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 polynomials. This method can be extended to compute nodes and weights for Gaussian quadrature on the unit circle and Gauss type quadrature rules with some fixed nodes.
A theory of spline quadrature rules for arbitrary continuity class in a closed interval $[a, b]$ with arbitrary nonuniform subintervals based on semi-classical orthogonal Jacobi polynomials is proposed. For continuity class $c \ge 2$ this theory depends on a conjecture.
The concept of Gauss quadrature can be generalized to approximate linear functionals with complex moments. Following the existing literature, this survey will revisit such generalization. It is well known that the (classical) Gauss quadrature for positive definite linear functionals is connected with orthogonal polynomials, and with the (Hermitian) Lanczos algorithm. Analogously, the Gauss quadrature for linear functionals is connected with formal orthogonal polynomials, and with the non-Hermitian Lanczos algorithm with look-ahead strategy; moreover, it is related to the minimal partial realiz…
The numerical integration of an analytical function $f(x)$ using a finite set of equidistant points can be performed by quadrature formulas like the Newton-Cotes. Unlike Gaussian quadrature formulas however, higher-order Newton-Cotes formulas are not stable, limiting the usable order of such formulas. Existing work showed that by the use of orthogonal polynomials, stable high-order quadrature formulas with equidistant points can be developed. We improve upon such work by making use of (orthogonal) Gram polynomials and deriving an iterative algorithm, together allowing us to reduce the space-co…
In this paper we illustrate that paraorthogonality on the unit circle $\mathbb{T}$ is the counterpart to orthogonality on $\mathbb{R}$ when we are interested in the spectral properties. We characterize quasi-paraorthogonal polynomials on the unit circle as the analogues of the quasi-orthogonal polynomials on $\mathbb{R}$. We analyze the possibilities of preselecting some of its zeros, in order to build positive quadrature formulas with prefixed nodes and maximal domain of validity. These quadrature formulas on the unit circle are illustrated numerically.
Gaussian Quadrature Rule using ε-Quasiorthogonality
Institute of Computational and Mathematical Engineering (ICME), Stanford University（斯坦福大学计算与数学工程研究所）；Mechanical Engineering Department, Stanford University（斯坦福大学机械工程系）
Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and expicitly list asymptotic expressions for the points and weights of Gaussian quadrature rules for three general classes of positive weight functions: analytic functions on a bounded interval with algebraic singularities at the endpoints, analytic weight functions on the halfline with exponential decay at infinity and an algebraic singularity at the finite endpoint, and analytic functions on the real line with exponential decay in both d…
Efficient mass and stiffness matrix assembly via weighted Gaussian quadrature rules for B-splines
BCAM – Basque Center for Applied Mathematics（巴斯克应用数学中心）；Curtin University（科廷大学）；Commonwealth Scientific and Industrial Research Organization (CSIRO)（联邦科学与工业研究组织）
The accumulated quadrature weights of Gaussian quadrature formulae constitute bounds on the integral over the intervals between the quadrature nodes. Classical results in this concern date back to works of Chebyshev, Markov and Stieltjes and are referred to as Separation Theorem of Chebyshev-Markov-Stieltjes (CMS Theorem). Similar separation theorems hold true for some classes of rational Gaussian quadrature. The Krylov subspace for a given matrix and initial vector is closely related to orthogonal polynomials associated with the spectral distribution of the initial vector in the eigenbasis of…
In this technical note, we explain how to construct Gaussian quadrature rules for efficiently and accurately computing integrals of S(x)f(x) where S(x) is the solar irradiance function tabulated in the ASTM standard and f(x) is an arbitary application-specific smooth function. This allows the integral to be computed accurately with a relatively small number of f(x) evaluations despite the fact that S(x) is non-smooth and wildly oscillatory. Julia software is provided to compute solar-weighted quadrature rules for an arbitrary bandwidth or number of points. We expect that this technique will be…
Generalized eigenvalue methods for Gaussian quadrature rules
Georgia Institute of Technology（佐治亚理工学院）；Technische Universität Berlin（柏林工业大学）；Uit – The Arctic University of Norway（挪威北极大学）；Universität Konstanz（康斯坦茨大学）；North Carolina State University（北卡罗来纳州立大学）
Optimization approaches to quadrature: new characterizations of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions
University of Tromsø（特罗姆瑟大学）；Universität Konstanz（康斯坦茨大学）
A new grid system on a sphere is proposed that allows for straight-forward implementation of both spherical-harmonics-based spectral methods and gridpoint-based multigrid methods. The latitudinal gridpoints in the new grid are equidistant and spectral transforms in the latitudinal direction are performed using Clenshaw-Curtis quadrature. The spectral transforms with this new grid and quadrature are shown to be exact within the machine precision provided that the grid truncation is such that there are at least 2N + 1 latitudinal gridpoints for the total truncation wavenumber of N. The new grid …
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 $S_{2N+1, 1}$, i.e. even and odd degree are presented. Also rules for the 2 missing spline classes $S_{2N-1, 0}$ and $S_{2N, 1}$ (the so called 1/2-rules), i.e. odd and even degree are presented. These quadrature rules are explicit in the sense, that they compute the nodes and their weights in the first/last boundary subinterval and, via a recursion the other nodes/weigh…
A new type of quadrature is developed. The Gaussian quadrature, for a given measure, finds optimal values of a function's argument (nodes) and the corresponding weights. In contrast, the Lebesgue quadrature developed in this paper, finds optimal values of function (value-nodes) and the corresponding weights. The Gaussian quadrature groups sums by function argument; it can be viewed as a $n$-point discrete measure, producing the Riemann integral. The Lebesgue quadrature groups sums by function value; it can be viewed as a $n$-point discrete distribution, producing the Lebesgue integral. Mathema…
We consider polynomial approximation over the interval $[-1,1]$ by regularized weighted discrete least squares methods with $\ell_2-$ or $\ell_1-$regularization, respectively. As the set of nodes we use Gauss quadrature points (which are zeros of orthogonal polynomials). The number of Gauss quadrature points is $N+1$. For $2L\leq2N+1$, with the aid of Gauss quadrature, we obtain approximation polynomials of degree $L$ in closed form without solving linear algebra or optimization problems. In fact, these approximation polynomials can be expressed in the form of the barycentric interpolation for…
Solving the population balance equation (PBE) for the dynamics of a dispersed phase coupled to a continuous fluid is expensive. Still, one can reduce the cost by representing the evolving particle density function in terms of its moments. In particular, quadrature-based moment methods (QBMMs) invert these moments with a quadrature rule, approximating the required statistics. QBMMs have been shown to accurately model sprays and soot with a relatively compact set of moments. However, significantly non-Gaussian processes such as bubble dynamics lead to numerical instabilities when extending their…
Stochastic collocation methods via $L_1$ minimization using randomized quadratures
Shanghai Normal University（上海师范大学）；University of Utah（犹他大学）；Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）
We study Gauss quadrature for Freud weights and derive worst case error estimates for functions in a family of associated Sobolev spaces. For the Gaussian weight $e^{-πx^2}$ these spaces coincide with a class of modulation spaces which are well-known in (time-frequency) analysis and also appear under the name of Hermite spaces. Extensions are given to more general sets of nodes that are derived from Marcinkiewicz-Zygmund inequalities. This generalization can be interpreted as a stability result for Gauss quadrature.
