Abstract
In 2016, Nedzhibov constructed a modification of the Weierstrass method for simultaneous computation of polynomial zeros. In this work, we obtain local and semilocal convergence theorems that improve and complement the previous results about this method. The semilocal result is of significant practical importance because of its computationally verifiable initial condition and error estimate. Numerical experiments to show the applicability of our semilocal theorem are also presented. We finish this study with a theoretical and numerical comparison between the modified Weierstrass method and the classical Weierstrass method.
Highlights
Throughout the present study, (K, | · |) denotes an arbitrary normed field and K[z] represents the ring of polynomials over K
We study the convergence of the modified Weierstrass method (4) regarding the function of initial conditions E : Kn → R+ defined by x−ξ
We are ready to state and prove the third main result of this paper. It is a theorem of significant practical importance since the initial condition and the error estimate are computationally verifiable
Summary
Throughout the present study, (K, | · |) denotes an arbitrary normed field and K[z] represents the ring of polynomials over K. In 1891, Weierstrass [1] established an iterative method for finding the root-vector of f. A local convergence analysis of the modified Weierstrass method (4) was presented in the papers [11,12,13,14,15,16]. (i) The modified Weierstrass iteration (4) is well defined and converges quadratically to the root-vector ξ of f. The modified Weierstrass iteration (4) is well defined and converges quadratically where ∆. Under the assumptions of Theorem 2 an assessment of the asymptotic error constant of the modified Weierstrass iteration (4) was provided in the following convergence theorem: Theorem 4 ([16] (Theorem 1)). Is well defined and converges quadratically to the root-vector ξ of f with error estimates (13).
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