A new homotopy for seeking all real roots of a nonlinear equation

Abstract

A new continuation method, which applies a new homotopy that is a combination of the fixed-point and Newton homotopies (FPN), is developed for seeking all real solutions to a nonlinear equation, written as f( x) = 0, without having to specify a bounded interval. First, the equation to be solved is multiplied by ( x − x 0), where x 0 is the starting value, which is set to zero unless the function does not exist at x 0, in which case x 0 becomes a tracking initiation point that can be set arbitrarily to any value where the function does exist. Next, the new function, ( x − x 0) f( x) = 0, is incorporated into the FPN homotopy. The initial step establishes a single bifurcation point from which all real roots can be found. The second step ensures a relatively simple continuation path that consists of just two branches that stem from the bifurcation point and prevents the formation of any isola. By tracking the two branches of the homotopy path, all real roots are located. Path tracking is carried out with MATLAB, using the continuation toolbox of CL_MATCONT, developed by Dhooge et al. (2006), based on the work of Dhooge, Govaerts, and Kuznetsov (2003), which applies Moore–Penrose predictor-corrector continuation to track the path, using convergence-dependent step-size control to negotiate turning points and other sharp changes in path curvature. This new method has been applied, without failure, to numerous nonlinear equations, including those with transcendental functions. As with other continuation methods, f( x)must have twice-continuous derivatives.