Location of a Physically Realized Solution for a System of Nonlinear Algebraic Equations

Date of Award


Degree Type


Degree Name

Doctor of Philosophy (PhD)



First Advisor

Joseph Kolibal

Advisor Department



In this dissertation we report on efforts to develop algorithmic tools that can provide useful insight to scientist and engineers. In this research effort we present the results of utilizing scientific computing methodologies to find feasible solutions for an engineering problem requiring the simulation of random vibration testing. The first emphasis was on solving a system of nonlinear algebraic equations that evolved from a physical definition for vibration testing of a previously un-solved avionics-engineering problem. The second emphasis was on addressing the fact that the mathematical model formulated from this problem produced multiple solutions. Our approach was to follow the software engineering principle of code reuse to find the appropriate solvers to solve our problem in lieu of developing a new method or implementing an existing algorithm. A survey of existing solvers was performed, which led to the development of a library that could be used to solve large problems quickly and easily. Similar tools implementing this concept were available for other areas such as systems of linear equations, and optimization but were lacking for the system of nonlinear algebraic equations. Benchmark testing provided the knowledge to identify the most robust solvers for application to the problem. We addressed the second emphasis by introducing a process of reducing the set of mathematically correct solutions to one that we proposed to be the unique physically realized solution. We have demonstrated with one possible method that identifying the solution is via a physical analogy with a problem possessing a known physical system. A software prototype is introduced utilizing the state-of-the-practice process for solving problems simular to ours.