Bifurcation and NonConvergence in the HansenPatrick RootFinding Method
 Publication Year:
 2014

 Bepress 62

 Bepress 21
 Repository URL:
 https://digitalcommons.csbsju.edu/honors_theses/37
 Author(s):
 Tags:
 Computer Science Student Work; Mathematics Student Work; Computer Sciences; Mathematics
thesis / dissertation description
The HansenPatrick RootFinding Method is a oneparameter family of cubically convergent rootfinding methods. The parameter is called alpha and can be any complex number. With a few different values of alpha, HansenPatrick becomes equivalent to other, more wellknown rootfinding methods. For example, when alpha equals 1, HansenPatrick becomes equivalent to Halley’s Method. There has been previous research into the dynamical systems that arise when varying the initial starting point or varying a family of functions. This paper deals with what happens when the initial point and function are fixed but the rootfinding method varies. We are interested in spurious cycles that can attract points which would ideally converge to a root of the function. We vary alpha near a known spurious cycle and track what happens to this cycle as alpha varies. Some results we obtain are the standard bifurcation diagram complete with cycle doubling and chaos, as well as Mandelbrot Sets by varying alpha in the complex direction.