A novel method for robust minimisation of univariate functions with quadratic convergence

We describe a novel method for minimisation of univariate functions which exhibits an essentially quadratic convergence and whose convergence interval is only limited by the existence of near maxima. Minimisation is achieved through a fixed-point iterative algorithm, involving only the first and second-order derivatives, that eliminates the effects of near inflexion points on convergence, as usually observed in other minimisation methods based on the quadratic approximation. Comparative numerical studies against the standard quadratic and Brent's methods demonstrate clearly the high robustness, high precision and convergence rate of the new method, even when a finite difference approximation is used in the evaluation of the second-order derivative.