On the Asymptotic Behaviour of a Certain Non-linear Difference Equation

Arising from the study of the convergence properties of a rationalapproximation method for determining a zero of the functionf(x)is a certain non-linear difference equation. This equation hasthe form v_n+1 = g_p–1(v_n)/g_p(v_n), Where g_p(v_n) is a polynomialin v_n whose coefficients depend on a parameter p, the orderof the zero of f The asymptotic behaviour of the differenceequation is studied and it is shown that if there is a limitorder of convergence it is always linear for multiple zeros.