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New Modification of Behl's Method Free from Second Derivative with an Optimal Order of Convergence

Wartono Wartono, Revia Agustiwari, Rahmawati Rahmawati

Abstract


Abstract

Behl’s method is one of the iterative methods to solve a nonlinear equation that converges cubically. In this paper, we modified the iterative method with real parameter β using second Taylor’s series expansion and reduce the second derivative of the proposed method using the equality of Chun-Kim and Newton Steffensen. The result showed that the proposed method has a fourth-order convergence for b = 0 and involves three evaluation functions per iteration with the efficiency index equal to 41/3 = 1.5874. Numerical simulation is presented for several functions to demonstrate the performance of the new method. The final results show that the proposed method has better performance as compared to some other iterative methods.

Keywords: efficiency index; third-order iterative method; Chun-Kim’s method; Newton-Steffensen’s method; nonlinear equation.

 

Abstrak

Metode Behl adalah salah satu metode iterasi yang digunakan untuk menyelesaikan persamaan nonlinear dengan orde konvergensi tiga. Pada artikel ini, modifikasi terhadap metode iterasi menggunakan ekspansi deret Taylor orde dua dengan parameter β  dan turunan kedua dihilangkan menggunakan penyetaraan dari metode Chun-Kim dan Newton-Steffensen. Hasil kajian menunjukkan bahwa metode iterasi yang diusulkan memiliki orde konvergensi empat untuk b = 0 dan melibatkan tiga evaluasi fungsi setiap iterasinya dengan indeks efisiensi sebesar 41/3 = 1,5874. Simulasi numerik dilakukan terhadap beberapa fungsi untuk menunjukkan performa modifikasi metode iterasi yang diusulkan. Hasil akhir menunjukkan bahwa metode iterasi tersebut mempunyai performa lebih baik dibandingkan dengan beberapa metode iterasi lainnya.

Kata kunci: indeks efisiensi; metode iterasi orde tiga; metode Chun-Kim; metode Newton- Steffensen; persamaan nonlinear.

References


S. Chapra and R. Canale, Numerical Methods for Engineering, New York: Mc-Graw-Hill, Inc., 2010.

J. Traub, Iterative Method do Solution of Equations, New York: Chelsea Publishing Company, 1964.

A. Melman, "Geometry, and convergence of Euler’s and Halley’s methods," SIAM Review, vol. 39, no. 4, pp. 728-735, 1997.

J. Sharma, "A family of third-order methods to solve nonlinear equations by quadratic curves approximation," Applied Mathematics, and Computation, vol. 184, pp. 210-215, 2007.

S. Amat, S. Busquier and J. M. Gutierrez, "Geometric construction of iterative function to solve nonlinear equations," Journal of Computational and Applied Mathematics, vol. 157, pp. 197-205, 2003.

S. Amat, S. Busquier, J. Guterrez and Hern, "On the global convergence of Chebyshev’s method," Journal of Computational and Applied Mathematics, vol. 220, pp. 17-21, 2008.

S. Abbasbandy, "Modified homotopy perturbation method for nonlinear equations and comparison with Adomian decomposition method," Applied Mathematics, and Computation,, vol. 172, pp. 431-438, 2006.

C. Chun, "Iterative methods improving Newton’s method by the decomposition method," Computers and Mathematics with Applications, vol. 50, pp. 1559-1568, 2005.

C. Chun and Y. Kim, "Several new third-order iterative methods for solving nonlinear equations," Acta Appl Math, vol. 109, pp. 1053-1063, 2010.

F. Shah and M. Noor, "Variational iteration technique and some methods for the approximate solution of nonlinear equations," Applied Mathematics and Informatics Sciences Letter, vol. 2, no. 3, pp. 85-93, 2014.

M. Baghat, "New two iterative methods for solving nonlinear equations," Journal of Mathematics Research, vol. 4, no. 3, p. 128–131, 2012.

Wartono and T. Nanda, "Modifikasi metode Baghat tanpa turunan kedua dengan orde konvergensi optimal," in Prosiding Seminar Nasional Teknologi Informasi, Komunikasi dan Industri IX, Pekanbaru, 2017.

H. Kung and J. Traub, "Optimal order of one-point and multipoint iteration," Journal of the Association for Computing Machinery, vol. 21, no. 4, p. 643–651, 1974.

R. Behl, V. Kanwar and K. Sharma, "Another simple way of deriving several iterative functions to solve nonlinear equations," Journal of Applied Mathematics, 2012.

J. Epperson, An Introduction to Numerical Methods and Analysis, New Jersey: John Willey & Son, Inc, 2013.

Wartono, M. Soleh, I. Suryani and Muhafzan, "Chebysev-Halley’s Method without second derivative of eighth-order of convergence," Global Journal of Pure and Applied Mathematics, vol. 12, no. 4, pp. 2987-2997, 2016.

J. Sharma, "A composite third-order Newton-Steffensen method for solving nonlinear equations," Applied Mathematics and Computation, vol. 169, pp. 242-246, 2005.


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DOI: 10.15408/inprime.v1i2.12787

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