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A Computationally Efficient Sixth-Order Method for Nonlinear Models

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Frontiers in Industrial and Applied Mathematics (FIAM 2021)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 410))

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Abstract

The aim of the present study is to develop an iterative scheme of high convergence order with minimal computational cost. With this objective, a three-step method has been designed by utilizing only two Jacobian matrices, single matrix inversion, and three function evaluations. Under some standard assumptions, the proposed method is found to possess the sixth order of convergence. The iterative schemes with these characteristics are hardly found in the literature. The analysis is carried out to assess the computational efficiency of the proposed method, and further, outcomes are compared with the efficiencies of existing ones. In addition, numerical experiments are performed by applying the method to some practical nonlinear problems. The entire analysis remarkably favors the new technique compared with existing counterparts in terms of computational efficiency, stability, and CPU time elapsed during execution.

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References

  1. Avazzadeh, Z., Heydari, M., Loghmani, G.B.: Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem. Appl. Math. Model. 35, 2374–2383 (2011). https://doi.org/10.1016/j.apm.2010.11.056

    Article  MATH  Google Scholar 

  2. Bahl, A., Cordero, A., Sharma, R., Torregrosa, J.R.: A novel bi-parametric sixth order iterative scheme for solving nonlinear systems and its dynamics. Appl. Math. Comput. 357, 147–166 (2019). https://doi.org/10.1016/j.amc.2019.04.003

    Article  MATH  Google Scholar 

  3. Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: Efficient high-order methods based on golden ratio for nonlinear systems. Appl. Math. Comput. 217, 4548–4556 (2011). https://doi.org/10.1016/j.amc.2010.11.006

    Article  MATH  Google Scholar 

  4. Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: Increasing the convergence order of an iterative method for nonlinear systems. Appl. Math. Lett. 25, 2369–2374 (2012). https://doi.org/10.1016/j.aml.2012.07.005

    Article  MATH  Google Scholar 

  5. Esmaeili, H., Ahmadi, M.: An efficient three-step method to solve system of nonlinear equations. Appl. Math. Comput. 266, 1093–1101 (2015). https://doi.org/10.1016/j.amc.2015.05.076

    Article  MATH  Google Scholar 

  6. Lotfi, T., Bakhtiari, P., Cordero, A., Mahdiani, K., Torregrosa, J.R.: Some new efficient multipoint iterative methods for solving nonlinear systems of equations. Int. J. Comput. Math. 92, 1921–1934 (2014). https://doi.org/10.1080/00207160.2014.946412

    Article  MATH  Google Scholar 

  7. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    MATH  Google Scholar 

  8. Ostrowski, A.M.: Solution of Equation and Systems of Equations. Academic Press, New York (1960)

    MATH  Google Scholar 

  9. Potra, F.A., Pták, V.: On a class of modified Newton processes. Numer. Funct. Anal. Optim. 2, 107–120 (1980). https://doi.org/10.1080/01630568008816049

    Article  MATH  Google Scholar 

  10. Sharma, J.R., Gupta, P.: An efficient fifth order method for solving systems of nonlinear equations. Comput. Math. Appl. 67, 591–601 (2014). https://doi.org/10.1016/j.camwa.2013.12.004

    Article  MATH  Google Scholar 

  11. Sharma, R., Sharma, J.R., Kalra, N.: A modified Newton-Özban composition for solving nonlinear systems. Int. J. Comput. Methods 17, 1950047 (2020). https://doi.org/10.1142/S0219876219500476

    Article  MATH  Google Scholar 

  12. Soleymani, F., Lotfi, T., Bakhtiari, P.: A multi-step class of iterative methods for nonlinear systems. Optim. Lett. 8, 1001–1015 (2014). https://doi.org/10.1007/s11590-013-0617-6

    Article  MATH  Google Scholar 

  13. Traub, J.F.: Iterative Methods for the Solution of Equations. Chelsea Publishing Company, New York (1982)

    MATH  Google Scholar 

  14. Wang, X., Kou, J., Gu, C.: Semilocal convergence of a sixth-order Jarratt method in Banach spaces. Numer. Algor. 57, 441–456 (2011). https://doi.org/10.1007/s11075-010-9438-1

    Article  MATH  Google Scholar 

  15. Wolfram, S.: The Mathematica Book. Wolfram Media, USA (2003)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, Punjab, 148106, India

    Janak Raj Sharma & Harmandeep Singh

Authors
  1. Janak Raj Sharma
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  2. Harmandeep Singh
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Corresponding author

Correspondence to Harmandeep Singh .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Institute of Technology, Nirma University, Ahmedabad, India

    Rajesh Kumar Sharma

  2. Dipartimento Di Matematica, University of Ferrara, Ferrara, Italy

    Lorenzo Pareschi

  3. Institute for Groundwater Studies, University of the Free State, Bloemfontein, South Africa

    Abdon Atangana

  4. Department of Mathematics, National Institute of Technology Rourkela, Rourkela, Odisha, India

    Bikash Sahoo

  5. Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, India

    Vijay Kumar Kukreja

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Sharma, J.R., Singh, H. (2023). A Computationally Efficient Sixth-Order Method for Nonlinear Models. In: Sharma, R.K., Pareschi, L., Atangana, A., Sahoo, B., Kukreja, V.K. (eds) Frontiers in Industrial and Applied Mathematics. FIAM 2021. Springer Proceedings in Mathematics & Statistics, vol 410. Springer, Singapore. https://doi.org/10.1007/978-981-19-7272-0_39

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