Quadrature Sums Research Articles

Ob approksimatsii poverkhnostnykh proizvodnykh funktsiy s primeneniem integral'nykh operatorov

Integral formulas are presented for approximating the surface gradient (of a scalar function given on a surface) and divergence (of a tangent vector field given on a surface) that are analogs of the well-known formulas for the derivatives of a function on a plane. Estimates of the error in the approximation of these functions are obtained. The question of subsequent approximation of the integrals that give expression for the surface gradient and divergence by quadrature sums over the values of the function under study at the nodes selected on the cells of the unstructured grid approximating the surface is also considered.

Length Scales in Bayesian Automatic Adaptive Quadrature

Two conceptual developments in the Bayesian automatic adaptive quadrature approach to the numerical solution of one-dimensional Riemann integrals [Gh. Adam, S. Adam, Springer LNCS 7125, 1–16 (2012)] are reported. First, it is shown that the numerical quadrature which avoids the overcomputing and minimizes the hidden floating point loss of precision asks for the consideration of three classes of integration domain lengths endowed with specific quadrature sums: microscopic (trapezoidal rule), mesoscopic (Simpson rule), and macroscopic (quadrature sums of high algebraic degrees of precision). Second, sensitive diagnostic tools for the Bayesian inference on macroscopic ranges, coming from the use of Clenshaw-Curtis quadrature, are derived.

On the problem of mutual deviation of certain quadrature sums of interpolation type

We obtained efficient computational formulae for quadrature sums that are optimal with respect to coefficients for arbitrary distribution of knots for certain classes of differentiable functions. Based on this, we found exact values of mutual deviation of interpolatory type quadrature sums.

FULL QUADRATURE SUMS FOR GENERALIZED POLYNOMIALS WITH FREUD WEIGHTS

Generalized nonnegative polynomials are defined as products of nonnegative polynomials raised to positive real powers. The generalized degree can be defined in a natural way. In this paper we extend quadrature sums involving pth powers of polynomials to those for generalized polynomials.

Quadrature sums and Lagrange interpolation for general exponential weights

We obtain forward and converse quadrature sum estimates associated with zeros of orthogonal polynomials for general exponential weights. These are then applied to establish mean convergence of Lagrange interpolation at zeros of these orthogonal polynomials. The results generalize earlier ones for even weights on (−1,1) or R .

Notes on Inequalities with Doubling Weights

Various important weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc., inequalities, have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumptions on the weights. In most of the cases this minimal assumption is the doubling condition. Sometimes, however, as in the weighted Nikolskii inequality, the slightly stronger A ∞condition is used. Throughout their paper the L p norm is studied under the assumption 1⩽ p<∞. In this note we show that their proofs can be modified so that many of their inequalities hold even if 0< p<1. The crucial tool is an estimate for quadrature sums for the pth power (0< p<∞ is arbitrary) of trigonometric polynomials established by Lubinsky, Máté, and Nevai. For technical reasons we discuss only the trigonometric cases.

A comparison of some approximate methods for solving the aerosol general dynamic equation

The general dynamic equation for aerosol evolution is converted into a set of differential equations for the rate of change of the moments, M m by multiplying the equation by v m and integrating over all v. The integrals over the size distribution appearing in these equations are approximated by quadrature sums over the distribution evaluated at discrete quadrature points. The results from different quadrature schemes are compared with accurate numerical values from a finite element method for a variety of coagulation mechanisms with and without gravitational removal. In general, three-point Laguerre or associated Laguerre quadrature, involving the numerical solution of three differential equations, produces results of comparable accuracy to more complicated schemes. The method is very easy to implement and typically produces values within a few percent of the accurate results. Increasing the number of quadrature points, or the number of differential equations, does not significantly improve the accuracy. For gravitational coagulation (both with and without gravitational removal) all quadrature schemes produce inaccurate results or fail completely.

