Abstract: In recent papers, Babolian & Delves^[2] and Belward^[3] described a Chebyshev series method for the solution of first kind integral equations. The expansion coefficients of the solution are determined as the solution of a mathematical programming problem.The method involves two regularization parameters, C_f and r, but values assigned to these parameters are heuristic in nature. Essah & Delves^[7] described an algorithm for setting these parameters automatically, but it has some difficulties. In this paper we describe three iterative algorithms for computing these parameters for singular and non-singular first kind integral equations. We give also error estimates which are cheap to compute. Finally, we give a number of numerical examples showing that these algorithms work well in practice.

Key words: Fredholm integral equations, Galerkin method, Regularization parameters, Error estimation, Ill-posed problems, Product of Chebyshev series