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## V. Conclusion 

In this paper, three new methods for the simultaneous approximation of all simple zeros of polynomials by utilizing the trapezoidal Newton’s method and the midpoint Newton’s method were proposed. It was proven that each method was of third order. By simultaneously approximating all simple zeros of polynomials and by comparing numerical experiments with various methods that are of third order, we obtained that the results of Method 1 and Method 2 are similar with that of previous methods. But we found out that the result of Method 3 are superior than that of any other methods. All methods we constructed in this paper are new and creative. It seems that these methods can be applied to various fields, and the study on the applications of Method 3 is now in progress. 

## References 

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[2] E. Durand, Solution num´ eriques des ´ equations alg´ ebraiques, Tom. I: ´ Equations du Type F ( x ) = 0, Racines d’un Polynˆ ome, Masson, Paris, 1960. 

[3] P. Henrici, Applied and Computational Complex Analysis, vol. 1 , John Wiley and Sons Inc., New York, 1974. 

[4] I. O. Kerner, Ein Gesamtschrittverfahren zur Berechnung der Nullstellen von Polynomen , Numer. Math. 8 (1966), 290-294. 

[5] M. S. Petkovi´ c, D. Herceg, I. Petkovi´ c, On a simultaneous method of Newton-Weierstrass’ type for finding all zeros for a polynomial , Appl. Math. Comput. 215 (2009), 2456-2463. 

[6] M. S. Petkovi´ c, L. D. Petkovi´ c, On a cubically convergent derivative free root finding method , Int. J. Comput. Math. 84 (2007), 505-513. 