Full quadrature sums for pth powers of polynomials with Freud weights

In this paper, we provide a solution of the quadrature sum problem of R. Askey for a class of Freud weights. Let r > 0, b ∈ (− ∞, 2]. We establish a full quadrature sum estimate ∑ j=1 n λ jn|PW| p(x jn)W −b↬C ∫ −∞ ∞ |PW| p(t)W 2−b(t) dt 1 ⩽ p < ∞, for every polynomial P of degree at most n + rn 1 3 , where W 2 is a Freud weight such as exp(−¦x¦ α) , α > 1, λ jn are the Christoffel numbers, x jn are the zeros of the orthonormal polynomials for the weight W 2, and C is independent of n and P. We also prove a generalisation, and that such an estimate is not possible for polynomials P of degree ⩽ m = m( n) if m(n) = n + ξ nn 1 3 , where ξ n → ∞ as n → ∞. Previous estimates could sum only over those x jn with ¦x jn¦ ⩽ σx 1n , some fixed 0 < σ < 1.

Chebyshev-type quadrature and partial sums of the exponential series

Chebyshev-type quadrature for the weight functions \[ w a ( t ) = 1 − a t π 1 − t 2 , − 1 &gt; t &gt; 1 , − 1 &gt; a &gt; 1 , {w_a}(t) = \frac {{1 - at}}{{\pi \sqrt {1 - {t^2}} }},\quad - 1 &gt; t &gt; 1,\quad - 1 &gt; a &gt; 1, \] is related to a problem concerning partial sums of the exponential series, namely the problem to extend the nth partial sum to a polynomial of degree 2N having all zeros on the circle | z | = | a | N |z| = |a|N . Using this connection, we show that the minimal number N of nodes needed for Chebyshev-type quadrature of degree n for w a ( t ) {w_a}(t) satisfies an inequality C 1 n ≤ N ≤ C 2 n {C_1}n \leq N \leq {C_2}n with positive constants C 1 , C 2 {C_1},{C_2} . As an application we prove that the minimal number N of nodes for Chebyshev-type quadrature of degree n on a torus embedded in R 3 {{\mathbf {R}}^3} satisfies an inequality C 1 n 2 ≤ N ≤ C 2 n 2 {C_1}{n^2} \leq N \leq {C_2}{n^2} .

Chebyshev-Type Quadrature and Partial Sums of the Exponential Series

Chebyshev-type quadrature for the weight functions \[ {w_a}(t) = \frac {{1 - at}}{{\pi \sqrt {1 - {t^2}} }},\quad - 1 < t < 1,\quad - 1 < a < 1,\] is related to a problem concerning partial sums of the exponential series, namely the problem to extend the nth partial sum to a polynomial of degree 2N having all zeros on the circle $|z| = |a|N$. Using this connection, we show that the minimal number N of nodes needed for Chebyshev-type quadrature of degree n for ${w_a}(t)$ satisfies an inequality ${C_1}n \leq N \leq {C_2}n$ with positive constants ${C_1},{C_2}$. As an application we prove that the minimal number N of nodes for Chebyshev-type quadrature of degree n on a torus embedded in ${{\mathbf {R}}^3}$ satisfies an inequality ${C_1}{n^2} \leq N \leq {C_2}{n^2}$.

О множителе Вейля для сходимости сумм Рима на

The paper is dedicated to the problem of almost everywhere convergence of the Riemann quadrature sums depending on one or two parameters for functions which are, in general, nonintegrable in the sense of Riemann. In particular, the following assertion is proved.

An asymptotic analysis of two algorithms for certain Hadamard finite-part integrals

Two algorithms for the approximate evaluation of certain Hadamard finite-part integrals, which date back to Grunwald (1867), have been analysed. It is shown that the quadrature sums are closely related to the finite-part integrals of the Bernstein polynomials. The stability of the algorithms is considered and two numerical examples are given

Quadrature Sums Involving pth Powers of Polynomials

R. Askey’s problem of estimating quadrature sums involving pth powers of polynomials $(0 < p < \infty )$ in terms of integrals, is solved. In its simplest form, the method can extend the large sieve of number theory to sums involving pth powers, rather than just squares, of trigonometric polynomials. Further, the method yields estimates whenever the abscissas in the quadrature have a suitable spacing and the weights have suitable bounds. In particular, it may be applied to generalized Jacobi weights and to Freud weights.

