INDIAN JOURNAL OF APPLIED RESEARCH X 397 Volume : 5 | Issue : 4 | April 2015 | ISSN - 2249-555X RESEARCH PAPER On the Convergence of a Fourth-Order Method for Simultaneous Finding Polynomial Zeros Slav I. Cholakov Milena D. Petkova Faculty of Mathematics and Informatics, University of Plovdiv. Plovdiv-4000, Bulgaria Faculty of Mathematics and Informatics, University of Plovdiv. Plovdiv-4000, Bulgaria Mathematics ABSTRACT In 2011, Petković, Rančić and Milošević [5] presented a new fourth-order iterative method for finding all zeros of a polynomial simultaneously. In this paper we establish a new local convergence theorem with error estimates for this method. In particular, an estimate of the radius of the convergence ball of the method is ob- tained. KEYWORDS simultaneous methods, polynomial zeros, local convergence, error estimate. 1. INTRODUCTION Throughout this paper ( ,| |) K ⋅ denotes an arbitrary normed field and [] Kz denotes the ring of polynomials over K . Let [] f Kz ∈ be a polynomial of degree 2 n ≥ . A vector n K ξ ∈ is called a root vector of f if 0 1 () ( ) n i i f z a z ξ = = − ∏ for all z K ∈ , where 0 a K ∈ . In the literature there are a lot of iterative methods for finding all zeros of polynomial simultaneously (see, e.g., Sendov, Andreev and Kjurkchiev [18], Kyurkchiev [2], McNamee [3], Petković [4] and references therein). In 2011, Petković, Rančić and Milošević [5] presented a new fourth-order iterative method for simultaneous finding polynomial zeros. Their method is defined by the following iteration (1) 1 , 1, 2, , k k x Tx k + = =  where the operator : n n T D K K ⊂ → is defined by ( ) 1 ( ), , () n Tx T x T x =  with (2) 2 2 2 ( ) ( ) () 2 (1 ) ( ) i i i i i i i i i i i f x u u S G f x Tx x u uS ′′ − − ′ = − − −       for 1, , i n =  , where ( ) ( ) i i i f x u f x = ′ , 1 i j i i j S x x ≠ = − ∑ , 2 1 ( ) i j i i j G x x ≠ = − ∑ . The domain D of T is the set { } : , ( ) 0, 1 0 . i j i i i n D x K x x for i j f x uS ′ = ∈ ≠ ≠ ≠ − ≠ Petković, Rančić and Milošević [5] proved the following asymptotic convergence theorem for the method (1). Theorem 1.1 ([5]). Let [] f Cz ∈ be a polynomial of degree 3 n ≥ which has n simple zeros in C . If an initial guess 0 n x C ∈ is sufficiently close to a root vector ξ of f , then the iteration (1) converges to ξ with order of convergence four. In this paper, we establish a new local convergence theorem with error estimates for the method (1) which improves Theorem 1.1. In particular, we obtain an estimate of the radius of convergence ball of this method.