Higher-Order Iteration Schemes for Solving Nonlinear Systems of Equations

We present a three-step family of iterative methods to solve systems of nonlinear equations. This family is a generalization of the well-known fourth-order King’s family to the multidimensional case. The convergence analysis of the methods is provided under mild conditions. The analytical discussion of the work is upheld by performing numerical experiments on some application oriented problems. Finally, numerical results demonstrate the validity and reliability of the suggested methods.

[1]  Xiaofeng Wang,et al.  An Efficient Sixth-Order Newton-Type Method for Solving Nonlinear Systems , 2017, Algorithms.

[2]  Alicia Cordero,et al.  Variants of Newton's method for functions of several variables , 2006, Appl. Math. Comput..

[3]  Alicia Cordero,et al.  A modified Newton-Jarratt’s composition , 2010, Numerical Algorithms.

[4]  Alicia Cordero,et al.  Approximation of artificial satellites' preliminary orbits: The efficiency challenge , 2011, Math. Comput. Model..

[5]  Alicia Cordero,et al.  Stability of King's family of iterative methods with memory , 2017, J. Comput. Appl. Math..

[6]  Alicia Cordero,et al.  Efficient high-order methods based on golden ratio for nonlinear systems , 2011, Appl. Math. Comput..

[7]  Eulalia Martínez,et al.  Convergence, efficiency and dynamics of new fourth and sixth order families of iterative methods for nonlinear systems , 2015, J. Comput. Appl. Math..

[8]  Alicia Cordero,et al.  Dynamics of iterative families with memory based on weight functions procedure , 2019, J. Comput. Appl. Math..

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

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

[11]  Ali Barati,et al.  A third-order Newton-type method to solve systems of nonlinear equations , 2007, Appl. Math. Comput..

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

[13]  D. K. R. Babajee,et al.  On Some Improved Harmonic Mean Newton-Like Methods for Solving Systems of Nonlinear Equations , 2015, Algorithms.

[14]  Herbert H. H. Homeier.  A modified Newton method with cubic convergence: the multivariate case , 2004 .

[15]  Alicia Cordero,et al.  Some new efficient multipoint iterative methods for solving nonlinear systems of equations , 2015, Int. J. Comput. Math..

[16]  Ali Barati,et al.  Super cubic iterative methods to solve systems of nonlinear equations , 2007, Appl. Math. Comput..

[17]  Vinay Kanwar,et al.  New two-parameter Chebyshev-Halley-like family of fourth and sixth-order methods for systems of nonlinear equations , 2016, Appl. Math. Comput..

[18]  Rakesh K. Kapania.  A pseudo-spectral solution of 2-parameter Bratu's equation , 1990 .

[19]  R. B. Simpson.  A Method for the Numerical Determination of Bifurcation States of Nonlinear Systems of Equations , 1975 .

[20]  Alicia Cordero,et al.  Chaos in King's iterative family , 2013, Appl. Math. Lett..

[21]  Alicia Cordero,et al.  Iterative methods of order four and five for systems of nonlinear equations , 2009, J. Comput. Appl. Math..

[22]  Miquel Grau-Sánchez,et al.  On the computational efficiency index and some iterative methods for solving systems of nonlinear equations , 2011, J. Comput. Appl. Math..

[23]  J. Sharma,et al.  Efficient Jarratt-like methods for solving systems of nonlinear equations , 2014 .

[24]  Ramandeep Behl,et al.  Some higher-order iteration functions for solving nonlinear models , 2018, Appl. Math. Comput..

[25]  D. K. R. Babajee.  On a two-parameter Chebyshev-Halley-like family of optimal two-point fourth order methods free from second derivatives , 2015 .

[26]  Alicia Cordero,et al.  New efficient methods for solving nonlinear systems of equations with arbitrary even order , 2016, Appl. Math. Comput..