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A class of efficient high‐order iterative methods with memory for nonlinear equations and their dynamics
Author(s) -
Howk Cory L.,
Hueso José L.,
Martínez Eulalia,
Teruel Carles
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4821
Subject(s) - mathematics , convergence (economics) , nonlinear system , computation , class (philosophy) , iterative method , simple (philosophy) , function (biology) , set (abstract data type) , mathematical optimization , focus (optics) , algorithm , computer science , artificial intelligence , philosophy , physics , epistemology , quantum mechanics , evolutionary biology , optics , economics , biology , programming language , economic growth
In this paper we obtain some theoretical results about iterative methods with memory for nonlinear equations. The class of algorithms we consider focus on incorporating memory without increasing the computational cost of the algorithm. This class uses for the predictor step of each iteration a quantity that has already been calculated in the previous iteration, typically the quantity governing the slope from the previous corrector step. In this way we do not introduce any extra computation, and more importantly, we avoid new function evaluations, allowing us to obtain high‐order iterative methods in a simple way. A specific class of methods of this type is introduced, and we prove the convergence order is 2 n +2 n −2 with n +1 function evaluations. An exhaustive efficiency study is performed to show the competitiveness of these methods. Finally, we test some specific examples and explore the effect that this predictor may have on the convergence set by setting a dynamical study.