Efficient Computation of Chebyshev Polynomials in Computer Algebra

In this article, we give an overview about the efficiency of theabove methods in the general purpose computer algebra systems Axiom, Macsyma, Maple, Mathematica, MuPAD and REDUCE. Primarily we study the implementation of the Chebyshev polynomials of the first kind as an example case. First, we consider the builtin implementations of the Chebyshev polynomials in these systems. Next we study the classical algorithms beginning with the slow ones, and leading to the efficient ones. Finally, we finish with an algorithm based on a divide and conquer approach which has a remarkable complexity. In particular, we will show that