Sains Malaysiana 49(7)(2020): 1755-1764

http://dx.doi.org/10.17576/jsm-2020-4907-25

 

Suatu Kelas Kaedah Optimum Bak Steffensen Bebas Terbitan untuk Punca Berganda
(A Class of Steffensen-Like Optimal Derivative-Free Method for Multiple Roots)

 

SYAHMI AFANDI SARIMAN & ISHAK HASHIM*

 

Jabatan Sains Matematik, Fakulti Sains dan Teknologi, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor Darul Ehsan, Malaysia

 

Diserahkan: 24 Mac 2020/Diterima: 10 April 2020

 

ABSTRAK

Objektif utama makalah ini adalah untuk mencari punca berganda bagi persamaan tak linear. Kaedah lelaran tiga tahap diubah suai menjadi bebas terbitan yang mengekalkan peringkat penumpuan optimum lapan. Skema lelaran bebas terbitan dibangunkan berdasarkan kaedah bak Steffensen dan konsep beza terhingga. Kaedah yang diubah suai ini selaras dengan penumpuan optimum mengikut konjektur Kung Traub yang dibuktikan melalui analisis penumpuan. Skema lelaran ini dapat bersaing dengan kaedah lelaran sedia ada daripada segi kebebasan terbitan. Indeks keberkesanan telah mencapai nilai  dan lebih baik daripada kaedah Newton klasik, . Beberapa ujian berangka dilakukan dalam menentukan keberkesanan skema lelaran yang dibangunkan bagi mencari punca berganda mahupun punca mudah.

Kata kunci: Bebas terbitan; kaedah lelaran; penumpuan optimum; persamaan tak linear; punca berganda

 

ABSTRACT

The main objective of this paper was to find multiple roots for nonlinear equations. The three-step iteration method is modified to be derivative free which maintains an optimum convergence of eight. The derivative-free iteration scheme was developed based on the Steffensen-like method and finite difference concept. The modified method satisfies the optimal convergence of Kung-Traub’s conjectures as shown in the convergence analysis. The iteration scheme can compete with the existing iteration methods in terms of free derivatives. The efficiency index has reached the value  and is better than the classical Newton method, . Numerical experiments have been done to determine the effectiveness of the iteration scheme in finding multiple roots and also simple roots.

Keywords: Derivative-free; iterative method; multiple roots; nonlinear equation; optimal convergence

 

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*Pengarang untuk surat-menyurat; email: ishak_h@ukm.edu.my

 

 

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