Quadrature Formulas For Integrals Research Articles

Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs

By combining a certain approximation property in the spatial domain, and weighted 𝓁2-summability of the Hermite polynomial expansion coefficients in the parametric domain obtained in Bachmayr et al. [ESAIM: M2AN 51 (2017) 341–363] and Bachmayr et al. [SIAM J. Numer. Anal. 55 (2017) 2151–2186], we investigate linear non-adaptive methods of fully discrete polynomial interpolation approximation as well as fully discrete weighted quadrature methods of integration for parametric and stochastic elliptic PDEs with lognormal inputs. We construct such methods and prove convergence rates of the approximations by them. The linear non-adaptive methods of fully discrete polynomial interpolation approximation are sparse-grid collocation methods which are certain sums taken over finite nested Smolyak-type indices sets of mixed tensor products of dyadic scale successive differences of spatial approximations of particular solvers, and of successive differences of their parametric Lagrange interpolating polynomials. The Smolyak-type sparse interpolation grids in the parametric domain are constructed from the roots of Hermite polynomials or their improved modifications. Moreover, they generate in a natural way fully discrete weighted quadrature formulas for integration of the solution to parametric and stochastic elliptic PDEs and its linear functionals, and the error of the corresponding integration can be estimated via the error in the Bochner space L1 (ℝ∞, V, γ) norm of the generating methods where γ is the Gaussian probability measure on ℝ∞ and V is the energy space. We also briefly consider similar problems for parametric and stochastic elliptic PDEs with affine inputs, and problems of non-fully discrete polynomial interpolation approximation and integration. In particular, the convergence rates of non-fully discrete polynomial interpolation approximation and integration obtained in this paper significantly improve the known ones.

An Euler-MacLaurin formula for polygonal sums

We prove an Euler-Maclaurin formula for double polygonal sums and, as a corollary, we obtain approximate quadrature formulas for integrals of smooth functions over polygons with integer vertices. Our Euler-Maclaurin formula is in the spirit of Pick's theorem on the number of integer points in an integer polygon and involves weighted Riemann sums, using tools from Harmonic analysis. Finally, we also exhibit a classical trick, dating back to Huygens and Newton, to accelerate convergence of these Riemann sums.

A truncated Clenshaw-Curtis formula approximates integrals over a semi-infinite interval

The Clenshaw-Curtis (C-C) rule is a quadrature formula for integrals on the finite interval [− 1,1] and known to be efficient for smooth integrands and suitable for constructing an automatic method owing to nice features, i.e., the C-C rule family is nested and the accuracy is easy to check. In this paper, for an integral on a semi-infinite interval $[0,\infty )$ with an integrand decaying exponentially as $x\to \infty $ , we propose a truncated formula of the C-C rule and present an automatic method to approximate the integral to the accuracy of double precision and its Matlab code. We reduce the interval $[0,\infty )$ to a finite interval [0,a], choosing a so that the ignored integral on $[a,\infty )$ is sufficiently small. To approximate the integral Ia on [0,a], we consider a wider interval [0,2a]. By a change of variables, we transform nodes of the C-C rule on [− 1,1] to those on [0,2a]. To approximate Ia, our formula uses the nodes belonging to [0,a]. Similarly, a truncated formula of the Gauss-Legendre (GLe) rule is available. For an analytic function f(z) on $[0,\infty )$ , we give an error analysis for our method. Using numerical examples, we compare our formula with the truncated GLe, Gauss-Laguerre and double exponential formulae in performance. Numerical examples show that our formula, as well as the truncated GLe formula, is efficient, particularly, for semi-infinitely oscillatory integrals.

Sharp quadrature formulas and inequalities between various metrics for rational functions

We obtain the sharp quadrature formulas for integrals of complex rational functions over circles, segments of the real axis, and the real axis itself. Among them there are formulas for calculating the L 2-norms of rational functions. Using the quadrature formulas for rational functions, in particular, for simple partial fractions and polynomials, we derive some sharp inequalities between various metrics (Nikol’skiĭ-type inequalities).

Local numerical integration on the sphere

Many applications in geomathematics as well as bio-medical applications require the analysis of an unknown target function of a large amount of data, which can be modeled as data on a subset of the surface of a sphere. An important ingredient of this analysis is to develop numerical integration schemes (quadrature formulas) to integrate spherical polynomials of as high a degree as possible exactly. Since many subsets of the sphere can be subdivided efficiently into spherical triangles, the problem reduces to computing quadrature formulas for integration on spherical triangles. In this paper, we present an algorithm for computing quadrature formulas based on “scattered data” on a triangle; i.e., without requiring a theoretically prescribed choice of the location of these points. We present several numerical examples to illustrate various features of our algorithm in the context of both integration and function approximation.

