Starting approximations for implicit Runge-Kutta methods applied to ordinary differential equations and differential-algebraic equations

Differential equations are powerful tools with which to model scientific phenomena. Ordinary differential equations (ODEs) are equations which involve a function and its derivative. If algebraic constraints are appended to one or more ODEs, the result is known as a system of differential-algebraic equations (DAEs). The type of DAE with which this work is concerned is particularly useful in constrained mechanical problems. Generally, a differential equation cannot be solved analytically, and so must be solved by a numerical method. IRK methods are a popular class of numerical methods for solving differential equations. These methods require a nonlinear system to be solved, usually via iteration. The initial iterate is called a starting approximation. The goal of this work is to find good starting approximations, meaning starting approximations which are near the true root. Solving the nonlinear system is the majority of the computational work in each step, so good starting approximations can significantly improve efficiency. The results contained within this thesis are proven analytically and supported by computational experiments.