Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: , a family of Steffensen-type methods of optimal order of convergence with two parameters is constructed by direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub (J. Assoc. Comput. Math. 1974, 21, 634–651) that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order                                        2m-1 . Furthermore, the family of Steffensen-type methods of super convergence is suggested by using arithmetic expressions for the parameters with memory but no additional new evaluation of the function. Their error equations, asymptotic convergence constants and convergence orders are obtained. Finally, they are compared with related root-finding methods in the numerical examples.
['Quan Zheng', 'Xin Zhao', 'Yufeng Liu']
I am writing a research paper on a idea An Optimal Biparametric Multipoint Family and Its Self-Acceleration with Memory for Solving Nonlinear Equations
[{"from": "human", "value": "I am writing a research paper on a idea An Optimal Biparametric Multipoint Family and Its Self-Acceleration with Memory for Solving Nonlinear Equations"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: , a family of Steffensen-type methods of optimal order of convergence with two parameters is constructed by direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub (J. Assoc. Comput. Math. 1974, 21, 634\u2013651) that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order                                        2m-1 . Furthermore, the family of Steffensen-type methods of super convergence is suggested by using arithmetic expressions for the parameters with memory but no additional new evaluation of the function. Their error equations, asymptotic convergence constants and convergence orders are obtained. Finally, they are compared with related root-finding methods in the numerical examples."}]