Research Article
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Year 2020, Volume: 49 Issue: 1, 425 - 438, 06.02.2020
https://doi.org/10.15672/hujms.459810

Abstract

References

  • [1] S. Abbasbandy, Improving Newton-Raphson method for nonlinear equations modified Adomian decomposition method, Appl. Math. Comput. 145, 887–893, 2003.
  • [2] G. Alefeld, On the convergence of Halley’s method, Amer. Math. Monthly, 8, 530–536, 1981.
  • [3] S. Amat and S. Busquier, Advances in iterative methods for nonlinear equations, Springer, 2016.
  • [4] G. Ardelean, A new third-order Newton-type iterative method for solving nonlinear equations, Appl. Math. Comput. 219, 9856–9864, 2013.
  • [5] J. Chen, A new family of exponential iteration methods with quadratic convergence of both diameters and points for enclosing zeros of nonlinear equations, Math. Comput. Model. 45, 544–552, 2007.
  • [6] C. Chun, A geometric construction of iterative formulas of order three, Appl. Math. Lett. 23, 512–516, 2010.
  • [7] C. Chun and Y. Ham, A one-parameter fourth-order family of iterative methods for nonlinear equations, Appl. Math. Comput. 189, 610–614, 2007.
  • [8] C. Chun and Y. Ham, Some fourth-order modifications of Newtons method, Appl. Math. Comput. 197, 654–658, 2008.
  • [9] A. Cordero and J.R. Torregrosa, Low-complexity root-finding iteration functions with no derivatives of any order of convergence, J. Comput. Appl. Math. 275, 502–515, 2015.
  • [10] A. Cordero, J.L. Hueso, E. Martinez and J.R. Torregrosa, Efficient high-order methods based on golden ratio for nonlinear systems, Appl. Math. Comput. 217, 4548–4556, 2011.
  • [11] M. Dehghan and M. Hajarian, Some derivative free quadratic and cubic convergence iterative formulaa for solving nonlinear equations, Comp. & App. Math. 29, 19–30, 2010.
  • [12] T. Fang, F. Guo and C.F.F. Lee, A new iteration method with cubic convergence to solve nonlinear algebraic equations, Appl. Math. Comput. 175, 1147–1155, 2006.
  • [13] W.J. Gilbert, Generalizations of Newton’s method, Fractals, 9, 251–262, 2001.
  • [14] Y.M. Hama, C. Chunb and S.G. Leeb, Some higher-order modifications of Newton’s method for solving nonlinear equations, J. Comput. Appl. Math. 222, 477–486, 2008.
  • [15] D. Herceg and D. Herceg, Sixth-order modifications of Newton’s method based on Stolarsky and Gini means, J. Comput. Appl. Math. 267, 244–253, 2014.
  • [16] A.S. Householder, Principles of numerical analysis, Dover, New-York, 1974.
  • [17] P. Jarratt, Some efficient fourth order multipoint methods for solving equations, BIT, 9, 119–124, 1969.
  • [18] R. Kalaba and A. Tishler, A generalized Newton algorithm using higher-order derivatives, J. Optim. Theory Appl. 39, 1–16, 1983.
  • [19] J. Kou, The improvements of modified Newton’s method, Appl. Math. Comput. 189, 602–609, 2007.
  • [20] G. Labelle, On extensions of the Newton-Raphson iterative scheme to arbitrary orders, Discrete Math. Theoret. Comput. Sci. proc. AN, 713–724, 2010.
  • [21] T.J. McDougalla and S.J. Wotherspoon, A simple modification of Newton’s method to achieve convergence of order $1+\sqrt{2}$, Appl. Math. Lett. 29, 20–25, 2014.
  • [22] M. Pakdemirli and H. Boyaci, Generation of root finding algorithms via perturbation theory and some formulas, Appl. Math. Comput. 184, 783–788, 2007.
  • [23] M.S. Petković, B. Neta, L.D. Petković, and J. Džunić, Multipoint methods for solving nonlinear equations, Elsevier, 2013.
  • [24] H. Ramosa and J.V. Aguiar, The application of Newton’s method in vector form for solving nonlinear scalar equations where the classical Newton method fails, J. Comput. Appl. Math. 275, 228–237, 2015.
  • [25] F.A. Shah and M.A. Noor, Some numerical methods for solving nonlinear equations by using decomposition technique, Appl. Math. Comput. 251, 378–386, 2015.
  • [26] M. Sharifi, D.K.R. Babajee and F. Soleymani, Finding the solution of nonlinear equations by a class of optimal methods, Computers and Mathematics with Applications, 63, 764–774, 2012.
  • [27] M.K. Singh, A six-order variant of Newtons method for solving nonlinear equations, Computational Methods in Science and Technology, 15, 185–193, 2009.
  • [28] H. Susanto and N. Karjanto, Newtons methods basins of attraction revisited, Appl. Math. Comput. 215, 1084–1090, 2009.
  • [29] J.F. Traub, Iterative methods for the solution of equations, Chelsea Publishing Company, New York, 1997.
  • [30] P. Wang, A third-order family of Newton-like iteration methods for solving nonlinear equations, J. Numer. Math. Stoch. 3, 13–19, 2011.

