Some improved inclusion methods for polynomial roots with Weierstrass' corrections

AbstractOne decade ago, the third order method without derivatives for the simultaneous inclusion of simple zeros of a polynomial was proposed in [1]. Following Nourein's idea [2], some modifications of this method with the increased convergence are proposed. The acceleration of convergence is attained by using Weierstrass' corrections without additional calculations, which provides a high computational efficiency of the modified methods. It is proved that their R-orders of convergence are asymptotically greater than 3.5. The presented interval methods are realized in circular complex arithmetic

A new higher-order family of inclusion zero-finding methods

AbstractStarting from a suitable fixed point relation, a new one-parameter family of iterative methods for the simultaneous inclusion of complex zeros in circular complex arithmetic is constructed. It is proved that the order of convergence of this family is four. The convergence analysis is performed under computationally verifiable initial conditions. An approach for the construction of accelerated methods with negligible number of additional operations is discussed. To demonstrate convergence properties of the proposed family of methods, two numerical examples results are given

A Family of Three-Point Methods of Ostrowski's Type for Solving Nonlinear Equations

A class of three-point methods for solving nonlinear equations of eighth order is constructed. These methods are developed by combining two-point Ostrowski's fourth-order methods and a modified Newton's method in the third step, obtained by a suitable approximation of the first derivative using the product of three weight functions. The proposed three-step methods have order eight costing only four function evaluations, which supports the Kung-Traub conjecture on the optimal order of convergence. Two numerical examples for various weight functions are given to demonstrate very fast convergence and high computational efficiency of the proposed multipoint methods

Famous puzzles of great mathematicians

This entertaining book presents a collection of 180 famous mathematical puzzles and intriguing elementary problems that great mathematicians have posed, discussed, and/or solved. The selected problems do not require advanced mathematics, making this book accessible to a variety of readers. Mathematical recreations offer a rich playground for both amateur and professional mathematicians. Believing that creative stimuli and aesthetic considerations are closely related, great mathematicians from ancient times to the present have always taken an interest in puzzles and diversions. The goal of thi

Comments on some recent methods for the simultaneous determination of polynomial zeros

AbstractIn this note we give some comments on the recent results concerning a simultaneous method of the fourth-order for finding complex zeros in circular interval arithmetic. The main discussion is directed to a rediscovered iterative formula and its modification, presented recently in (Sun and Kosmol (J. Comput. Appl. Math. 130 (2001) 293)). The presented comments include some critical parts of the papers (Jpn. J. Indust. Appl. Math. 15 (1998) 295) and (Sun and Kosmol, 2001) which treat the same subject