A Note on the “Constructing” of Nonstationary Methods for Solving Nonlinear Equations with Raised Speed of Convergence

In this paper we give methodological survey of "contempo- rary methods" for solving the nonlinear equation f(x) = 0. The reason for this review is that many authors in present days rediscovered such classi- cal methods. Here we develop one methodological schema for constructing nonstationary methods with a preliminary chosen speed of convergence.

[1]  Ibrahim Abu-Alshaikh,et al.  Two-point iterative methods for solving nonlinear equations , 2006, Appl. Math. Comput..

[2]  M. Frontini,et al.  Third-order methods from quadrature formulae for solving systems of nonlinear equations , 2004, Appl. Math. Comput..

[3]  Nenad Ujevic,et al.  A method for solving nonlinear equations , 2006, Appl. Math. Comput..

[4]  Danfu Han,et al.  The construction of rational iterative methods for solving nonlinear equations , 2008, Appl. Math. Comput..

[5]  Sukhjit Singh Dhaliwal,et al.  On a class of quadratically convergent iteration formulae , 2005, Appl. Math. Comput..

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

[7]  Muhammad Aslam Noor,et al.  Three-step iterative methods for nonlinear equations , 2006, Appl. Math. Comput..

[8]  J. M. Gutiérrez,et al.  A family of Chebyshev-Halley type methods in Banach spaces , 1997, Bulletin of the Australian Mathematical Society.

[9]  F. Potra,et al.  Nondiscrete induction and iterative processes , 1984 .

[10]  Changbum Chun,et al.  A note on some iterative schemes for nonlinear equations , 2007, Appl. Math. Comput..

[11]  Yitian Li,et al.  Modified Halley's method free from second derivative , 2006, Appl. Math. Comput..

[12]  José Luis Díaz-Barrero,et al.  An improvement of the Euler-Chebyshev iterative method , 2006 .

[13]  Ahmet Yasar Özban,et al.  Some new variants of Newton's method , 2004, Appl. Math. Lett..

[14]  Nazir Ahmad Mir,et al.  Some quadrature based three-step iterative methods for non-linear equations , 2007, Appl. Math. Comput..

[15]  José Luis Díaz-Barrero,et al.  An improvement to Ostrowski root-finding method , 2006, Appl. Math. Comput..

[16]  Xiuhua Wang,et al.  Some modifications of Newton's method with fifth-order convergence , 2007 .

[17]  Miquel Grau.  An Improvement to the Computing of Nonlinear Equation Solutions , 2004, Numerical Algorithms.

[18]  Changbum Chun.  A one-parameter family of quadratically convergent iteration formulae , 2007, Appl. Math. Comput..

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

[20]  Changbum Chun,et al.  Some higher-order modifications of Newton's method for solving nonlinear equations , 2008 .

[21]  M. Koçak,et al.  A class of iterative methods with third-order convergence to solve nonlinear equations , 2008 .

[22]  Qingbiao Wu,et al.  Three-step iterative methods with eighth-order convergence for solving nonlinear equations , 2009 .

[23]  M. Z. Dauhoo,et al.  An analysis of the properties of the variants of Newton's method with third order convergence , 2006, Appl. Math. Comput..

[24]  Alicia Cordero,et al.  Variants of Newton's Method using fifth-order quadrature formulas , 2007, Appl. Math. Comput..

[25]  Changbum Chun,et al.  Some improvements of Jarratt's method with sixth-order convergence , 2007, Appl. Math. Comput..

[26]  Mário Basto,et al.  A new iterative method to compute nonlinear equations , 2006, Appl. Math. Comput..

[27]  Gyurhan Nedzhibov.  On a few Iterative Methods for Solving Nonlinear Equations , 2002 .

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

[29]  Muhammad Aslam Noor,et al.  Two-step iterative methods for nonlinear equations , 2006, Applied Mathematics and Computation.

[30]  Miodrag S. Petković,et al.  Point estimation theory and its applications , 1997 .

[31]  Li Yitian,et al.  Letter to the Editor: Third-order modification of Newton's method , 2007 .

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

[33]  Changbum Chun.  A one-parameter family of third-order methods to solve nonlinear equations , 2007, Appl. Math. Comput..

[34]  Muhammad Aslam Noor,et al.  Fourth-order convergent iterative method for nonlinear equation , 2006, Appl. Math. Comput..

[35]  Yitian Li,et al.  An improvement of the Jarratt method , 2007, Appl. Math. Comput..

[36]  M. Javidi,et al.  Iterative methods to nonlinear equations , 2007, Appl. Math. Comput..

[37]  Amir Sapir,et al.  A nonstationary iterative second-order method for solving nonlinear equations , 2007, Appl. Math. Comput..

[38]  Changbum Chun.  Some variants of King's fourth-order family of methods for nonlinear equations , 2007, Appl. Math. Comput..

[39]  Yitian Li,et al.  Modified Chebyshev-Halley methods with sixth-order convergence , 2007, Appl. Math. Comput..

[40]  M. Frontini,et al.  Modified Newton's method with third-order convergence and multiple roots , 2003 .

