SOME ACCELERATIONS OF THE CONVERGENCE OF CERTAIN CLASS OF SEQUENCES - Popov, Blagoj S. (Matematichki Bilten, 1992)
The sequence \({U_n+1/U_n}\) of the ratios of consecutive numbers \(U_n\), \(n=0,1,2,…\) defined by \(aU_n+1+bU_n+cU_n-1=0\) with \(U_0=0\), \(U_1=1\) converges to the root \(λ_1\) of \(f(x)=ax^2+bx+c=0\), supposing \(|λ_1|>|λ_2|\). Newton’s method for the equation \(f(x)=0\) with initial approximation 1 produces the subsequence \({U_2n+1/U_2n}\). The Halley’s iteration method for this equation produces the subsequence \({U_3n+1/U_3n}\). Applying the Newton’s modified method and the Schröder’s iteration method we obtain similar subsequences.