Abstract
In this paper, we have obtained three optimal order Newton’s like methods of order four, eight, and sixteen for solving nonlinear algebraic equations. The convergence analysis of all the optimal order methods is discussed separately. We have discussed the corresponding conjugacy maps for quadratic polynomials and also obtained the extraneous fixed points. We have considered several test functions to examine the convergence order and to explain the dynamics of our proposed methods. Theoretical results, numerical results, and fractal patterns are in support of the efficiency of the optimal order methods.
Highlights
We display the numerical results of some test problems to examine the efficiency of proposed new optimal order methods
The fourth-order Maheshwari method and fourth-order proposed method are diverging for the functions f 2 at both points −10 and 50, while the proposed 16th order method is best among all the methods by taking the least number of iterations
Numerical results reveal that the proposed 8th and 16th order optimal methods are more effective than the other methods at the point considered, as they are converging to the root with high speed
Summary
Several authors have made modifications and refinements of Newton’s method to speed it up or to find a better method by using different techniques i.e., adding functions terms, derivatives terms, and/or by variations in the points of iterations [1,2]. We have obtained three optimal order (four, eight, and sixteen) iterative methods using a new different technique. This technique is more simple to previous one developed by. We have developed the optimal order methods by repeated applications of Newton’s method and thereafter approximated the derivative term in Newton’s method by a suitable polynomial to reduce the functions evaluations and obtained the optimal order methods. We have discussed the dynamics of our methods and found that the basin of attraction for different order methods is in support of numerical results
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