A Modification on Newton's Method for Solving Systems of Nonlinear Equations
In this paper, we are concerned with the further study for system of nonlinear equations. Since systems with inaccurate function values or problems with high computational cost arise frequently in science and engineering, recently such systems have attracted researcher-s interest. In this work we present a new method which is independent of function evolutions and has a quadratic convergence. This method can be viewed as a extension of some recent methods for solving mentioned systems of nonlinear equations. Numerical results of applying this method to some test problems show the efficiently and reliability of method.
[1] R.L. Burden, J.D. Faires, Numerical Analysis, 7th ed., PWS Publishing
Company, Boston, 2001.
[2] J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations
in Several variables, Academic Press, 1970.
[3] E. Babolian, J. Biazar, A.R. Vahidi, Solution of a system of
nonlinear equations by Adimian decomposition method, Appl. Math.
Comput. Vol. 150, 2004, pp. 847-854.
[4] Frontini, E. Sormani, Third-order methods from quadrature formulae for
solving systems of nonlinear equations, Appl. Math. Comput, Vol. 149,
2004, pp. 771-782.
[5] A. Cordero, J.R. Torregrosa, Variants of Newton's method for functions
of several variables, Appl. Math. Comput., Vol. 183, 2006, pp.199-208.
[6] F. Freudensten, B. Roth, Numerical solution of systems of nonlinear
equations, J. ACM, Vol. 10 , 1963, pp. 550-556.
[7] M. Grau-S├ínchez, J.M. Peris, J.M. Gutiérrez, Accelerated iterative
methods for finding solutions of a system of nonlinear equations, Appl.
Math. Comput. Vol. 190 , 2007, pp. 1815-1823.
[8] H.H.H. Homeier, A modified Newton method with cubic convergence:
The multivariate case, J. Comput. Appl. Math. 169 , 2004), pp 161-169.
[9] J. Kou, A third-order modification of Newton method for systems of
nonlinear equations, Appl. Math. Comput, Vol. 191 , 2007, pp. 117-121.
[10] L.F. Shampine, R.C. Allen, S. Pruess, Fundamentals of Numerical
Computing, John Wiley and Sons, New York, 1997.
[11] M. Kupferschmid, J.G. Ecker, A note on solution of nonlinear
programming problems with imprecise function and gradient values ,
Math. Program. Study, Vol. 31 , 1987, pp. 129-138.
[12] M.N. Vrahatis, T.N. Grapsa, O. Ragos, F.A. Zafiropoulos, On the
localization and computation of zeros of Bessel functions, Z. Angew. Math. Mech, Vol 77 (6, pp, 1997, pp. 467-475.
[13] M.N. Vrahatis, G.D. Magoulas, V.P. Plagianakos, From linear to
nonlinear iterative methods, Appl. Numer. Math, Vol. 45 , No.1, 2003, pp. 59-77.
[14] M.N. Vrahatis, O. Ragos, F.A. Zafiropoulos, T.N. Grapsa, Locating and
computing zeros of Airy functions, Z. Angew. Math. Mech, Vol. 76, No.7, 1996, pp. 419-422.
[15] W. Chen, A Newton method without evaluation of nonlinear function
values, CoRR cs.CE/9906011, 1999.
[16] T.N. Grapsa, E.N. Malihoutsaki, Newton's method without direct
function evaluations, in: E. Lipitakis (Ed.), Proceedings of 8th Hellenic European Research on Computer Mathematics & its Applications-Conference, HERCMA 2007, 2007.
[17] E.N. Malihoutsaki, I.A. Nikas, T.N. Grapsa, Improved Newton's method
without direct function evaluations, Journal of Computational and
Applied Mathematics, Vol. 227 , 2009, pp. 206-212.
[18] T.N. Grapsa, Implementing the initialization-dependence and the
singularity difficulties in Newton's method, Tech. Rep. 07-03, Division
of Computational Mathematics and Informatics, Department of
Mathematics, University of Patras, 2007.
[19] T.N. Grapsa, M.N. Vrahatis, A dimension-reducing method for solving
systems of nonlinear equations in ¶Çü£n , Int. J. Comput. Math, Vol. 32 ,
1990, pp. 205-216.
[20] T.N. Grapsa, M.N. Vrahatis, A dimension-reducing method for
unconstrained optimization, J. Comput. Appl. Math, Vol. 66, No.1-2,
1996, pp. 239-253.
[21] D.G. Sotiropoulos, J.A. Nikas, T.N. Grapsa, Improving the efficiency of
a polynomial system solver via a reordering technique, in: D.T. Tsahalis (Ed.), Proceedings of 4th GRACM Congress on Computational
Mechanics, Vol. III, 2002.
[1] R.L. Burden, J.D. Faires, Numerical Analysis, 7th ed., PWS Publishing
Company, Boston, 2001.
