A Two-Point Newton Method Suitable for Nonconvergent Cases and with Super-Quadratic Convergence

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., ). 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. 1. Introduction Iterative procedures for solutions of equations are routinely employed in many science and engineering problems. Starting with the classical Newton methods, a number of methods for finding roots of equations have come to exist, each of which has its own advantages and limitations. The Newton method of root finding is based on the iterative formula: Newton’s method displays a faster quadratic convergence near the root while it requires evaluation of the function and its derivative at each step of the iteration. However, when the derivative evaluated is zero, Newton method stalls. For low values of the derivative, the Newton iteration offshoots away from the current point of iteration and may possibly converge to a root far away from the intended domain. For certain forms of equations, Newton method diverges or oscillates and fails to converge to the desired root. In addition, the convergence of Newton method can be slow near roots of multiplicity although modifications can be made to increase the rate of convergence [1]. Modifications of the Newton method with higher order convergence have been proposed that require also evaluation of a function and its derivatives. An example of such methods is a third order convergence method by Weerakoon and Fernando [2] that requires evaluation of one