[41]  Muhammad Aslam Noor,et al.  Fifth-order iterative methods for solving nonlinear equations , 2007, Appl. Math. Comput..

[42]  Guoping He,et al.  A cubically convergent Newton-type method under weak conditions , 2008 .

[43]  Miquel Grau-Sánchez,et al.  A variant of Cauchy's method with accelerated fifth-order convergence , 2004, Appl. Math. Lett..

[44]  Vejdi I. Hasanov,et al.  On some families of multi-point iterative methods for solving nonlinear equations , 2006, Numerical Algorithms.

[45]  Yitian Li,et al.  Modified Chebyshev's method free from second derivative for non-linear equations , 2007, Appl. Math. Comput..

[46]  Vinay Kanwar,et al.  A family of third-order multipoint methods for solving nonlinear equations , 2006, Appl. Math. Comput..

[47]  J. R. Sharma,et al.  A composite third order Newton-Steffensen method for solving nonlinear equations , 2005, Appl. Math. Comput..

[48]  Arif Rafiq.  A note on "New classes of iterative methods for nonlinear equations" and "Some iterative methods free from second derivatives for nonlinear equations" , 2007, Appl. Math. Comput..

[49]  Xiuhua Wang,et al.  Some variants of Ostrowski's method with seventh-order convergence , 2007 .

[50]  Changbum Chun,et al.  A one-parameter fourth-order family of iterative methods for nonlinear equations , 2007, Appl. Math. Comput..

[51]  Xinyuan Wu,et al.  New high-order convergence iteration methods without employing derivatives for solving nonlinear equations , 2001 .

[52]  Changbum Chun,et al.  A simply constructed third-order modifications of Newton's method , 2008 .

[53]  José Antonio Ezquerro,et al.  Chebyshev-like methods and quadratic equations , 1999 .

[54]  Gyurhan H. Nedzhibov.  An acceleration of iterative processes for solving nonlinear equations , 2005, Appl. Math. Comput..

[55]  Yitian Li,et al.  The improvements of Chebyshev-Halley methods with fifth-order convergence , 2007, Appl. Math. Comput..

[56]  Beny Neta,et al.  On Popovski's method for nonlinear equations , 2008, Appl. Math. Comput..

[57]  G. Alefeld,et al.  Introduction to Interval Computation , 1983 .

[58]  Changbum Chun,et al.  Some fourth-order modifications of Newton's method , 2008, Appl. Math. Comput..

[59]  J. M. Gutiérrez,et al.  Geometric constructions of iterative functions to solve nonlinear equations , 2003 .

[60]  Herbert H. H. Homeier.  A modified Newton method for rootfinding with cubic convergence , 2003 .

[61]  Changbum Chun,et al.  Some sixth-order variants of Ostrowski root-finding methods , 2007, Appl. Math. Comput..

[62]  Changbum Chun.  A two-parameter third-order family of methods for solving nonlinear equations , 2007, Appl. Math. Comput..

[63]  Peng Wu,et al.  A family of iterative methods with higher-order convergence , 2006, Appl. Math. Comput..

[64]  Maria Italia Gualtieri,et al.  A New Iterative Method for the Computation of the Solutions of Nonlinear Equations , 2001, Numerical Algorithms.

[65]  Yitian Li,et al.  A family of fifth-order iterations composed of Newton and third-order methods , 2007, Appl. Math. Comput..

[66]  A. B. Kasturiarachi,et al.  Leap-frogging Newton's method , 2002 .

[67]  Changbum Chun,et al.  A new iterative method for solving nonlinear equations , 2006, Appl. Math. Comput..

[68]  Miguel Ángel Hernández,et al.  An acceleration of Newton's method: Super-Halley method , 2001, Appl. Math. Comput..

[69]  Changbum Chun,et al.  Computers and Mathematics with Applications , 2022 .

[70]  Arnold Neumaier,et al.  Introduction to Numerical Analysis , 2001 .

[71]  Xiaojian Zhou.  A class of Newton's methods with third-order convergence , 2007, Appl. Math. Lett..

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

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

[74]  H. H. H. Homeiera,et al.  On Newton-type methods with cubic convergence , 2004 .

[75]  Jisheng Kou,et al.  The improvements of modified Newton's method , 2007, Appl. Math. Comput..

[76]  Yitian Li,et al.  Fourth-order iterative methods free from second derivative , 2007, Appl. Math. Comput..

[77]  Arif Rafiq.  A note on "A family of methods for solving nonlinear equations" , 2008, Appl. Math. Comput..

[78]  Qingbiao Wu,et al.  New family of seventh-order methods for nonlinear equations , 2008, Appl. Math. Comput..

[79]  Changbum Chun,et al.  Some third-order families of iterative methods for solving nonlinear equations , 2007, Appl. Math. Comput..

[80]  Changbum Chun,et al.  Construction of third-order modifications of Newton's method , 2007, Appl. Math. Comput..

[81]  Janak Raj Sharma,et al.  A family of modified Ostrowski methods with accelerated sixth order convergence , 2007, Appl. Math. Comput..

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

[83]  G. Nedzhibov,et al.  On Analytic Iterative Functions for Solving Nonlinear Equations and Systems of Equations , 2004, NAA.