Continued Fraction Error Analysis And Rational Interpolation In The Continued Fraction Method

The theorem of continued fractions is an old branch of mathematics, it is applied more and more widely with the development of science and technology, especially on rational interpolants. The applications of continued fractions are concerned nearly with its convergence.There are two main parts in my thesis: getting truncation error bounds of special continued fractions and analysing continued fractions algorithm on rational interpolants.At first, I analysed the convergence of Sleszynski-Pringsheim and Worpitzky continued fractions using the backward recurrence algorithm of continued fractions, and obtained good truncation error bounds of them.While, about the applications of continued fractions on rational interpolants, an algorithm using continued fractions, which was called inverse difference-continued fractions method, can get rational interpolants functions easily. In the thesis, I analysed the existence of inverse difference φ(x₀,x₁,··· ,x_n) , to judge if the rational interpolants problems can be solved by the method;further more, I analysed the appearance of inaccessible points (x_k,y_k) when inverse difference exists, and obtained the necessary and sufficient condition about it.