# Continuation Methods for Solving Nonlinear Equations

Background: To solve a nonlinear equation F(x,λ)=0 computationally, we must use an iterative method such as Newton’s Method.  Most iterative nonlinear solvers are very sensitive to the choice of initial iterate, which makes it difficult to solve some problems. When the problem depends on a natural parameter λ, a continuation method can be employed to compute solutions of  F(x,λ)=0 for different values of λ, building a “bridge” between a hard problem and an easier problem. A more sophisticated continuation uses an arclength-like parameter s to trace the solution of  F(x,λ)=0.

Prof. Howell’s research on continuation methods focuses on the type of constraint that the arclength parameter s must satisfy. The most popular approach requires that successive solutions of F(x,λ)=0 lie on a trajectory that is orthogonal to the previous solution direction. However, this method has limitations when the solution has high curvature. Another constraint requires successive iterates to lie on a sphere around the previous solution, thereby circumventing some of the difficulties encountered by the orthogonal constraint. However, this spherical constraint is not as commonly used.

Research Problems: There are several problems that can be investigated by undergraduates:

• Is there a way to mathematically analyze how well the different constraints will perform on problems with a high curvature?
• What effect does the choice of constraint have on the linear systems of equations that have to be solved at each step?
• Are there even better constraints for arclength continuation?

What Courses/Skills Do I Need To Have Taken?  MATH 120 (Introductory Calculus) and MATH 203 (Linear Algebra) is enough to get started!  Experience in MATLAB is helpful but not required.

When Can I Work on the Project?  You can work on the project in your spare time in the Fall or Spring semesters, or up to full-time during the summer.

Is Funding Available?  There has been no grant funding set aside for this project, however there may be grants that we can apply for once we formalize a project.