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

Here γM (x) ∼ Q − 1 4 (x), QM (x) ∼ Q 1 2 (x) and ǫM (x) → 0 as x → ∞, while M(≥ 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 diagonalizations of the coefficient matrices in a sequence of related differential systems. The main contribution of [3] to (1. 1) was to show that, for a general class of coefficients Q, the magnitude of the error term ǫM (x) for large x decreases as M increases. This feature of ǫM (x) was then exploited in [3] for the case where Q(x) = q(x)−λ and (1. 2) becomes the usual Sturm-Liouville equation with spectral parameter λ . The smallness of ǫM (x) for a suitable choice of M (such as M = 6) leads to an efficient numerical algorithm for estimating the Titchmarsh-Weyl function m(λ) with precise global error bounds. In [3], the case where q(x) = −x (0 < α ≤ 2) and Reλ ≥ −1 was considered in detail. In this paper, we develop these ideas for higher-order differential equations. Thus we consider the generalisation of ( 1. 1) for the n-th order equation