, 55:6 | Cite as

Unified convergence analysis for Picard iteration in n-dimensional vector spaces

  • Petko D. Proinov


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).


Picard iteration Successive approximations Local convergence Semilocal convergence Error estimates Newton method 

Mathematics Subject Classification

65J15 47J25 47H10 54H25 65H05 



This research is supported by Grant FP17-FMI-008 of University of Plovdiv Paisii Hilendarski.


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Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and InformaticsUniversity of Plovdiv Paisii HilendarskiPlovdivBulgaria

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