Abstract
In this paper, we provide three types of general convergence theorems for Picard iteration in n-dimensional vector spaces over a valued field. These theorems can be used as tools to study the convergence of some particular Picard-type iterative methods. As an application, we present a new semilocal convergence theorem for the one-dimensional Newton method for approximating all the zeros of a polynomial simultaneously. This result improves in several directions the previous one given by Batra (BIT Numer Math 42:467–476, 2002).
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This research is supported by Grant FP17-FMI-008 of University of Plovdiv Paisii Hilendarski.
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Proinov, P.D. Unified convergence analysis for Picard iteration in n-dimensional vector spaces. Calcolo 55, 6 (2018). https://doi.org/10.1007/s10092-018-0251-x
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DOI: https://doi.org/10.1007/s10092-018-0251-x
Keywords
- Picard iteration
- Successive approximations
- Local convergence
- Semilocal convergence
- Error estimates
- Newton method