Application of the Chebyshev Polynomials in Root-Finding Problem

Koay, Kenneth Ting Foong (2022) Application of the Chebyshev Polynomials in Root-Finding Problem. Final Year Project (Bachelor), Tunku Abdul Rahman University College.

Text Kenneth Koay Ting Foong_FullText.pdf Restricted to Registered users only Download (3MB)

Abstract

This research is about obtaining all possible real roots of a continuous function ()fx in the target interval [ , ] x a b  all at once. We can approximate the possible real roots on the interval x[a,b] by just three main steps. The first step is to obtain the coefficients of the Chebyshev interpolation via iteration. We use linear transformation to obtain the shifted Chebyshev polynomials of the first kind to form the complete truncated Chebyshev polynomial of the first kind. The second step is to use the coefficients of the interpolated function in the formation of a nn Chebyshev-Frobenius companion matrix. The third step is to calculate the eigenvalues  of the nn Chebyshev-Frobenius companion matrix. The real eigenvalues  will be mapped to the interval [ , ] ab using a linear transformation method. The results of the real eigenvalues  after linear transformation are the possible approximated roots in the interval [ , ] x a b  . We will also compare the roots of the function ()fx that we obtain using the proposed Chebyshev Root-Finding algorithm and classical iteration methods such as the Newton-Raphson method. For evaluating the sum of Chebyshev polynomials, we can use the Clenshaw algorithm instead of usually substituting the value of x . The idea of approximating all possible real roots in an interval all at once can link different fields of mathematics such as Numerical Method and Linear Algebra.