to Root Finding. In: Slamanig, D., Tsigaridas, E., Zafeirakopoulos, Z. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2019. Lecture Notes in Computer Science(), vol 11989. Springer, Cham. https://doi.org/10.1007/978-3-030-43120-4_4
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DOI: https://doi.org/10.1007/978-3-030-43120-4_4
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