A simple algorithm for high order Newton iteration formulae and some new variants

Year 2020, Volume: 49 Issue: 1, 425 - 438, 06.02.2020
https://doi.org/10.15672/hujms.459810

Abstract

The high order Newton iteration formulas are revisited in this paper. Translating the nonlinear root finding problem into a fixed point iteration involving an unknown general function whose root is searched, a double Taylor series is undertaken regarding the root and the root finding function. Based on the error analysis of the expansion, a simple algorithm is later proposed to construct Newton iteration formulae of any order commencing from the traditional linearly convergent fixed point iteration method and quadratically convergent Newton-Raphson method of frequently at the disposal of the scientific community. It is shown that the well-known variants like the Halley's method or Haouseholder's methods of high order can be reproduced from the general case outlined here. Some further rare single-step classes of any order are shown to be derivable from the presented algorithm. Finally, some new higher order accurate variants are also offered taking into account multi-step compositions which demand less computation of higher derivatives. The efficiency, accuracy and performance of the proposed methods and also their potential advantages over the classical ones are numerically demonstrated and discussed on some well-documented examples from the open literature.

References

  • [1] S. Abbasbandy, Improving Newton-Raphson method for nonlinear equations modified Adomian decomposition method, Appl. Math. Comput. 145, 887–893, 2003.
  • [2] G. Alefeld, On the convergence of Halley’s method, Amer. Math. Monthly, 8, 530–536, 1981.
  • [3] S. Amat and S. Busquier, Advances in iterative methods for nonlinear equations, Springer, 2016.
  • [4] G. Ardelean, A new third-order Newton-type iterative method for solving nonlinear equations, Appl. Math. Comput. 219, 9856–9864, 2013.
  • [5] J. Chen, A new family of exponential iteration methods with quadratic convergence of both diameters and points for enclosing zeros of nonlinear equations, Math. Comput. Model. 45, 544–552, 2007.
  • [6] C. Chun, A geometric construction of iterative formulas of order three, Appl. Math. Lett. 23, 512–516, 2010.
  • [7] C. Chun and Y. Ham, A one-parameter fourth-order family of iterative methods for nonlinear equations, Appl. Math. Comput. 189, 610–614, 2007.
  • [8] C. Chun and Y. Ham, Some fourth-order modifications of Newtons method, Appl. Math. Comput. 197, 654–658, 2008.
  • [9] A. Cordero and J.R. Torregrosa, Low-complexity root-finding iteration functions with no derivatives of any order of convergence, J. Comput. Appl. Math. 275, 502–515, 2015.
  • [10] A. Cordero, J.L. Hueso, E. Martinez and J.R. Torregrosa, Efficient high-order methods based on golden ratio for nonlinear systems, Appl. Math. Comput. 217, 4548–4556, 2011.
  • [11] M. Dehghan and M. Hajarian, Some derivative free quadratic and cubic convergence iterative formulaa for solving nonlinear equations, Comp. & App. Math. 29, 19–30, 2010.
  • [12] T. Fang, F. Guo and C.F.F. Lee, A new iteration method with cubic convergence to solve nonlinear algebraic equations, Appl. Math. Comput. 175, 1147–1155, 2006.
  • [13] W.J. Gilbert, Generalizations of Newton’s method, Fractals, 9, 251–262, 2001.
  • [14] Y.M. Hama, C. Chunb and S.G. Leeb, Some higher-order modifications of Newton’s method for solving nonlinear equations, J. Comput. Appl. Math. 222, 477–486, 2008.
  • [15] D. Herceg and D. Herceg, Sixth-order modifications of Newton’s method based on Stolarsky and Gini means, J. Comput. Appl. Math. 267, 244–253, 2014.
  • [16] A.S. Householder, Principles of numerical analysis, Dover, New-York, 1974.
  • [17] P. Jarratt, Some efficient fourth order multipoint methods for solving equations, BIT, 9, 119–124, 1969.
  • [18] R. Kalaba and A. Tishler, A generalized Newton algorithm using higher-order derivatives, J. Optim. Theory Appl. 39, 1–16, 1983.
  • [19] J. Kou, The improvements of modified Newton’s method, Appl. Math. Comput. 189, 602–609, 2007.
  • [20] G. Labelle, On extensions of the Newton-Raphson iterative scheme to arbitrary orders, Discrete Math. Theoret. Comput. Sci. proc. AN, 713–724, 2010.
  • [21] T.J. McDougalla and S.J. Wotherspoon, A simple modification of Newton’s method to achieve convergence of order $1+\sqrt{2}$, Appl. Math. Lett. 29, 20–25, 2014.
  • [22] M. Pakdemirli and H. Boyaci, Generation of root finding algorithms via perturbation theory and some formulas, Appl. Math. Comput. 184, 783–788, 2007.
  • [23] M.S. Petković, B. Neta, L.D. Petković, and J. Džunić, Multipoint methods for solving nonlinear equations, Elsevier, 2013.
  • [24] H. Ramosa and J.V. Aguiar, The application of Newton’s method in vector form for solving nonlinear scalar equations where the classical Newton method fails, J. Comput. Appl. Math. 275, 228–237, 2015.
  • [25] F.A. Shah and M.A. Noor, Some numerical methods for solving nonlinear equations by using decomposition technique, Appl. Math. Comput. 251, 378–386, 2015.
  • [26] M. Sharifi, D.K.R. Babajee and F. Soleymani, Finding the solution of nonlinear equations by a class of optimal methods, Computers and Mathematics with Applications, 63, 764–774, 2012.
  • [27] M.K. Singh, A six-order variant of Newtons method for solving nonlinear equations, Computational Methods in Science and Technology, 15, 185–193, 2009.
  • [28] H. Susanto and N. Karjanto, Newtons methods basins of attraction revisited, Appl. Math. Comput. 215, 1084–1090, 2009.
  • [29] J.F. Traub, Iterative methods for the solution of equations, Chelsea Publishing Company, New York, 1997.
  • [30] P. Wang, A third-order family of Newton-like iteration methods for solving nonlinear equations, J. Numer. Math. Stoch. 3, 13–19, 2011.
There are 30 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Mustafa Turkyilmazoglu 0000-0003-0412-4580

