Improved Muller method and Bisection method with,global and asymptotic superlinear convergence of both point and interval for solving nonlinear equations

A new and improved version of Muller method and Bisection method with global and asymptotic superlinear convergence for finding a simple root x* of a nonlinear equation f(x) = 0 in the interval [a, b] is proposed in this paper. The new iteration procedure combines Muller method with Bisection method to generate simultaneously two sequences {x(n)} which goes to x* and {[a(n), b(n)]} which encloses x*. The global and superlinear convergence for the both sequences {x(r)} and {b(n) - a(n)} are analyzed. The asymptotic efficiency index of the improved Muller method and Bisection method for the both sequences {x(n)} and {b(n) - a(n)} proves to be 1.84 approximately on certain conditions, in the sense of Ostrowski. As a result, the new and improved version of Muller method and Bisection method preserve their respective nice property and remove their respective defect. The new version has been tested on a series of elementary functions. The numerical results show that the new version of Muller method and Bisection method proposed in this paper is more effective compared with the traditional version for solving nonlinear equations. For the computation of multiple zeros a effective strategy is discussed. (c) 2004 Published by Elsevier Inc.