Improvement of a convergence condition for the Durand-Kerner iteration

A new semilocal convergence theorem for the Weierstrass method for finding zeros of a polynomial simultaneously

“…Batra [13] has proved that the Weierstrass method is convergent under the condition W (z 0 ) ∞ < δ(z 0 )/(2n). The following corollary improves Batra's result as well as previous results [4,9,10,11,12].…”

“…In 1962, Dochev [2,3] proved the first local convergence theorem for the Weierstrass method. Since 1980 a number of authors [4,5,6,7,8,9,10,11,12,13,14,15] have obtained semilocal convergence theorems for the Weierstrass method from data at one point (point estimation). In this note we present a new semilocal convergence theorem for the Weierstrass method which improves and generalizes all these results.…”

“…is faster if h k is closer to 1. In 1995, Wang and Zhao[8] considered the SOR method(13) with h k defined by…”

Newton's method and the Computational Complexity of the Fundamental Theorem of Algebra

“…Following the paradigm of Smale's point estimates we derive convergence conditions using attainable data from the iteration process (see, e.g., [1]). …”

The convergence of a family of parallelzero-finding methods

“…According to the results of the papers [8][9][10][11][12][13][14], it turned out that suitable initial conditions, providing a guaranteed convergence of iterative methods for the simultaneous determination of polynomial zeros, are of the form of the inequality w (°) < c~d <°), (1.1) where c~ is the quantity which depends only on the polynomial degree n. Moreover, Wang and Zhao [8] came to form (1.1) in a quite natural way by applying their improvement of Smale's results for Newton's method. In Section 2 we give a short review of basic operations of circular complex arithmetic, necessary for finding bounds of some complex quantities.…”