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Approximate diagonalization in differential systems and an effective algorithm for the computation of the spectral matrix

Published online by Cambridge University Press:  01 May 1997

B. M. BROWN ,
M. S. P. EASTHAM ,
D. K. R. McCORMACK  and
W. D. EVANS
B. M. BROWN
Affiliation:
Department of Computer Science, Cardiff University of Wales, Cardiff, CF2 3XF
M. S. P. EASTHAM
Affiliation:
Department of Computer Science, Cardiff University of Wales, Cardiff, CF2 3XF
D. K. R. McCORMACK
Affiliation:
Department of Computer Science, Cardiff University of Wales, Cardiff, CF2 3XF
W. D. EVANS
Affiliation:
School of Mathematics, University of Wales College of Cardiff, Senghennydd Road, Cardiff, CF2 4AG
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Abstract

1. Introduction

In a recent paper [3], an extended Liouville–Green formula

formula here

was developed for solutions of the second-order differential equation

formula here

Here γM(x)∼Q−¼(x), QM(x)∼Q−½(x) and εM(x)→0 as x→∞, while M([ges ]2) is an integer and γM and QM can be defined in terms of Q and its derivatives up to order M−1. The general form of (1·1) had been obtained previously by Cassell [5], [6], [7] and Eastham [10], [11, section 2·4]. In particular, the proof of (1·1) in [10] and [11] depended on the formulation of (1·2) as a first-order system and then on a process of M repeated diagonalization of the coefficient matrices in a sequence of related differential systems.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 121 , Issue 3 , May 1997 , pp. 495 - 517
Copyright
Cambridge Philosophical Society 1997

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