Rational approximation of discrete data with asymptomatic behaviour

This thesis is concerned with the least-squares approximation of discrete data that
appear to exhibit asymptotic behaviour. In particular, we consider using rational
functions as they are able to display a number of types of asymptotic behaviour. The
research is biased towards the development of simple and easily implemented algorithms
that can be used for this purpose. We discuss a number of novel approximation
forms, including the Semi-Infinite Rational Spline and the Asymptotic Polynomial.
The Semi-Infinite Rational Spline is a piecewise rational function, continuous across
a single knot, and may be defined to have different asymptotic limits at ±∞. The
continuity constraints at the knot are implicit in the function definition, and it can be
fitted to data without the use of constrained optimisation algorithms. The Asymptotic
Polynomial is a linear combination of weighted basis functions, orthogonalised
with respect to a rational weight function of nonlinear approximation parameters.
We discuss an efficient and numerically stable implementation of the Gauss-Newton
method that can be used to fit this function to discrete data. A number of extensions
of the Loeb algorithm are discussed, including a simple modification for fitting Semi-
Infinite Rational Splines, and a new hybrid algorithm that is a combination of the
Loeb algorithm and the Lawson algorithm (including its Rice and Usow extension),
for fitting ℓp rational approximations. In addition, we present an extension of the Rice
and Usow algorithm to include ℓp approximation for values p < 2. Also discussed is
an alternative representation of a polynomial ratio denominator, that allows pole free
approximations to be fitted to data with the use of unconstrained optimisation methods.
In all cases we present a large number of numerical applications of these methods
to illustrate their usefulness.