Related papers: Quadrature formulas from rational approximations
We present several new quadrature formulas in the triangle for exact integration of polynomials. The points were computed numerically with a cardinal function algorithm which imposes that the number of quadrature points $N$ be equal to the…
The five (5) families of quadrature rules with periods of one or two intervals for the real line and spline classes $C^0$, $C^1$ are presented. The formulae allow one to calculate the points or weights of these quadrature rules in a very…
Neural Networks have been widely used to solve Partial Differential Equations. These methods require to approximate definite integrals using quadrature rules. Here, we illustrate via 1D numerical examples the quadrature problems that may…
The present paper is devoted to construction of an optimal quadrature formula for approximation of Fourier integrals in the Hilbert space $W_2^{(1,0)}[a,b]$ of non-periodic, complex valued functions. Here the quadrature sum consists of…
Kernel quadratures and other kernel-based approximation methods typically suffer from prohibitive cubic time and quadratic space complexity in the number of function evaluations. The problem arises because a system of linear equations needs…
This manuscript reviews theoretical results and applications related to quadratic forms in Gaussian random variables. It summarizes definitions, canonical representations, exact and approximate distributional results, numerical inversion…
An efficient numerical quadrature is proposed for the approximate calculation of the potential energy in the context of pseudo potential electronic structure calculations with Daubechies wavelet and scaling function basis sets. Our…
A novel development is given of the theory of Gaussian quadrature, not relying on the theory of orthogonal polynomials. A method is given for computing the nodes and weights that is manifestly independent of choice of basis in the space of…
A novel algorithm for the computation of the quadratic numerical range is presented and exemplified yielding much better results in less time compared to the random vector sampling method. Furthermore, a bound on the probability for the…
We introduce a new type of quadrature, known as approximate Gaussian quadrature (AGQ) rules using {\epsilon}-quasiorthogonality, for the approximation of integrals of the form \int f(x)d \alpha(x). The measure {\alpha}(\cdot) can be…
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:…
Several applications of the QR-AAA algorithm, a greedy scheme for vector-valued rational approximation, are presented. The focus is on demonstrating the flexibility and practical effectiveness of QR-AAA in a variety of computational…
We provide explicit expressions for quadrature rules on the space of $C^1$ quintic splines with uniform knot sequences over finite domains. The quadrature nodes and weights are derived via an explicit recursion that avoids an intervention…
Compound distributions allow construction of a rich set of distributions. Typically they involve an intractable integral. Here we use a quadrature approximation to that integral to define the quadrature compound family. Special care is…
It is shown that quadratic constraints are compatible with the geometric integrability scheme of the multidimensional quadrilateral lattice equation. The corresponding Ribaucour reduction of the fundamental transformation of quadrilateral…
In this paper the properties of R\'edei rational functions are used to derive rational approximations for square roots and both Newton and Pad\'e approximations are given as particular cases. As a consequence, such approximations can be…
This article provides a simple geometric interpretation of the quadratic formula. The geometry helps to demystify the formula's complex appearance and casts it into a much simpler existence, thus potentially benefits early algebra students.
Approximation using Fourier features is a popular technique for scaling kernel methods to large-scale problems, with myriad applications in machine learning and statistics. This method replaces the integral representation of a…
Many real life problems can be reduced to the solution of a complex exponentials approximation problem which is usually ill posed. Recently a new transform for solving this problem, formulated as a specific moments problem in the plane, has…
We obtain cubature formulas of volume potentials over bounded domains combining the basis functions introduced in the theory of approximate approximations with their integration over the tangential-halfspace. Then the computation is reduced…