In two recent papers [1, 2],
we have developed an algorithm for the solution of
linear differential systems of the type
on an interval [X, ∞). Here Z is an
n-component vector, ρ is a scalar factor, D is a
constant diagonal matrix and R is a perturbation matrix such that
x→∞, for some δ>0.The algorithm
implements a repeated transformation process by
means of which (1·1) is transformed into other systems whose perturbation
are of successively smaller orders of magnitude as x→∞.
When the perturbation reaches a prescribed accuracy, the Levinson
asymptotic theorem [4, 5] provides the
solution of the final system in the process, the solution also possessing
The corresponding solution of (1·1) is then obtained by transforming
The algorithm has two main aspects. The first is the algebraic one of
symbolically the matrix terms which appear in the transformation
process. In  and ,
the algorithm was set up in such a way that this aspect is implemented
symbolic algebra system Mathematica. The other aspect is the numerical
deriving from the algebra the values of the solutions of (1·1),
given ρ, D and R.
Included here are the determination of explicit error-bounds and procedures
handling large numbers of matrix terms.
In this paper we develop these ideas in two different but not unrelated
First, we replace the constant matrix D by one which has entries
of differing orders
of magnitude as x→∞. Our methods are
sufficiently indicated by two orders of magnitude, and thus we consider
where formula here
with constant and diagonal D˜ and D, while
λ is a scalar factor such that λ(x)→∞
x→∞. The amplification of our previous algorithm to
cover this new situation is
dealt with in Sections 2–4 below. At the end of Section 3, we state
asymptotic theorem in the form that we need, with the necessary accuracy
incorporated. As in  and ,
ρ(x) is xa (α>−1).
However, the value
α=−1 also occurs in applications, and our
second extension is to include α=−1 in
the scope of the algorithm. This matter is the subject of Section 5.