Electronic Thesis and Dissertation Repository


Master of Science


Applied Mathematics


Robert M. Corless


We compare two different root-finding methods, eigenvalue methods and homotopy methods, using three test problems: Mandelbrot polynomials, Fibonacci-Mandelbrot polynomials, and Narayana-Mandelbrot polynomials. For the eigenvalue methods, using both MATLAB and Maple, we computed the eigenvalues of a specialized recursively-constructed, supersparse, upper Hessenberg matrix, inspired by Piers Lawrence's original construction for the Mandelbrot polynomials, for all three families of polynomials. This led us to prove that this construction works in general. Therefore, this construction is genuinely a new kind of companion matrix. For the homotopy methods, we used a special-purpose homotopy, in which we used an equivalent differential equation to solve for the roots of all three families of polynomials. To solve these differential equations, we used our own ode solver, based on MATLAB's ode45 routine, which has pole-vaulting capabilities. We had two versions of this ode solver: one in MATLAB, and the other in C++ that implements Bailey's ARPREC package.