Some generalized recurrence relations for nonlinear equations by using decomposition technique with application of fractal geometry

Nonlinear equations frequently appear in diverse fields of applied sciences, where real-world phenomena cannot be accurately represented by linear models. Therefore, developing efficient numerical methods to approximate the roots of such equations remain a challenging and intellectually stimulating task. These methods are crucial in physics, engineering and computer science for solving nonlinear equations. In response to the growing demands of real-time systems, complicated simulations and high-performance computing, this article introduces few novel root-finding methods that significantly improve the convergence order of the traditional approaches. Accelerated decomposition technique is to diversify different classes of iterative methods. Newly derived methods are compared with existing methods numerically as well as graphically. Polynomiography is employed to visualize the basins of attraction, providing insight into the convergence behavior and stability of the methods. The results indicate that the new algorithms not only overcome the limitations of existing techniques but also offer a visually intuitive understanding of root-finding processes. This study presents innovative root-finding methods that utilize accelerated decomposition techniques. The proposed methods demonstrate a significant improvement in convergence order compared to traditional approaches Through numerical and graphical comparisons, the newly derived methods are shown to outperform existing methods. <DispFormula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>=</mml:mo> <mml:mspace></mml:mspace> <mml:msub><mml:mi>x</mml:mi> <mml:mi>n</mml:mi></mml:msub> <mml:mo>-</mml:mo> <mml:mfrac><mml:mrow><mml:mi>f</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:msub><mml:mi>x</mml:mi> <mml:mi>n</mml:mi></mml:msub> <mml:mo>)</mml:mo></mml:mrow> <mml:mi>p</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:msub><mml:mi>x</mml:mi> <mml:mi>n</mml:mi></mml:msub> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> <mml:mrow><mml:msup><mml:mi>p</mml:mi> <mml:mo>'</mml:mo></mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:msub><mml:mi>x</mml:mi> <mml:mi>n</mml:mi></mml:msub> <mml:mo>)</mml:mo></mml:mrow> <mml:mi>f</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:msub><mml:mi>x</mml:mi> <mml:mi>n</mml:mi></mml:msub> <mml:mo>)</mml:mo></mml:mrow> <mml:mo>+</mml:mo> <mml:msup><mml:mi>f</mml:mi> <mml:mo>'</mml:mo></mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:msub><mml:mi>x</mml:mi> <mml:mi>n</mml:mi></mml:msub> <mml:mo>)</mml:mo></mml:mrow> <mml:mi>p</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:msub><mml:mi>x</mml:mi> <mml:mi>n</mml:mi></mml:msub> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:math></DispFormula>.