Sequences of transformations and triangular recursion schemes, with applications in numerical analysis

摘要

In many fields of numerical analysis there appear transformations of the form Tkv = Σv + k i = v αk i,v Ti. When v varies, a sequence of transformations is obtained. This approach covers, for example, the E− and Θ-algorithms, the recursion formulae for B-splines, Bernstein polynomials and orthogonal polynomials, Padé approximants, the divided difference scheme and projection methods. In this paper it will be proved that such transformations can be written as a ratio of determinants and can be recursively computed by a triangular recursion scheme. The reciprocal of these results also holds. Furthermore, we will show that Tvk can be represented in terms of a complex contour integral. Throughout the paper we will study several examples in some detail, and it will turn out that the application of our general theory leads to interesting new results in the special cases. Among others, we will derive a new determinantal representation formula for B-splines, a recurrence relation for generalized Bernstein polynomials, a generalization of the E-algorithm and we will prove that the Θ-algorithm can be represented as a quotient of determinants.