Related papers: Information Theory and Quadrature Rules

Quadrature formulas for $\int_a^b f(x) dx$ where derivative terms need only be evaluated at $a$ and $b$ in the composite rule are identified. Error bounds are given when $f:[a,b]\to\mathbb{R}$ satisfies $f^{(n-1)}$ is absolutely continuous…

Simple proofs of the midpoint, trapezoidal and Simpson's rules are proved for numerical integration on a compact interval. The integrand is assumed to be twice continuously differentiable for the midpoint and trapezoidal rules, and to be…

When integrating functions that have poles outside the interval of integration, but are regular otherwise, it is suggested that the quadrature rule in question ought to integrate exactly not only polynomials (if any), but also suitable…

A method of deriving quadrature rules has been developed which gives nodes and weights for a Gaussian-type rule which integrates functions of the form: f(x,y,t) = a(x,y,t)/((x-t)^2+y^2) + b(x,y,t)/([(x-t)^2+y^2]^{1/2}) +…

Despite extensive research on symmetric polynomial quadrature rules for triangles, as well as approaches to their calculation, few studies have focused on non-polynomial functions, particularly on their integration using symmetric triangle…

Closed formulae for all Gaussian or optimal, 1-parameter quadrature rules in a compact interval [a, b] with non uniform, asymmetric subintervals, arbitrary number of nodes per subinterval for the spline classes $S_{2N, 0}$ and $S_{2N+1,…

This work is devoted to the study of integration with respect to binomial measures. We develop interpolatory quadrature rules and study their properties. Local error estimates for these rules are derived in a general framework.

A straightforward 3-point quadrature formula of closed type is derived that improves on Simpson's rule. Just using the additional information of the integrand's derivative at the two endpoints we show the error is sixth order in grid…

Let M(f) denote the Midpoint Rule and T(f) the Trapezoidal Rule for estimating integral_a^b f(x) dx. Then Simpson's Rule = tM(f) + (1-t)T(f), where t = 2/3. We generalize Simpson's Rule to multiple integrals as follows. Let D be some…

We study a new simple quadrature rule based on integrating a $C^1$ quadratic spline quasi-interpolant on a bounded interval. We give nodes and weights for uniform and non-uniform partitions. We also give error estimates for smooth functions…

The information in an individual finite object (like a binary string) is commonly measured by its Kolmogorov complexity. One can divide that information into two parts: the information accounting for the useful regularity present in the…

By introducing a parameter, we give a unified generalization of some quadrature rules, which not only unify the recent results about error bounds for generalized mid-point, trapezoid and Simpson's rules, but also give some new error bounds…

We consider probabilistic theories in which the most elementary system, a two-dimensional system, contains one bit of information. The bit is assumed to be contained in any complete set of mutually complementary measurements. The…

We develop quadrature rules for the isogeometric analysis of wave propagation and structural vibrations that minimize the discrete dispersion error of the approximation. The rules are optimal in the sense that they only require two…

Proper scoring rules are methods for encouraging honest assessment of probability distributions. Just like likelihood, a proper scoring rule can be applied to supply an unbiased estimating equation for any statistical model, and the theory…

Kolmogorov argued that the concept of information exists also in problems with no underlying stochastic model (as Shannon's information representation) for instance, the information contained in an algorithm or in the genome. He introduced…

Numerical integration is encountered in all fields of numerical analysis and the engineering sciences. By now, various efficient and accurate quadrature rules are known; for instance, Gauss-type quadrature rules. In many applications,…

The standard design principle for quadrature formulas is that they should be exact for integrands of a given class, such as polynomials of a fixed degree. We show how this principle fails to predict the actual behavior in four cases:…

Symmetric polynomial quadrature rules for triangles are commonly used to efficiently integrate two-dimensional domains in finite-element-type problems. While the development of such rules focuses on the maximum degree a given number of…

Filon-Simpson quadrature rules are derived for integrals of the type \int_a^b dx f(x) sin(xy)/(xy) and \int_a^b dx f(x) 4 sin^2(xy/2)/(xy)^2 which are needed in applications of the worldline variational approach to Quantum Field Theory.…