The results in this paper are motivated by two analogies. First, m-harmonic functions in Rn are extensions of the univariate algebraic polynomials of odd degree 2m−1. Second, Gauss’ and Pizzetti’s mean value formulae are natural multivariate analogues of the rectangular and Taylor’s quadrature formulae, respectively. This point of view suggests that some theorems concerning quadrature rules cou...

We show that the weights of extended Gauss-Legendre quadrature rules are all positive.

Recently Laurie presented a new algorithm for the computation of (2n+1)-point Gauss-Kronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order 2n+ 1 from certain mixed moments, and then computes a partial spectral factorization. We describe a new algorithm that does not require the entries of the tridiagonal matrix to b...

Applying a few steps of the Arnoldi process to a large nonsymmetric matrix A with initial vector v is shown to induce several quadrature rules. Properties of these rules are discussed, and their application to the computation of inexpensive estimates of the quadratic form 〈f, g〉 := v∗(f(A))∗g(A)v and related quadratic and bilinear forms is considered. Under suitable conditions on the functions ...

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 ...

In the companion paper (Bellet et al., 2021), a spherical harmonic subspace associated to Cubed Sphere has been introduced. This is further analyzed here. particular, it permits define new based quadrature. quadrature inherits rotational invariance properties of subspace. Contrary Gaussian quadrature, where set nodes and weights solution nonlinear system, only are unknown Despite this conceptua...

The Jacobi matrix of the (2n+1)-point Gauss-Kronrod quadrature rule for a given measure is calculated efficiently by a five-term recurrence relation. The algorithm uses only rational operations and is therefore also useful for obtaining the Jacobi-Kronrod matrix analytically. The nodes and weights can then be computed directly by standard software for Gaussian quadrature formulas.

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...

We construct a quadrature formula with n+ 1 angles and positive weights, exact in the (2n+1)-dimensional space of trigonometric polynomials of degree ≤ n on intervals with length smaller than 2π. We apply the formula to the construction of product Gaussian quadrature rules on circular sectors, zones, segments and lenses. 2000 AMS subject classification: 65D32.