Details of Grant 

EPSRC Reference:	GR/S22134/01
Title:	Efficient Evans function calculations via Neumann and Magnus expansions
Principal Investigator:	Malham, Dr SJA
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department:	S of Mathematical and Computer Sciences
Organisation:	Heriot-Watt University
Scheme:	First Grant Scheme Pre-FEC
Starts:	01 October 2003	Ends:	30 September 2006	Value (£):	125,517
EPSRC Research Topic Classifications:	Numerical Analysis
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
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Panel History:
Summary on Grant Application Form
A practical problem when integrating systems of linear differential equations, is that if we wish to sample the solution for a different value of an inherent parameter, then we must re-integrate. This is particularly inefficient when we want to accurately sample the solution over a continuous widespread set of parameter values. Though continuity methods resolve this issue locally, they still involve some degree of re-integration. A particular application we have in mind is that of evaluating the Evans function for different values of the spectral parameter, which involves repeated integration of the spectral problem. In this project we propose to extensively study a set of new, recently proposed, efficient numerical algorithms based on Neumann and Magnus expansions, that completely avoid the need for re-integration. The basic idea is that we expand either the Neumann or Magnus series solution for such systems as a power series in the parameter(s) in question. The coefficients of the series can be precomputed to any required accuracy. Then we evaluate the series for the parameter values we wish to sample. This proposal is intended to develop and extend these ideas which, to give one example, will revolutionize Evans function calculations and the direct construction of the pure-point spectrum of linear operators---repeated integration of the spectral problem will no longer be necessary.
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Organisation Website:	http://www.hw.ac.uk