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 appendices
A Doctoral Thesis. Submitted in partial fulfillment of the requirements for the award of Doctor of Philosophy of Loughborough University.