Local and Semilocal Convergence of a Family of Multi-point Weierstrass-Type Root-Finding Methods
- PDF / 609,139 Bytes
- 20 Pages / 439.37 x 666.142 pts Page_size
- 11 Downloads / 220 Views
Local and Semilocal Convergence of a Family of Multi-point Weierstrass-Type Root-Finding Methods Petko D. Proinov
and Milena D. Petkova
Abstract. Weierstrass (Sitzungsber K¨ onigl Preuss Akad Wiss Berlin II: 1085–1101, 1891) introduced his famous iterative method for numerical finding all zeros of a polynomial simultaneously. Kyurkchiev and Ivanov (Ann Univ Sofia Fac Math Mech 78:132–136, 1984) constructed a family of multi-point root-finding methods which are based on the Weierstrass method. The purpose of this research is threefold: (1) to develop a new simple approach for the study of the local convergence of the multi-point simultaneous iterative methods; (2) to present a new local convergence result for this family which improves in several directions the result of Kyurkchiev and Ivanov; (3) to provide semilocal convergence results for Kyurkchiev–Ivanov’s family of iterative methods. Mathematics Subject Classification. Primary 65H05; Secondary 12Y05. Keywords. Iterative methods, simultaneous methods, polynomial zeros, local convergence, semilocal convergence, error estimates.
1. Introduction The calculation of zeros of polynomials is the oldest mathematical problem. There are many classical and modern numerical methods for solving polynomial equations (see [5,6,18–20,31,32] and references therein). In 1891, Weierstrass [35] introduced the first iterative method for simultaneous computation of all zeros of a complex polynomial. Moreover, he provided a semilocal convergence analysis of his method and, as a consequence, he gave the first constructive proof of the fundamental theorem of algebra which states that the field of complex numbers C is algebraically closed. Further development of Weierstrass’ semilocal convergence result was given in [27]. A detailed convergence analysis of the Weierstrass method can be found in [24,27]. In 1913, This work was supported by the National Science Fund of the Bulgarian Ministry of Education and Science under Grant DN 12/12.
0123456789().: V,-vol
107
Page 2 of 20
P. D. Proinov and M. D. Petkova
MJOM
K¨ ursch´ ak [17] noted that the Weierstrass method can be considered in valued fields and he gave the first application of this method to the theory of valued fields (see also [27]). Let K be a valued field, K[z] be the ring of polynomials over K, and let f ∈ K[z] be a polynomial of degree n ≥ 2. The Weierstrass method is defined in Kn by the following iteration: x(k+1) = x(k) − Wf (x(k) ),
k = 0, 1, 2, . . . ,
(1.1)
where x(0) ∈ Kn , the Weierstrass correction Wf : D ⊂ Kn → Kn is defined as follows: f (xi ) , (1.2) Wf (x) = (W1 (x), . . . , Wn (x)) with Wi (x) = a0 j = i (xi − xj ) a0 is the leading coefficient of f , and D is the set of all vectors in Kn with pairwise distinct components. In 1960–1966, the Weierstrass method was rediscovered by Durand [10] (in implicit form), Dochev [9], Kerner [14], and Preˇsi´c [21]. For this reason, the method (1.1) is also known as ‘Durand–Kerner method’, ‘Weierstrass–Dochev method’, etc. Nowadays, there are many iterative methods for find
Data Loading...
