Calculation of the weights of interpolatory quadratures

Summary

We present an algorithm for the stable evaluation of the weights of interpolatory quadratures with prescribed simple or multiple knots and compare its performance with that obtained by directly solving, using the method proposed by Galimberti and Pereyra [1], the confluent Vandermonde system of linear equations satisfied by the weights. Elsewhere Kautsky [5] has described a property which relates the weights of interpolatory quadratures to the principal vectors of certain non-derogatory matrices. Using this property one can get the information about the weight functionw of the approximated integral implicitly through the (symmetric tridiagonal) Jacobi matrix associated with the polynomials orthonormal with respect tow. The results indicate that the accuracy of the method presented is much higher than that achieved by solving the Vandermonde system directly.