Symbolic and numerical methods for Hamiltonian systems

This thesis concentrates on two main areas. Traditional numerical methods for ordinary differential equations and new state-of-the-art techniques developed by taking advantage of recent developments in symbolic computation. Computer algebra has been an essential tool, both in the research itself and in the implementation of the resulting algorithms. The numerical methods developed are primarily intended for use with Hamiltonian systems, but many find uses in solving other forms of ordinary differential equations. New order condition theory for deriving symplectic Runge-Kutta methods applicable to Hamiltonian problems is presented. This process is automated using computer-based derivation. New and efficient methods are then derived. Alternative numerical methods based on classical generating function techniques are also given. It is proven that these classical methods can be generated to arbitrarily high orders. It is further shown that generation of these methods can also be automated via the use of computer algebra. Numerical examples are presented where the efficiency and accuracy of the methods developed is demonstrated. Qualitative comparisons with standard established techniques are also gi ven. The symbolic tools developed have been partitioned into a suite of individually documented packages. The symbolic packages are used throughout the main body of the thesis where appropriate, with implementation details and detailed usage instructions given in a set of appendice