Scott Allen Burns
A Monomial-Based Method for Solving Systems of Algebraic Equations
Systems of nonlinear algebraic equations arise frequently in science and engineering, either directly from a discrete model or indirectly from a discretization of a continuous model. In exceptional cases, such systems can be solved in closed-form, but generally, numerical methods provide the only hope of obtaining a solution. The majority of numerical nonlinear equation solvers are based on linearization techniques. Newton's method, for example, generates a sequence of easily-solved linear subproblems that usually produces an increasingly refined approximation of one of the solutions of the nonlinear system.
This project reconsiders the basic strategy of linearization. It investigates an alternative monomial approximation of the nonlinear system deriving from the arithmetic-geometric mean inequality. A logarithmic transformation of variables converts the monomial system into an easily-solved linear system. Four invariance properties of the monomial method, not shared by Newton's method, have been identified. These invariance properties are responsible for enhanced performance of the monomial method in comparison to Newton's method.
The immediate goal of this project is to clearly identify the classes of problems that are solved most effectively by the monomial method. To this end, three specific areas will be investigated: (1) the generalization of the monomial method to non-algebraic functional forms; (2) application of the monomial method to other engineering disciplines; and (3) evaluation of the effectiveness of the monomial method for solving ill-conditioned and badly-scaled problems.