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# Evaluation of Chebyshev polynomials on intervals and application to root finding

1 GAMBLE - Geometric Algorithms and Models Beyond the Linear and Euclidean realm
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
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.
Keywords :
Document type :
Conference papers

https://hal.inria.fr/hal-02405752
Contributor : Guillaume Moroz Connect in order to contact the contributor
Submitted on : Wednesday, December 11, 2019 - 7:02:01 PM
Last modification on : Thursday, March 17, 2022 - 10:08:41 AM
Long-term archiving on: : Thursday, March 12, 2020 - 10:23:45 PM

### Files

macis2019_chebyshev.pdf
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### Identifiers

• HAL Id : hal-02405752, version 1
• ARXIV : 1912.05843

### Citation

Viviane Ledoux, Guillaume Moroz. Evaluation of Chebyshev polynomials on intervals and application to root finding. Mathematical Aspects of Computer and Information Sciences 2019, Nov 2019, Gebze, Turkey. ⟨hal-02405752⟩

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