Perturbed Padé-type Approximation

In the past decades, we witnessed great progress in function approximation regarding its theory research as well as practical applications. Function approximation is not only the basis of numerical analysis but also plays a significant role in numerical solution of differential equations. To be specific, function approximation focuses on the problem of how to approach the complicated functions on the given domain by taking advantage of some simple functions. Generally speaking, we adopt three kinds of simple functions, namely, polynomial functions, piecewise polynomial functions and rational functions, which are easy to calculate on the computer to accomplish the approximation. Admittedly, the methods mentioned above are powerful tools for function approximation to some degree. However, the effect of polynomial approximation is mediocre when functions to be approximated have singularities. Under this circumstance, we should better use rational function approximation to improve the effect.Practically, the investigation on rational approximation is meaningful and this is why it drew much attention from many researchers in the past 20 years. Systematically, the methods of obtaining rational approximation include: (1) Pade approximation; (2) Chebyshev-Pade approximation; (3) continued fractions. Nevertheless, the traditional methods we referred above have their shortcomings, say, they contain time-consuming calculations and the calculation algorithms lack inheritance. In this thesis, we introduce two novel approximation ways, i.e., perturbed Pade approximation with many perturbation parameters and one perturbation parameter, respectively. As for the perturbed Pade approximation with one perturbation parameter, we derive the formulae for odd functions and even ones and investigate the relationship in calculation formulae between their original functions and derivatives. And then, we come up with perturbed Chebyshev-Pade approximation with a perturbation parameter which can make up for the shortcomings of perturbed Pade approximation. In addition, the advantages of this perturbed Chebyshev-Pade approximation over the traditional ones lie in three aspects: (1) we can obtain new approximation formulae through adjusting the perturbation parameter without re-calculating; (2) Under almost the same precision, our methods need less time-complexity than traditional ones; (3) We present corresponding error estimation so that we can roughly derive the right approximation formulae within the error tolerance. Finally, we illustrate the results by means of some numerical examples.