Is Gauss quadrature better than Clenshaw-Curtis?

We consider the question of whether Gauss quadrature, which is very famous, is more powerful than the much simpler Clenshaw-Curtis quadrature, which is less well-known. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following Elliott and O'Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of $\log((z+1)/(z-1))$ in the complex plane. Gauss quadrature corresponds to Pad\'e approximation at $z=\infty$. Clenshaw-Curtis quadrature corresponds to an approximation whose order of accuracy at $z=\infty$ is only half as high, but which is nevertheless equally accurate near $[-1,1]$

Error Estimations of Turan Formulas with Gori-Micchelli and Generalized Chebyshev Weight Functions

S. Li in [Studia Sci. Math. Hungar. 29 (1994) 71-83] proposed a Kronrod type extension to the well-known Turan formula. He showed that such an extension exists for any weight function. For the classical Chebyshev weight function of the first kind, Li found the Kronrod extension of Turan formula that has all its nodes real and belonging to the interval of integration, [-1, 1]. In this paper we show the existence and the uniqueness of the additional two cases - the Kronrod exstensions of corresponding Gauss-Turan quadrature formulas for special case of Gori-Micchelli weight function and for generalized Chebyshev weight function of the second kind, that have all their nodes real and belonging to the integration interval [-1, 1]. Numerical results for the weight coefficients in these cases are presented, while the analytic formulas of the nodes are known