On quadrature formulae based on derivative collocation

The recent article “A new type of weighted quadrature rules and its relation to orthogonal polynomials” by Masied-Jamei [M. Masjed-Jamei, A new type of weighted quadrature rules and its relation to orthogonal polynomials, Appl. Math. Comput. 188 (2007) 154-165] introduces quadrature rules based on the evaluation of the derivative(s) of the integrand function rather the function itself. The approach appears useful when a number of derivatives, including the integrand, vanish at a point λ, leading to increased order of accuracy compared to standard Gaussian rules. It is also shown by Masjed-Jamei (2007) how the nodes and weights of the resulting quadrature formula relate to nodes and weights of standard Gaussian quadratures applied to a weight function w to be determined by solving a specific system of integral equations. We give here an explicit expression for w and provide strategies for the practical computation of the quadrature nodes and weights. Additional comments on the examples used by Masjed-Jamei (2007) as well as a generalization involving multiple λ's, are also included.