A note on positive quadrature rules

Quadrature Rules for Fuzzy Integrals

A Note on Finite Quadrature Rules with a Kind of Freud Weight Function

We introduce a finite class of weighted quadrature rules with the weight function |x|−2a exp −1/x2 on −∞,∞ as ∫−∞|x|−2a exp −1/x2 f x dx ∑n i 1 wif xi Rn f , where xi are the zeros of polynomials orthogonal with respect to the introduced weight function, wi are the corresponding coefficients, andRn f is the error value. We show that the above formula is valid only for the finite values of n. In...

A constraint on extensible quadrature rules

When the worst case integration error in a family of functions decays as n−α for some α > 1 and simple averages along an extensible sequence match that rate at a set of sample sizes n1 < n2 < · · · < ∞, then these sample sizes must grow at least geometrically. More precisely, nk+1/nk ≥ ρmust hold for a value 1 < ρ < 2 that increases with α. This result always rules out arithmetic sequences but ...

A historical note on Gauss-Kronrod quadrature

The idea of Gauss–Kronrod quadrature, in a germinal form, is traced back to an 1894 paper of R. Skutsch. The idea of inserting n+1 nodes into an n-point Gaussian quadrature rule and choosing them and the weights of the resulting (2n+1)-point quadrature rule in such a manner as to maximize the polynomial degree of exactness is generally attributed to A.S. Kronrod [2], [3]. This is entirely justi...

Quadrature rules on unbounded interval

After some remarks on the convergence order of the classical gaussian formula for the numerical evaluation of integrals on unbounded interval, the authors develop a new quadrature rule for the approximation of such integrals of interest in the practical applications. The convergence of the proposed algorithm is considered and some numerical examples are given.