Work

Optimal Approximation of the 1/ x Function using Chebyshev Polynomials and Magic Constants

Year: 2024

Type: article

Abstract: In this article we analyze low-cost accurate approximation of the function \(1/x\) using Chebyshev polynomials of the first kind and minimizing number of elementary operations in computer codes (in particular, by using the so-called magic constants). It is shown that Newton-Raphson iterative method is not optimal and a new approach is proposed. We prove that optimal Chebyshev polynomials can be factorized in terms of Chebyshev polynomials of lower order which leads to new optimal iteration schemes. We also construct a family of new algorithms by dividing the considered interval into sub-intervals where different magic constants and multiplicative factors are used (in order to increase the accuracy). Theoretical considerations and proofs are completed with numerical tests on three types of microcontroller processors.

Source: ACM Transactions on Mathematical Software

DOI: 10.1145/3708472

Authors: Cezary J. Walczyk, Leonid V. Moroz, Volodymyr Samotyy, Jan L. Cieśliński

Institutions: University of Białystok, Warsaw University of Technology, Cracow University of Technology

Language: English

Topic: Numerical Methods and Algorithms

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

OpenAlex ID: W4405483755

FWCI: 0.401

Cites: 30

Cited by: 2

Related to: 10