Generalized multistep Steffensen iterative method. Solving the model of a photomultiplier device

Abstract

It is well known that the Steffensen-type methods approximate the derivative appearing in Newton's scheme by means of the first-order divided difference operator. The generalized multistep Steffensen iterative method consists of composing the method with itself m times. Specifically, the divided difference is held constant for every m steps before it is updated. In this work, we introduce a modification to this method, in order to accelerate the convergence order. In the proposed, scheme we compute the divided differences in first and second step and use the divided difference from the second step in the following m−1 steps. We perform an exhaustive study of the computational efficiency of this scheme and also introduce memory to this method to speed up convergence without performing new functional evaluations. Finally, some numerical examples are studied to verify the usefulness of these algorithms.

Keywords:

• Iterative processes
• convergence order
• computational efficiency index
• divided differences
• derivative-free iterative processes

AMS 2020 SUBJECT CLASSIFICATIONS:

• 45G10
• 47H17
• 65J15

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was partially supported by Ministerio de Economía y Competitividad [grant number PGC2018-095896-B-C22].