Mean convergence of Lagrange interpolation for Freud's weights with application to product integration rules

The connection between convergence of product integration rules and mean convergence of Lagrange interpolation in L p (1 < p < ∞) has been thoroughly analysed by Sloan and Smith [37]. Motivated by this connection, we investigate mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials associated with Freud weights on R . Our results apply to the weights exp(− x m /2), m = 2, 4, 6 …, and for the Hermite weight ( m = 2) extend results of Nevai [28] and Bonan [2] in at least one direction. The results are sharp in L p , 1 < p ⩽ 2. As a consequence, we can improve results of Smith, Sloan and Opie [38] on convergence of product integration rules based on the zeros of the orthogonal polynomials associated with the Hermite weight. In the process, we prove a new Markov-Stieltjes inequality for Gauss quadrature sums, and solve a problem of Nevai on how to estimate certain quadrature sums.

Application of quadrature rules for Cauchy-type integrals to the generalized Poincaré-Bertrand formula

The classical Poincaré-Bertrand transposition formula for the inversion of the order of integration in repeated Cauchy-type integrals is generalized in accordance with a new interpretation of Cauchy-type integrals. Next, the Gauss-Jacobi quadrature rule is applied, in a particular case of the generalized Poincaré-Bertrand formula, to both members of this formula and it is proved that this formula still remains valid (after the approximation of the integrals by quadrature sums). Two simple applications of this result, one concerning the convergence of a quadrature rule for repeated Cauchy-type integrals, and the other the numerical solution of singular integral equations, are made. Further generalizations and applications of the present results follow easily.

A polynomial quadrature method for half-range neutron transport problems

The properties of the two orthonormal polynomial sets associated with the two half-range weight functions in neutron transport problems are further investigated. Transformation formulae from one set to the other are generalized and thence a limit expression for one of the weight functions is obtained in the form of a polynomial quotient. The two highest algebraic degree of precision quadrature systems related to the two orthonormal polynomial sets are then applied (with special provisos) to singular-type transport integrals and to the representation, in terms of quadrature sums, of the extended orthogonality and completeness relations for the Case eigenfunctions. The quadrature formalism is finally exemplified on the standard half-range transport problems. Tables for nodes and quadrature coefficients are given, and an appendix deals with the interconnection between the moments of different order of a weight function.

Neutron transport half-range orthogonal polynomial sets and related quadrature sums and properties

The half-range orthogonality property for neutron transport eigenfunctions leads to the definition of two weight functions. Each of these yields a set of orthonormal polynomials and related nodes and quadrature coefficients. The polynomials of different set, same order, are here shown to be linked by singular integral transformations. As a consequence, they obey almost identical recursion formulae. The transformations also generate cross-set relations involving nodes and quadrature coefficients of different set and equal order. Calculation procedures are described and examples of numerical tables for nodes and quadrature coefficients presented. Appendices deal with the reduction to the fullrange case and the limit behaviour of nonasymptotic quadrature sums.

Quadrature sums of highest algebraic degree of precision for neutron transport integrals

A Gaussian-type quadrature formula for neutron transport integrals is here re-established according to the orthogonal-polynomial method. Limit properties of the asymptotic and nonasymptotic parts of the quadrature sum are also obtained, together with another quadrature formula of the highest algebraic degree of precision, for purely scattering media. Numerical-application examples are given in the appendix.

Approximation by Bernstein type rational functions

holds for sufficiently large n's provided a.=n -1/3, b.=n 2/3. Here co depends only on A and ~, and co2a (.) is the modulus of continuity o f f ( x ) on the interval [0, 2A]. As it was noted in [1], the convergence of R.(f , x) holds under the more general conditions a.=b./n~O, b . ~ o (rQ~oo) as well. The aim of the present paper is to improve the estimate (1) by an appropriate choice of a. and b. in the case when f (x ) satisfies some more restrictive conditions. Furthermore we shall show that these results can be applied to approximate certain improper integrals by quadrature sums of positive coefficients based on finite number of equidistant nodes. First we assume that f (x ) is uniformly continuous in [0, oo); then the modulus of continuity cos(. ) o f f ( x ) exists on the entire positive half-axis.