A superlinear Scaling Factor Regula Falsi root finder that detects the simple or multiple character of the root

Abstract

In this work a new method of Scaling Factor Regula Falsi type is developed, by using parabolic interpolation. It has global convergence and a high computational efficiency for simple roots. The method, not being a hybrid one, allows changing the function used for calculating the scaling factor at every iteration, making possible to switch to other more adequate methods for multiple roots (e.g., a generalised Illinois one). As an important feature of the method, it allows determining with no additional calculations, whether the root is simple or multiple. The developed algorithm has been tested with numerous functions extracted from the bibliography, performing, in most cases, better than some routines (like brentq, brenth and toms748) found in common numerical libraries. Fully operative routines of the new method are provided as supplementary files in four different programming languages: python, lua, C and Fortran90.