An Enhancement of the Bisection Method Average Performance Preserving Minmax Optimality

Abstract

We identify a class of root-searching methods that surprisingly outperform the bisection method on the average performance while retaining minmax optimality. The improvement on the average applies for any continuous distributional hypothesis. We also pinpoint one specific method within the class and show that under mild initial conditions it can attain an order of convergence of up to 1.618, i.e., the same as the secant method. Hence, we attain both an improved average performance and an improved order of convergence with no cost on the minmax optimality of the bisection method. Numerical experiments show that, on regular functions, the proposed method requires a number of function evaluations similar to current state-of-the-art methods, about 24% to 37% of the evaluations required by the bisection procedure. In problems with non-regular functions, the proposed method performs significantly better than the state-of-the-art, requiring on average 82% of the total evaluations required for the bisection method, while the other methods were outperformed by bisection. In the worst case, while current state-of-the-art commercial solvers required two to three times the number of function evaluations of bisection, our proposed method remained within the minmax bounds of the bisection method.