On Berinde's method for comparing iterative processes

TL;DR

This paper examines Berinde's method for comparing the convergence speed of iterative processes, proving its effectiveness and discussing its influence in the literature.

Contribution

It provides a proof that Berinde's comparison method reliably indicates which iterative process converges faster.

Findings

Berinde's method accurately compares convergence speeds

The paper confirms the method's validity through proof

It discusses the method's impact on related research

Abstract

In the literature there are several methods for comparing two convergent iterative processes for the same problem. In this note we have in view mostly the one introduced by Berinde in [Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators, Fixed Point Theory Appl. 2004, no. 2, 97--105] because it seems to be very successful. In fact, if IP1 and IP2 are two iterative processes converging to the same element, then IP1 is faster than IP2 in the sense of Berinde. The aim of this note is to prove this almost obvious assertion and to discuss briefly several papers that cite the mentioned Berinde's paper and use his method for comparing iterative processes.