Families of optimal multipoint methods for solving nonlinear equations: A survey

Multipoint iterative root-solvers belong to the class of the most powerful methods for solving nonlinear equations since they overcome theoretical limits of one-point methods concerning the convergence order and computational efficiency. Although the construction of these methods has occurred in the 1960s, their rapid development have started in the first decade of the 21-st century. The most important class of multipoint methods are optimal methods which attain the convergence order 2n using n + 1 function evaluations per iteration. In this paper we give a review of optimal multipoint methods of the order four (n = 2), eight (n = 3) and higher (n > 3), some of which being proposed by the authors. All of them possess as high as possible computational efficiency in the sense of the Kung-Traub hypothesis (1974). Numerical examples are included to demonstrate a very fast convergence of the presented optimal multipoint methods.

[1]  Amit Kumar Maheshwari,et al.  A fourth order iterative method for solving nonlinear equations , 2009, Appl. Math. Comput..

[2]  Miodrag S. Petkovic,et al.  A family of three-point methods of optimal order for solving nonlinear equations , 2010, J. Comput. Appl. Math..

[3]  Qingbiao Wu,et al.  A new family of eighth-order iterative methods for solving nonlinear equations , 2009, Appl. Math. Comput..

[4]  A. Ostrowski.  Solution of equations and systems of equations , 1967 .

[5]  Miodrag S. Petkovic,et al.  A note on some recent methods for solving nonlinear equations , 2007, Appl. Math. Comput..

[6]  Miodrag S. Petkovic,et al.  Iterative methods based on the signum function approach for solving nonlinear equations , 2009, Numerical Algorithms.

[7]  Miodrag S. Petković,et al.  A one parameter square root family of two-step root-finders , 2007, Appl. Math. Comput..

[8]  Miodrag S. Petkovic,et al.  Sigmoid-like functions and root finding methods , 2008, Appl. Math. Comput..

[9]  Changbum Chun,et al.  A family of composite fourth-order iterative methods for solving nonlinear equations , 2007, Appl. Math. Comput..

[10]  J. Traub.  Iterative Methods for the Solution of Equations , 1982 .

[11]  Miodrag S. Petkovic,et al.  ON OPTIMAL MULTIPOINT METHODS FOR SOLVING NONLINEAR EQUATIONS 1 , 2009 .

[12]  Changbum Chun,et al.  Some fourth-order iterative methods for solving nonlinear equations , 2008, Appl. Math. Comput..

[13]  Yitian Li,et al.  A composite fourth-order iterative method for solving non-linear equations , 2007, Appl. Math. Comput..

[14]  Sunethra Weerakoon,et al.  A variant of Newton's method with accelerated third-order convergence , 2000, Appl. Math. Lett..

[15]  Beny Neta,et al.  On a family of multipoint methods for non-linear equations , 1981 .

[16]  Beong In Yun,et al.  A non-iterative method for solving non-linear equations , 2008, Appl. Math. Comput..

[17]  H. T. Kung,et al.  Optimal Order of One-Point and Multipoint Iteration , 1974, JACM.

[18]  R. F. King.  A Family of Fourth Order Methods for Nonlinear Equations , 1973 .

[19]  Miodrag S. Petkovic,et al.  On a General Class of Multipoint Root-Finding Methods of High Computational Efficiency , 2010, SIAM J. Numer. Anal..