On the development of iterative methods for multiple roots

There are very few optimal fourth order methods for solving nonlinear algebraic equations having roots of multiplicity m. Here we compare 4 such methods, two of which require the evaluation of the (m-1)^s^t root. We will show that such computation does not affect the overall cost of the method.

[1]  Beny Neta,et al.  High-order nonlinear solver for multiple roots , 2008, Comput. Math. Appl..

[2]  Beny Neta,et al.  Extension of Murakami's high-order non-linear solver to multiple roots , 2010, Int. J. Comput. Math..

[3]  Beny Neta,et al.  Author's Personal Copy Computers and Mathematics with Applications Some Fourth-order Nonlinear Solvers with Closed Formulae for Multiple Roots , 2022 .

[4]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[5]  G. Humbert.  Sur la méthode d'approximation d'Hermite , 1916 .

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

[7]  Beny Neta,et al.  Multipoint Methods for Solving Nonlinear Equations , 2012 .

[8]  Xiangke Liao,et al.  A new fourth-order iterative method for finding multiple roots of nonlinear equations , 2009, Appl. Math. Comput..

[9]  Ke Chen,et al.  Applied Mathematics and Computation , 2022 .

[10]  Changbum Chun,et al.  On a family of Laguerre methods to find multiple roots of nonlinear equations , 2013, Appl. Math. Comput..

[11]  Baoqing Liu,et al.  A new family of fourth-order methods for multiple roots of nonlinear equations , 2013 .

[12]  Ernst Schröder,et al.  Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen , 1870 .

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

[14]  Liao Xiangke,et al.  A new fourth-order iterative method for finding multiple roots of nonlinear equations , 2009 .

[15]  A. Ostrowski.  Solution of equations in Euclidean and Banach spaces , 1973 .

[16]  Gerald S. Shedler,et al.  Parallel numerical methods for the solution of equations , 1967, CACM.

[17]  Yongzhong Song,et al.  Constructing higher-order methods for obtaining the multiple roots of nonlinear equations , 2011, J. Comput. Appl. Math..