A probabilistic model for quadrature rules

摘要

The approximate formula of quadrature rules is usually indicated as ∫abf(x)dx≃∑i=1nwif(xi),where xi are the integration nodes and wi are the corresponding coefficients. During last decades, various cases of above type formula, such as Gauss quadrature rules, Newton–Cotes rules and so on, have been introduced to somehow estimate xi and wi. In this paper, we propose a quadrature model based on a probabilistic approach to estimate the related nodes and coefficients. For this purpose, we apply the normal distribution, as a special case, and present an error analysis for the obtained probabilistic formulas. Some numerical examples are given in this way to show the efficiency of the proposed model.