A common framework for modified Regula Falsi methods and new methods of this kind

Abstract

We show that the modified Regula Falsi methods based on a scaling factor (Scaling Factor Regula Falsi, SFRF) have a common framework, and that the scaling factor depends only on two dimensionless parameters. This common framework allows for the comparison of different methods and their possible combination, without hybridization for treating some difficult cases, such as multiple roots. With this framework we prove, for all SFRF methods, the global convergence of the successive approximations to the root, and also the convergence of the bracketing interval radius to zero, for both simple and multiple roots. We show that SFRF methods only occasionally need a small scaling factor to make the radius of the interval go to zero. This way, the accumulation of the approximations to the root near an interval limit that some improvements of Regula Falsi methods suffer from is cured. As an example, new SFRF methods are exposed. One is specific for multiple roots of known multiplicity, and greatly outperforms pure bisection. Another, developed for simple roots, compares well with other methods belonging to numerical libraries that are widely used by the scientific community, and for multiple roots the new method outperforms them. The new framework could also allow the development of other SFRF methods overperforming the previously known ones.