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Generalized compound quadrature formulae for finite-part integrals. (English) Zbl 0871.41021
Summary: We investigate the error term of the dth degree compound quadrature formulae for finite part integrals of the form 0 1 x -p f(x)dx where p and p1. We are mainly interested in error bounds of the form |R[f]|c|f (s) | with best possible constant c. It is shown that, for p and n uniformly distributed nodes, the error behaves as O(n p-s-1 ) for fC s [0,1], p-1<sd+1. In a previous paper we have shown that this is not true for p. As an improvement, we consider the case of non-uniformly distributed nodes. Here, we show that for all p1 and fC s [0,1], an O(n -s ) error estimates can be obtained in theory by a suitable choice of the nodes. A set of nodes with this property is stated explicitly. In practice this graded mesh causes stability problems which are computationally expensive to overcome.

MSC:
41A55Approximate quadratures