AN EFFICIENT DERIVATIVE FREE ITERATIVE METHOD FOR SOLVING SYSTEMS OF NONLINEAR EQUATIONS

We present a derivative free method of fourth order convergence for solving systems of nonlinear equations. The method consists of two steps of which first step is the well-known Traub’s method. First-order divided difference operator for functions of several variables and direct computation by Taylor’s expansion are used to prove the local convergence order. Computational efficiency of new method in its general form is discussed and is compared with existing methods of similar nature. It is proved that for large systems the new method is more efficient. Some numerical tests are performed to compare proposed method with existing methods and to confirm the theoretical results.

[1]  Miodrag S. Petkovic,et al.  Remarks on "On a General Class of Multipoint Root-Finding Methods of High Computational Efficiency" , 2011, SIAM J. Numer. Anal..

[2]  Hongmin Ren,et al.  A class of two-step Steffensen type methods with fourth-order convergence , 2009, Appl. Math. Comput..

[3]  Stephen Wolfram,et al.  The Mathematica Book , 1996 .

[4]  Miquel Grau-Sánchez,et al.  Frozen divided difference scheme for solving systems of nonlinear equations , 2011, J. Comput. Appl. Math..

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

[6]  J. F. Steffensen.  Remarks on iteration. , 1933 .

[7]  C. Kelley.  Solving Nonlinear Equations with Newton's Method , 1987 .

[8]  Jingya Li,et al.  An optimal Steffensen-type family for solving nonlinear equations , 2011, Appl. Math. Comput..

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

[10]  Miquel Grau-Sánchez,et al.  A technique to choose the most efficient method between secant method and some variants , 2012, Appl. Math. Comput..

[11]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[12]  Miodrag S. Petkovic,et al.  On generalized multipoint root-solvers with memory , 2012, J. Comput. Appl. Math..

[13]  Xiaofeng Wang,et al.  A family of Steffensen type methods with seventh-order convergence , 2013, Numerical Algorithms.

[14]  Miodrag S. Petković,et al.  Families of optimal multipoint methods for solving nonlinear equations: A survey , 2010 .

[15]  Vincent Lefèvre,et al.  MPFR: A multiple-precision binary floating-point library with correct rounding , 2007, TOMS.

[16]  Puneet Gupta,et al.  Some efficient derivative free methods with memory for solving nonlinear equations , 2012, Appl. Math. Comput..

[17]  Miodrag S. Petkovic,et al.  Derivative free two-point methods with and without memory for solving nonlinear equations , 2010, Appl. Math. Comput..

[18]  A. Householder.  Solution of Equations and Systems of Equations (A. M. Ostrowski) , 1967 .

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

[20]  Peng Zhao,et al.  A variant of Steffensen's method of fourth-order convergence and its applications , 2010, Appl. Math. Comput..

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