Choice of specific interpolation points in quadrature formula for integrals containing Bessel functions

A method for calculation of integrals containing Bessel functions is suggested which based on linear interpolation applied to the function entering the integral along with the Bessel function. A specific choice of interpolation points allows excluding the contribution into integral of all interpolation points besides the first one. The study of errors leads to determination of bounds of the considered integral.

Quadrature formulas for integration of multivariate trigonometric polynomials on spherical triangles

We describe an explicit construction of quadrature rules exact for integrating multivariate trigonometric polynomials of a given coordinatewise degree on a spherical triangle. The theory is presented in the more general setting of quadrature formulas on a compact subset of the unit hypersphere, $${\mathbb {S}^q}$$ , embedded in the Euclidean space $${\mathbb {R} ^{q+1}}$$ . The number of points at which the polynomials are sampled is commensurate with the dimension of the polynomial space.

Quadrature formulas for integrals transforms generated by orthogonal polynomials

By using the three-term recurrence equation satisfied by a family of orthogonal polynomials, their asymptotic expressions and bilinear generating functions, we obtain quadrature formulas for the integral transforms generated by the classical orthogonal polynomials. These integral transforms, related to the so-called Poisson integrals, correspond to a fractional Fourier transform in the case of Hermite polynomials, a Bessel transform in the case of Laguerre polynomials and an Appell transform in the case of Jacobi polynomials. We also give a nonclassical example in which a quadrature formula for an integral transform generated by Bateman–Pasternak polynomials is obtained. The kernel of this transform is related to a product of Gauss hypergeometric functions.

A quadrature formula for integrals with nearby singularities

AbstractThe purpose of this paper is to propose a new quadrature formula for integrals with nearby singularities. In the boundary element method, the integrands of nearby singular boundary integrals vary drastically with the distance between the field and the source point. Especially, field variables and their derivatives at a field point near a boundary cannot be computed accurately. In the present paper a quadrature formula for 𝓁‐isolated singularities near the integration interval, based on Lagrange interpolatory polynomials, is obtained. The error estimation of the proposed formula is also given. Quadrature formulas for regular and singular integrals with conjugate poles are derived. Numerical examples are given and the proposed quadrature rules present the expected polynomial accuracy. Copyright © 2006 John Wiley & Sons, Ltd.

Orthogonal Laurent polynomials and quadrature formulas for unbounded intervals

We study the convergence of quadrature formulas for integrals over the positive real line with an arbitrary (possibly complex) distribution function. The nodes of the quadrature formulas are the ze...

Orthogonal Laurent polynomials and quadrature formulas for unbounded intervals: II. Interpolatory rules

We study the convergence of quadrature formulas for integrals over the positive real line with an arbitrary (possibly complex) distribution function. The nodes of the quadrature formulas are the zeros of orthogonal Laurent polynomials with respect to an auxiliary distribution function and a certain nesting. The quadratures are called interpolatory (product) formulas. The class of functions for which convergence holds is characterized in terms of the moments of the auxiliary distribution function. We also include the convergence analysis of related two-point Padé-type approximants to the Stieltjes transform of the given distribution function. Finally, some illustrative numerical examples are also given.

Orthogonal Laurent Polynomials and Quadrature Formulas for Unbounded Intervals: I. Gauss-Type Formulas

We study the convergence of quadrature formulas for integrals over the positive real line with an arbitrary distribution function. The nodes of the quadrature formulas are the zeros of orthogonal Laurent polynomials with respect to the distribution function and with respect to a certain nesting. This ensures a maximal domain of validity and the quadratures are therefore called Gauss-type formulas. The class of functions for which convergence holds is characterized in terms of the moments of the distribution function. Moreover, error estimates are given when f satisfies certain continuity conditions. Finally, these results are applied to the family of distributions d’(x) = x exp{ (x 1 + x 2 )}dx, 1, 2 1/2, 2 R.

Numerical Solution of a System of Boundary Integral Equations with a Logarithmic Singularity for Problems in the Theory of Shells with Holes

A quadrature formula for integrals having a logarithmic singularity is investigated. The formula permits the method of mechanical quadratures to be used in solving a system of boundary integral equations with an analogous singularity in problems associated with the theory of perforated shells.

