Quadrature formulae for singular functions

Abstract

We are interested in the computation of a singular weight integral. We can get round the numerical difficulties which arise from this singularity in two steps. First, the function to be integrated has been chosen within a space of sufficiently smooth functions, which already neutralizes the singularity, even in the case of equidistant nodes. In the second step, in order to get a greater accuracy, which is the main purpose of this paper, we can make use of the quasi-optimal nodes found in the piecewise polynomial approximation theory. Moreover, these quasi-optimal nodes are defined in such a way that we can apply the Richardson extrapolation, and further improve the results. Examples are given in order to show how the results are improved and to establish the limitations of our method.

Keywords:

• Integration
• weight
• Hilbert space
• reproducing kernel
• nodes
• quasi-optimal nodes
• Richardson's extrapolation
• Hermite-kind formula

C.R. Categories:

• 5.1.6.