Quadrature rules estimate ∫ 1 0 f(x) dx when f is defined by a table of n+1 values. Every binary string of length n defines a quadrature rule by choosing which endpoint of each interval represents the interval. The standard rules, such as Simpson’s Rule, correspond to strings of low Kolmogorov complexity, making it possible to define new quadrature rules with no smoothness assumptions, as well ...

We introduce two direct quadrature methods based on linear rational interpolation for solving general Volterra integral equations of the second kind. The first, deduced by a direct application of linear barycentric rational quadrature given in former work, is shown to converge at the same rate as the rational quadrature rule but is costly on long integration intervals. The second, based on a co...

Abstract. Gauss quadrature is a popular approach to approximate the value of a desired integral determined by a measure with support on the real axis. Laurie proposed an (n+1)-point quadrature rule that gives an error of the same magnitude and of opposite sign as the associated n-point Gauss quadrature rule for all polynomials of degree up to 2n + 1. This rule is referred to as an anti-Gauss ru...

We investigate the possibility of using two-dimensional Romberg integration to approximate integrals, over the square 0 < x, y < 1, of integrand functions of the form g(x, y)l(x y) where g(x, y) is, for example, analytic in x and y. We show that Romberg integration may be properly justified so long as it is based on a diagonally symmetric rule and function values on the singular diagonal, if re...

In this paper, anti-Gaussian quadrature rules for trigonometric polynomials are introduced. Special attention is paid to an even weight function on [-?, ?). The main properties of such proved and a numerical method their construction presented. That based relations between nodes weights the rule algebraic polynomials. Some examples included. Also, we compare our with other available methods.

The solution of the radiation transfer equation for the Earth's atmosphere needs to account for the re ectivity of the ground. When using the spherical harmonics method, the solution for this term involves an integral with a particular measure that presents numerical challenges. We are interested in computing a high order Gauss quadrature rule for this measure. We show that the two classical al...

The existence and uniqueness of the Gaussian interval quadrature formula with respect to the Hermite weight function on R is proved. Similar results have been recently obtained for the Jacobi weight on [−1, 1] and for the generalized Laguerre weight on [0,+∞). Numerical construction of the Gauss–Hermite interval quadrature rule is also investigated, and a suitable algorithm is proposed. A few n...

This paper considers the problem of constructive approximation of a continuous function on the unit sphere Sr−1 ⊆ Rr by a spherical polynomial from the space Pn of all spherical polynomials of degree ≤ n. In particular, for r = 3 it is shown that the hyperinterpolation approximation Lnf (in which the Fourier coefficients in the exact L2 orthogonal projection Pnf are approximated by a positive w...

In order to approximate the Riemann–Stieltjes integral ∫ b a f (t) dg (t) by 2–point Gaussian quadrature rule, we introduce the quadrature rule ∫ 1 −1 f (t) dg (t) ≈ Af ( − √ 3 3 ) + Bf (√ 3 3 ) , for suitable choice of A and B. Error estimates for this approximation under various assumptions for the functions involved are provided as well.