2012/10/21 by Ababu Teklemariam Tiruneh, Tiruneh, Ababu Teklemariam
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #FOS: Mathematics #Fractional Differential Equations Solutions #Iterative Methods for Nonlinear Equations #Matrix Theory and Algorithms #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1210.5766
openalex publication_date 2012/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An iterative formula based on Newton Method alone is presented for the\niterative solutions of equations that ensures convergence in cases where the\ntraditional Newton Method may fail to converge to the desired root. In\naddition, the method has super quadratic convergence of order 2.414. Newton\nmethod is said to fail in certain cases leading to oscillation, divergence to\nincreasingly large number or off-shooting away to another root further from the\ndesired domain or off shooting to an invalid domain where the function may not\nbe defined. In addition when the derivative at the iteration point is zero,\nNewton method stalls. In most of these cases, hybrids of several methods such\nas Newton, bisection and secant methods are suggested as substitute methods and\nNewton method is essentially blended with other methods or altogether\nabandoned. This paper argues that a solution is still possible in most of these\ncases by the application of Newton Method alone without resorting to other\nmethods and with the same computational effort, two functional evaluations per\niteration, like the traditional Newton method. In addition, the proposed\nmodified formula based on Newton method has better convergence characteristics\nthan the traditional Newton method.\n