Publication Date February 6, 2020
Published in Issue Year 2020 Volume: 49 Issue: 1

Cite

APA Turkyilmazoglu, M. (2020). A simple algorithm for high order Newton iteration formulae and some new variants. Hacettepe Journal of Mathematics and Statistics, 49(1), 425-438. https://doi.org/10.15672/hujms.459810
AMA Turkyilmazoglu M. A simple algorithm for high order Newton iteration formulae and some new variants. Hacettepe Journal of Mathematics and Statistics. February 2020;49(1):425-438. doi:10.15672/hujms.459810
Chicago Turkyilmazoglu, Mustafa. “A Simple Algorithm for High Order Newton Iteration Formulae and Some New Variants”. Hacettepe Journal of Mathematics and Statistics 49, no. 1 (February 2020): 425-38. https://doi.org/10.15672/hujms.459810.
EndNote Turkyilmazoglu M (February 1, 2020) A simple algorithm for high order Newton iteration formulae and some new variants. Hacettepe Journal of Mathematics and Statistics 49 1 425–438.
IEEE M. Turkyilmazoglu, “A simple algorithm for high order Newton iteration formulae and some new variants”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 1, pp. 425–438, 2020, doi: 10.15672/hujms.459810.
ISNAD Turkyilmazoglu, Mustafa. “A Simple Algorithm for High Order Newton Iteration Formulae and Some New Variants”. Hacettepe Journal of Mathematics and Statistics 49/1 (February 2020), 425-438. https://doi.org/10.15672/hujms.459810.
JAMA Turkyilmazoglu M. A simple algorithm for high order Newton iteration formulae and some new variants. Hacettepe Journal of Mathematics and Statistics. 2020;49:425–438.
MLA Turkyilmazoglu, Mustafa. “A Simple Algorithm for High Order Newton Iteration Formulae and Some New Variants”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 1, 2020, pp. 425-38, doi:10.15672/hujms.459810.
Vancouver Turkyilmazoglu M. A simple algorithm for high order Newton iteration formulae and some new variants. Hacettepe Journal of Mathematics and Statistics. 2020;49(1):425-38.

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