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Rational approximation of discrete data with asymptomatic behaviour

Cooper, Philip (2007) Rational approximation of discrete data with asymptomatic behaviour. Doctoral thesis, University of Huddersfield.

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    Abstract

    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.

    Item Type: Thesis (Doctoral)
    Subjects: Q Science > Q Science (General)
    Q Science > QA Mathematics > QA75 Electronic computers. Computer science
    Q Science > QA Mathematics
    Schools: School of Computing and Engineering
    Related URLs:
    Depositing User: Sara Taylor
    Date Deposited: 25 Sep 2008 16:48
    Last Modified: 28 Jul 2010 19:26
    URI: http://eprints.hud.ac.uk/id/eprint/2026

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