A Two-Point Newton Method Suitable for Nonconvergent Cases and with Super-Quadratic Convergence. (19th March 2013)

• Record Type: Journal Article
• Authors: Tiruneh, Ababu Teklemariam; Ndlela, W. N.; Nkambule, S. J.
• Other Names: Benzi Michele Academic Editor.
• Is Part Of: Advances in numerical analysis. Volume 2013(2013)
• Publication Date: 2013-03-19
• Subjects: Numerical analysis -- Periodicals; Numerical analysis; Periodicals; Electronic journals; 518
• DOI: 10.1155/2013/687382
• Languages: English
• ISSNs: 1687-9562

Abstract

Abstract : An iterative formula based on Newton's method alone is presented for the iterative solutions of equations that ensures convergence in cases where the traditional Newton Method may fail to converge to the desired root. In addition, the method has super-quadratic convergence of order 2.414 (i.e., 1 + 2 ). Newton method is said to fail in certain cases leading to oscillation, divergence to increasingly large number, or offshooting away to another root further from the desired domain or offshooting to an invalid domain where the function may not be defined. In addition when the derivative at the iteration point is zero, Newton method stalls. In most of these cases, hybrids of several methods such as Newton, bisection, and secant methods are suggested as substitute methods and Newton method is essentially blended with other methods or altogether abandoned. This paper argues that a solution is still possible in most of these cases by the application of Newton method alone without resorting to other methods and with the same computational effort (two functional evaluations per iteration) like the traditional Newton method. In addition, the proposed modified formula based on Newton method has better convergence characteristics than the traditional Newton method.