Abstract
We present new modifications to Newton's method for solving nonlinear equations. The analysis of convergence shows that these methods have fourth-order convergence. Each of the three methods uses three functional evaluations. Thus, according to Kung-Traub's conjecture, these are optimal methods. With the previous ideas, we extend the analysis to functions with multiple roots. Several numerical examples are given to illustrate that the presented methods have better performance compared with Newton's classical method and other methods of fourth-order convergence recently published.
Highlights
One of the most important problems in numerical analysis is solving nonlinear equations
In [14], Cordero and Torregrosa presented a family of Steffensen-type methods of fourth-order convergence for solving nonlinear smooth equations by using a linear combination of divided differences to achieve a better approximation to the derivative
The new methods require less number of functional evaluations. This means that the new methods have better efficiency in computing process than Newton’s method as compared to other methods, and the method FAM3 produces the best results
Summary
One of the most important problems in numerical analysis is solving nonlinear equations. All previous methods use the second derivative of the function to obtain a greater order of convergence. In [14], Cordero and Torregrosa presented a family of Steffensen-type methods of fourth-order convergence for solving nonlinear smooth equations by using a linear combination of divided differences to achieve a better approximation to the derivative. The three new methods (for simple roots) in this paper use three functional evaluations and have fourth-order convergence; they are optimal methods and their efficiency index is E = 1.587, which is greater than the efficiency index of Newton’s method, which is E = 1.414. In the case of multiple roots, the method developed here is cubically convergent and uses three functional evaluations without the use of second derivative of the function.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
