High Order Accelerating Method For Quadratic Sequences

Aitken’s method is usually used for accelerating sequences with linear convergence. However, with sequences of order 2 it is not clear that this method accelerate them. In this paper, we are going to suppose that {pn}n=0∞ is a sequence on R of order 2 and we want to accelerate it. Here, we prove that Aiken’s method decelerate such sequences. Therefore, we propose a new method for accelerating such a kind of sequences. As a corollary, we show that for quadratically convergent sequences {pn}n=0∞, produced by a function G i.e., pn+1=Gpn, this acceleration allows us to find methods of order of convergence 5. In particular, if the sequence is produced by the Newton’s Method, we obtain a method with convergence order 5 and efficiency index 54≈1.4953. Numerical examples with arbitrary precision are provided where especial cases are considered.