SOME ACCELERATIONS OF THE CONVERGENCE OF CERTAIN CLASS OF SEQUENCES

Abstract

The sequence \(\{U_{n+1}/U_n\}\) of the ratios of consecutive numbers \(U_n\), \(n=0,1,2,\ldots\) defined by \(aU_{n+1}+bU_n+cU_{n-1}=0\) with \(U_0=0\), \(U_1=1\) converges to the root \(\lambda_1\) of \(f(x)=ax^2+bx+c=0\), supposing \(|\lambda_1|>|\lambda_2|\). 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.