Gauss–Radau formulae for Jacobi and Laguerre weight functions

Explicit expressions are obtained for the weights of the Gauss–Radau quadrature formula for integration over the interval [−1, 1] relative to the Jacobi weight function (1− t) α (1+ t) β , α>−1, β>−1. The nodes are known to be the eigenvalues of a symmetric tridiagonal matrix, which is also obtained explicitly. Similar results hold for Gauss–Radau quadrature over the interval [0, ∞) relative to the Laguerre weight t α e −t , α>−1.

Numerical integration over a disc. A new Gaussian quadrature formula

We construct a quadrature formula for integration on the unit disc which is based on line integrals over $n$ distinct chords in the disc and integrates exactly all polynomials in two variables of total degree $2n-1$ .

Rates of convergence of multipoint rational approximants and quadrature formulas on the unit circle

In this paper, multipoint rational approximants to the Riesz-Herglotz transform of a Borel measure μ, supported on [−π, π] are considered. We give estimates for the rate of convergence of these approximants. These estimates are obtained from the asymptotic behaviour of a sequence of rational functions, orthogonal with respect to μ and having prescribed poles 1 α k , where { α k } 1 ∞ is a sequence of complex numbers compactly contained in the open unit disk. As an application we also give the rate of convergence of rational Szegő quadrature formulas for integrals with respect to μ.

Error estimates for generalized compound quadrature formulas

We define quadrature formulas for integrals with weight functions by applying a given approximation method locally. This allows the generalisation of different quadrature formulas, e.g., the compound Newton-Cotes formulas, Gauss summation formulas, or Gregory's formulas, to the case of weighted integrals, as well as to construct new quadrature formulas, and to derive error estimates for all these quadrature formulas. The estimates considered here are mainly of the form |R[f]|≤∥f (r) ∥, provided the underlying approximation method is exact for polynomials of degree <r (R[f] is the quadrature error)

Abscissae and weights for the Gauss-Laguerre quadrature formula

A program is described which calculates the abscissae and weights for the Gauss-Laguerre quadrature formula for integrals of the form ∫∈e -x x α f( x)d x very rapidly and with high accuracy even in the case of many abscissae. The abscissae are given by the zeros of the Laguerre polynomials, which are found by the Newton-Raphson method with suitable initial approximations. The program is useful for the problem in the case of an integrand f( x) oscillating rapidly.

Numerical Quadrature Rules for Some Infinite Range Integrals

Recently the present author has given a new approach to numerical quadrature and derived new numerical quadrature formulas for finite range integrals with algebraic and/or logarithmic endpoint singularities. In the present work this approach is used to derive new numerical quadrature formulas for integrals of the form $\smallint _0^\infty {x^\alpha }{e^{ - x}}f(x) dx$ and $\smallint _0^\infty {x^\alpha }{E_p}(x)f(x) dx$, where ${E_p}(x)$ is the exponential integral. It turns out the new rules are of interpolatory type, their abscissas are distinct and lie in the interval of integration and their weights, at least numerically, are positive. For fixed $\alpha$ the new integration rules have the same set of abscissas for all p. Finally, the new rules seem to be at least as efficient as the corresponding Gaussian quadrature formulas. As an extension of the above, numerical quadrature formulas for integrals of the form $\smallint _{ - \infty }^{ + \infty }|x{|^\beta }{e^{ - {x^2}}}f(x) dx$ too are considered.

Numerical quadrature rules for some infinite range integrals

Recently the present author has given a new approach to numerical quadrature and derived new numerical quadrature formulas for finite range integrals with algebraic and/or logarithmic endpoint singularities. In the present work this approach is used to derive new numerical quadrature formulas for integrals of the form ∫ 0 ∞ x α e − x f ( x ) d x \smallint _0^\infty {x^\alpha }{e^{ - x}}f(x)\,dx and ∫ 0 ∞ x α E p ( x ) f ( x ) d x \smallint _0^\infty {x^\alpha }{E_p}(x)f(x)\,dx , where E p ( x ) {E_p}(x) is the exponential integral. It turns out the new rules are of interpolatory type, their abscissas are distinct and lie in the interval of integration and their weights, at least numerically, are positive. For fixed α \alpha the new integration rules have the same set of abscissas for all p. Finally, the new rules seem to be at least as efficient as the corresponding Gaussian quadrature formulas. As an extension of the above, numerical quadrature formulas for integrals of the form ∫ − ∞ + ∞ | x | β e − x 2 f ( x ) d x \smallint _{ - \infty }^{ + \infty }|x{|^\beta }{e^{ - {x^2}}}f(x)\,dx too are considered.