A Two-Point Newton Method suitable for non-convergent Cases and with Super-Quadratic Convergence

TL;DR

This paper introduces a modified Newton method that guarantees convergence in cases where the traditional Newton method fails, and achieves super-quadratic convergence without hybridization.

Contribution

A new iterative formula based solely on Newton's method is proposed, ensuring convergence and super-quadratic convergence in challenging cases where traditional methods fail.

Findings

Ensures convergence where Newton method fails

Achieves super-quadratic convergence of order 2.414

Requires same computational effort as traditional Newton method

Abstract

An iterative formula based on Newton 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. Newton method is said to fail in certain cases leading to oscillation, divergence to increasingly large number or off-shooting away to another root further from the desired domain or off shooting 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.