Date of Award
Master of Science (MS)
Broyden’s method is a quasi-Newton iterative method used to find roots of non-linear systems of equations. Research has shown and improved the rate of convergence for special cases and specific applications of the method. However, there is limited literature regarding the well-posedness of the method. In practice, a numerical method must reliably converge to the appropriate root. This paper will discuss the domain of attraction for the roots of a system found by using Broyden’s method. A method of approximating the radius of convergence of a root will be described which considers the largest disk centered at the root such that all values within the disk converge to the root. Literature on Broyden’s method has conflicting claims about the initial approximation of the Jacobian. Plots will demonstrate the effect of the initial guess of the Jacobian for the iterative scheme. In this paper, the importance of using a finite difference approximation for the initial guess of the Jacobian will be shown through examples of 2 × 2 systems of equations.
Bonthron, Michael, "Analyzing Domain of Convergence for Broyden’s Method" (2022). College of Science and Health Theses and Dissertations. 462.