A Function-Conditioned Neural Framework for High-Order Caputo Fractional King–Neta Root Estimator with Computational Analysis

Fractional iterative methods provide a flexible extension of the classical root-finding methods for nonlinear equations. These methods can also offer control through the fractional-order parameter and enhance convergence and robustness for challenging nonlinear models. In the present work, three high-order multipoint schemes are presented in the Caputo fractional framework, extending the classical multipoint schemes of King and Neta to the fractional setting within a fractional Newton framework. The proposed schemes are the two-step Caputo fractional King method (CFKM2), the three-step Caputo fractional King method (CFKM3), and the three-step Caputo fractional Neta method (CFNM3). By employing the local convergence analysis based on the generalized Caputo Taylor expansion, it is demonstrated that the methods achieve the convergence orders of 4ϱ, 8ϱ and 6ϱ, respectively. The main advantage of CFNM3 is that it does not require any extra evaluation of the Caputo derivative in the last correction step, which reduces the computational cost. The proposed schemes are numerically compared in terms of convergence behavior, accuracy, and computational efficiency for different fractional orders. In addition to the local convergence analysis, the study further explores the global dynamics of CFKM3 using complex basins of attraction, polynomiographs and fractional-order sensitivity maps. The basin data is then input into a single function-conditioned neural network trained by a BFGS quasi-Newton optimization routine to estimate the coordinates of attracting roots. When ϱ=0.999, the three representative complex test problems show local convergence of CFKM3 in just two iterations, and full-grid convergence rates are between 85.81% and 90.24%. The Caputo derivative-based descriptors are found to be the most important for the network performance according to the predictive analysis.