High order accelerating method for quadratic sequences

• Rodrigo Castro ^[1] ; Oscar A. Restrepo ^[2] ; Luis Echeverri ^[2]
  1. [1] Universidad de Valparaíso

     Universidad de Valparaíso

     Valparaíso, Chile

     
  2. [2] Universidad de Antioquia

     Universidad de Antioquia

     Colombia

     
• Localización: SeMA Journal: Boletín de la Sociedad Española de Matemática Aplicada, ISSN-e 2254-3902, ISSN 2254-3902, Vol. 81, Nº. 2, 2024, págs. 283-290
• Idioma: inglés
• DOI: 10.1007/s40324-023-00327-3
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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 = G (pn), 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 4 √ 5 ≈ 1. 4953. Numerical examples with arbitrary precision are provided where especial cases are considered.