[2] J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations
in Several variables, Academic Press, 1970.
[3] E. Babolian, J. Biazar, A.R. Vahidi, Solution of a system of
nonlinear equations by Adimian decomposition method, Appl. Math.
Comput. Vol. 150, 2004, pp. 847-854.
[4] Frontini, E. Sormani, Third-order methods from quadrature formulae for
solving systems of nonlinear equations, Appl. Math. Comput, Vol. 149,
2004, pp. 771-782.
[5] A. Cordero, J.R. Torregrosa, Variants of Newton's method for functions
of several variables, Appl. Math. Comput., Vol. 183, 2006, pp.199-208.
[6] F. Freudensten, B. Roth, Numerical solution of systems of nonlinear
equations, J. ACM, Vol. 10 , 1963, pp. 550-556.
[7] M. Grau-S├ínchez, J.M. Peris, J.M. Gutiérrez, Accelerated iterative
methods for finding solutions of a system of nonlinear equations, Appl.
Math. Comput. Vol. 190 , 2007, pp. 1815-1823.
[8] H.H.H. Homeier, A modified Newton method with cubic convergence:
The multivariate case, J. Comput. Appl. Math. 169 , 2004), pp 161-169.
[9] J. Kou, A third-order modification of Newton method for systems of
nonlinear equations, Appl. Math. Comput, Vol. 191 , 2007, pp. 117-121.
[10] L.F. Shampine, R.C. Allen, S. Pruess, Fundamentals of Numerical
Computing, John Wiley and Sons, New York, 1997.
[11] M. Kupferschmid, J.G. Ecker, A note on solution of nonlinear
programming problems with imprecise function and gradient values ,
Math. Program. Study, Vol. 31 , 1987, pp. 129-138.
[12] M.N. Vrahatis, T.N. Grapsa, O. Ragos, F.A. Zafiropoulos, On the
localization and computation of zeros of Bessel functions, Z. Angew. Math. Mech, Vol 77 (6, pp, 1997, pp. 467-475.
[13] M.N. Vrahatis, G.D. Magoulas, V.P. Plagianakos, From linear to
nonlinear iterative methods, Appl. Numer. Math, Vol. 45 , No.1, 2003, pp. 59-77.
[14] M.N. Vrahatis, O. Ragos, F.A. Zafiropoulos, T.N. Grapsa, Locating and
computing zeros of Airy functions, Z. Angew. Math. Mech, Vol. 76, No.7, 1996, pp. 419-422.
[15] W. Chen, A Newton method without evaluation of nonlinear function
values, CoRR cs.CE/9906011, 1999.
[16] T.N. Grapsa, E.N. Malihoutsaki, Newton's method without direct
function evaluations, in: E. Lipitakis (Ed.), Proceedings of 8th Hellenic European Research on Computer Mathematics & its Applications-Conference, HERCMA 2007, 2007.
[17] E.N. Malihoutsaki, I.A. Nikas, T.N. Grapsa, Improved Newton's method
without direct function evaluations, Journal of Computational and
Applied Mathematics, Vol. 227 , 2009, pp. 206-212.
[18] T.N. Grapsa, Implementing the initialization-dependence and the
singularity difficulties in Newton's method, Tech. Rep. 07-03, Division
of Computational Mathematics and Informatics, Department of
Mathematics, University of Patras, 2007.
[19] T.N. Grapsa, M.N. Vrahatis, A dimension-reducing method for solving
systems of nonlinear equations in ¶Çü£n , Int. J. Comput. Math, Vol. 32 ,
1990, pp. 205-216.
[20] T.N. Grapsa, M.N. Vrahatis, A dimension-reducing method for
unconstrained optimization, J. Comput. Appl. Math, Vol. 66, No.1-2,
1996, pp. 239-253.
[21] D.G. Sotiropoulos, J.A. Nikas, T.N. Grapsa, Improving the efficiency of
a polynomial system solver via a reordering technique, in: D.T. Tsahalis (Ed.), Proceedings of 4th GRACM Congress on Computational
Mechanics, Vol. III, 2002.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:62537", author = "Jafar Biazar and Behzad Ghanbari", title = "A Modification on Newton's Method for Solving Systems of Nonlinear Equations", abstract = "In this paper, we are concerned with the further study for system of nonlinear equations. Since systems with inaccurate function values or problems with high computational cost arise frequently in science and engineering, recently such systems have attracted researcher-s interest. In this work we present a new method which is independent of function evolutions and has a quadratic convergence. This method can be viewed as a extension of some recent methods for solving mentioned systems of nonlinear equations. Numerical results of applying this method to some test problems show the efficiently and reliability of method.
", keywords = "System of nonlinear equations.", volume = "3", number = "10", pages = "862-5", }
