Applications of Chebyshev Polynomials in Numerical Computation

Abstract

This study is concerned with Chebyshev polynomials and their applications in numerical computation. The basic properties of the first kind Chebyshev polynomials in the interval [-1, 1], are used extensively. Three different applications of the first kind Chebyshev polynomials are studied. First, a sufficient condition for convergence of Chebyshev semi-iterative methods applied to the numerical solution of algebraic linear systems is proved. The convergence condition depends on the bounds on the eigenvalues of a square matrix. Second, the problem of approximating a given function by Chebyshev polynomials is considered. The approximating polynomials are used to predict the value of the function at Chebyshev zeros “roots”. Third, the Gauss - Chebyshev quadrature method for the numerical integration of a given function over a finite range is applied. It consists essentially of expanding the integral in a series of Chebyshev polynomials.