Calcolo

, 55:6

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

• Petko D. Proinov
Article

## 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).

## Keywords

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

## Mathematics Subject Classification

65J15 47J25 47H10 54H25 65H05

## Notes

### Acknowledgements

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

## References

1. 1.
Banach, S.: Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales. Fund. Math. 3, 133–181 (1922)
2. 2.
Batra, P.: Simultaneous point estimates for Newton’s method. BIT Numer. Math. 42, 467–476 (2002)
3. 3.
Berinde, V.: Iterative Approximation of Fixed Points. Vol. 1912 of Lecture Notes in Mathematics. Springer, Berlin (2007)Google Scholar
4. 4.
Cholakov, S.I., Vasileva, M.T.: A convergence analysis of a fourth-order method for computing all zeros of a polynomial simultaneously. J. Comput. Appl. Math. 321, 270–283 (2017)
5. 5.
Engler, A.J., Prestel, A.: Valued Fields, Springer Monographs in Mathematics. Springer, Berlin (2005)
6. 6.
Ivanov, S.I.: A unified semilocal convergence analysis of a family of iterative algorithms for computing all zeros of a polynomial simultaneously. Numer. Algorithms 75, 1193–1204 (2017)
7. 7.
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)
8. 8.
Proinov, P.D.: A generalization of the Banach contraction principle with high order of convergence of successive approximations. Nonlinear Anal. 67, 2361–2369 (2007)
9. 9.
Proinov, P.D.: New general convergence theory for iterative processes and its applications to Newton–Kantorovich type theorems. J. Complex. 26, 3–42 (2010)
10. 10.
Proinov, P.D.: A unified theory of cone metric spaces and its application to the fixed point theory. Fixed Point Theory Appl. 2013(103), 1–38 (2013)
11. 11.
Proinov, P.D.: General convergence theorems for iterative processes and applications to the Weierstrass root-finding method. J. Complex. 33, 118–144 (2016)
12. 12.
Proinov, P.D.: Relationships between different types of initial conditions for simultaneous root finding methods. Appl. Math. Lett. 52, 102–111 (2016)
13. 13.
Proinov, P.D.: A general semilocal convergence theorem for simultaneous methods for polynomial zeros and its applications to Ehrlich’s and Dochev-Byrnev’s methods. Appl. Math. Comput. 284, 102–114 (2016)
14. 14.
Proinov, P.D.: On the local convergence of Ehrlich method for numerical computation of polynomial zeros. Calcolo 53, 413–426 (2016)
15. 15.
Proinov, P.D., Cholakov, S.I.: Semilocal convergence of Chebyshev-like root-finding method for simultaneous approximation of polynomial zeros. Appl. Math. Comput. 236, 669–682 (2014)
16. 16.
Proinov, P.D., Ivanov, S.I.: On the convergence of Halley’s method for simultaneous computation of polynomial zeros. J. Numer. Math. 23, 379–394 (2015)
17. 17.
Proinov, P.D., Petkova, M.D.: A new semilocal convergence theorem for the Weierstrass method for finding zeros of a polynomial simultaneously. J. Complex. 30, 366–380 (2014)
18. 18.
Proinov, P.D., Vasileva, M.T.: On the convergence of high-order Ehrlich-type iterative methods for approximating all zeros of a polynomial simultaneously. J. Inequal. Appl. 2015(336), 1–25 (2015)
19. 19.
Proinov, P.D., Vasileva, M.T.: On a family of Weierstrass-type root-finding methods with accelerated convergence. Appl. Math. Comput. 273, 957–968 (2016)
20. 20.
Smale, S.: Newton’s method estimates from data at one point. In: Ewing, R.E., Gross, K.I., Martin, C.F. (eds.) The Merging of Disciplines, pp. 185–196. Springer, New York (1986)Google Scholar
21. 21.
Traub, J.F.: Iterative Methods for the Solution of Equations, 2nd edn. Chelsea Publishing Company, New York (1982)
22. 22.
Wang, X.H., Han, D.F.: On dominating sequence method in the point estimate and Smale theorem. Sci. China Ser. A 33, 135–144 (1990)
23. 23.
Werner, W.: On the simultaneous determination of polynomial roots. Lecture Notes Math. 953, 188–202 (1982)
24. 24.
Weierstrass, K.: Neuer Beweis des Satzes, dass jede ganze rationale Function einer Veränderlichen dargestellt werden kann als ein Product aus linearen Functionen derselben Veränderlichen. Sitzungsber. Königl. Preuss. Akad. Wiss. Berlin II, 1085–1101 (1891)

© 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