Algorithms for the computation of approximations by algebraic functions
The application of Pade approximation to problems in mathematical physics was introduced by Balrnr and Gammel (Baker and Gammel (1961)). Pade approximations, the rational analogue of the Taylor polynomial approximation, could be expected to represent a wider range of behaviour than simple polynomial approximation. Various generalizations of this approach include integral approximants (Hunter and Baker (1979), and Rehr, Joyce and Guttmann (1980)) applied to the theory of critical phenomena, and algebraic approximants (Brak, Guttmann and Enting (1990)) occurring in lattice theory. In considering approximation from the more general classes of integral or algebraic functions, there are two separate aspects that merit attention. These are the properties of the approximation and the computation of the approximation. The local approximating properties of the quadratic algebraic function have been studied (Brookes and Mclnnes (1990)) and some qualitative results have been reported (Brookes (1990)). The results have been extended to general integral functions (Mclnnes (1989)) and general algebraic functions (Mc!nnes (1991)). The main results established were that a clear formulation of the approximation problem leads to the existence of a unique approximating algebraic form which determines the polynomial coefficients of the algebraic equation for the algebraic function, and the existence of a unique distinguished algebraic function which is a solution of this algebraic equation. In addition the order of approximation is quantified in a variety of circumstances. However the computation of the general algebraic form which defines the polynomial coefficients of the implicit equation for the algebraic function approximation may involve severe numerical problems, and seems to have received little direct attention. The objective of this paper is to review existing algorithms for this computation and to consider a new approach. In Section 2 the general formulation of approximation by algebraic functions is summarized. Some existing algorithms for the computation of the algebraic form are reviewed in Section 3 and a new algorithm is described in Section 4.
SubjectsField of Research::01 - Mathematical Sciences::0101 - Pure Mathematics::010101 - Algebra and Number Theory
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