General convergence theorems for iterative processes and applications to the Weierstrass root-finding method

TL;DR

This paper establishes general convergence theorems for iterative processes in cone metric spaces and applies these results to analyze the Weierstrass root-finding method for polynomials, improving existing convergence results.

Contribution

It introduces new convergence theorems for Picard iteration in cone metric spaces and applies them to enhance understanding of the Weierstrass method's convergence.

Findings

Convergence theorems for Picard iteration in cone metric spaces

Detailed convergence analysis of the Weierstrass method

Generalizations of existing convergence results

Abstract

In this paper, we prove some general convergence theorems for the Picard iteration in cone metric spaces over a solid vector space. As an application, we provide a detailed convergence analysis of the Weierstrass iterative method for computing all zeros of a polynomial simultaneously. These results improve and generalize existing ones in the literature.