| Citation: |
Shahid Abdullah, Neha Choubey, Suresh Dara. DYNAMICAL ANALYSIS OF OPTIMAL ITERATIVE METHODS FOR SOLVING NONLINEAR EQUATIONS WITH APPLICATIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3349-3376. doi: 10.11948/20240009
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DYNAMICAL ANALYSIS OF OPTIMAL ITERATIVE METHODS FOR SOLVING NONLINEAR EQUATIONS WITH APPLICATIONS
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Abstract
In this study, we introduced a new family of two- and three-step iterative methods for solving non-linear equations. The proposed methods adhere to the Kung and Traub conjecture, making them optimal as they require only three function evaluations for a fourth-order method and four function evaluations for an eighth-order method per cycle. To achieve a fourth-order method, we employed linear combination technique merging Xiaojan's method with Yu and Xu's method, while for an eighth-order method, we utilized weight function approach. The convergence criteria of the proposed schemes are thoroughly covered in the two primary theorems. Through comparative analysis with existing methods using various nonlinear models and test functions, we conducted extensive numerical investigations to demonstrate the superior performance and efficacy of our proposed techniques. Furthermore, we explored the fractal behavior of our technique and other existing methods by considering different forms of complex functions within the basins of attraction.
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References
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Shahid Abdullah, Neha Choubey, Suresh Dara. DYNAMICAL ANALYSIS OF OPTIMAL ITERATIVE METHODS FOR SOLVING NONLINEAR EQUATIONS WITH APPLICATIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3349-3376. doi: 10.11948/20240009
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Figure 1.
Basins of attraction for
$ CC4 $ $ CH4 $ $ JJ4 $ $ AN4 $ $ MH4 $ $ SQ4 $ $ SA4 $ $ p_1(z) $ -
Figure 2.
Basins of attraction for
$ CC4 $ $ CH4 $ $ JJ4 $ $ AN4 $ $ MH4 $ $ SQ4 $ $ SA4 $ $ p_2(z) $ -
Figure 3.
Basins of attraction for
$ CC4 $ $ CH4 $ $ JJ4 $ $ AN4 $ $ MH4 $ $ SQ4 $ $ SA4 $ $ p_3(z) $ -
Figure 4.
Basins of attraction for
$ JH8 $ $ JM8 $ $ SP8 $ $ SJ8 $ $ MH8 $ $ RS8 $ $ SA8 $ $ p_1(z) $ -
Figure 5.
Basins of attraction for
$ JH8 $ $ JM8 $ $ SP8 $ $ SJ8 $ $ MH8 $ $ RS8 $ $ SA8 $ $ p_2(z) $ -
Figure 6.
Basins of attraction for
$ JH8 $ $ JM8 $ $ SP8 $ $ SJ8 $ $ MH8 $ $ RS8 $ $ SA8 $ $ p_3(z) $ -
Figure 7.
Graphical comparisons of basins of attraction for
$ CC4 $ $ CH4 $ $ JJ4 $ $ AN4 $ $ SQ4 $ $ SA4 $ -
Figure 8.
Graphical comparisons of basins of attraction for
$ JH8 $ $ JM8 $ $ SP8 $ $ SJ8 $ $ MH8 $ $ RS8 $ $ SA8 $ -
Figure 9.
Comparison of fourth-order methods based on the error in consecutive iterations,
$ |w_n- w_{n-1}| $ $ g_1(w) $ $ g_8(w) $ -
Figure 10.
Comparison of eighth-order methods based on the error in consecutive iterations,
$ |w_n- w_{n-1}| $ $ g_1(w) $ $ g_8(w) $