Quadrature Formulas

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Quadrature Formulas

formulas for approximating definite integrals by means of the values of the integrand at a finite number of points. The most common quadrature formulas have the form

where x₁x₂, …, x_n are the nodes of the quadrature formula, A₁, A₂ . . . ,A_n are its coefficients, and R_n is the remainder term. For example,

where a ≤ ξ ≤ b (trapezoidal rule). Sometimes for mulas for mechanical, or numerical, quadrature are also called quadrature formulas.

REFERENCE

Krylov, V. I. Priblizhennoe vychislenie integralov, 2nd ed. Moscow, 1967.

References in periodicals archive ?

In the case of standard Gaussian quadrature formulas (with simple or multiple nodes), L[infinity]-error bounds are considered by Gautschi and Varga [3], Schira [18], Milovanovic and Spalevic [8], Pejcev and Spalevic [14], and others.

ERROR BOUNDS FOR KRONROD EXTENSION OF GENERALIZATIONS OF MICCHELLI-RIVLIN QUADRATURE FORMULA FOR ANALYTIC FUNCTIONS

In sparse grids, quadrature formulas of high order for some dimensions are combined with quadrature formulas of lower order for the remaining dimensions in a way that the interpolation error is equivalent to that of full tensor products.

Time-Dependent Reliability-Based Design Optimization of Vibratory Systems

They show how to use exact-difference schemes as a starting point for constructing finite-difference schemes suitable for implementation on a computer, and show that combining exact-difference schemes with modern quadrature formulas results in numerical algorithms that are highly efficient.

Exact Finite-Difference Schemes

To integrate an arbitrary function h(x) over the element [[OMEGA].sup.e], the symmetric quadrature formulas, described by Zhang et al.

The Implementation of Spectral Element Method in a CAE System for the Solution of Elasticity Problems on Hybrid Curvilinear Meshes

In addition, one can make use of Filon's quadrature formulas (see [26], p.

Complete solution of the Marchenko equation for a simple model system/Martsenko vorrandi taielik lahendus lihtsa mudelsusteemi jaoks

In these references authors use methods on an estimate by basis function such as wavelets, polynomials, and so forth or use some quadrature formulas. But these technics usually have convergence rate of polynomial order with respect to M where M represents the cardinal of terms of sum in the expansion or the cardinal of points of the quadrature formula.

On solution of integrodifferential equation with delay parameter by sine basis functions

In order to model the triangles with curved edges in the boundary of the area more accurately, curvilinear coordinate transformation is used to map a curved triangle to a straight one; then the high order symmetrical Gaussian quadrature formulas over the triangles are adopted to evaluate the finite Fourier integration (1) more accurately; finally the 2D NUFFT [20] algorithm helps to make the computational complexity similar to 2D FFT.

An accurate conformal Fourier transform method for 2D discontinuous functions

Now if f is as above, then we have the one-point and two-points open Newton-Cotes quadrature formulas

Some weighted generalizations of open Newton-Cotes type inequalities for mappings of bounded variation and their applications *

Quadrature formulas based on rational interpolation.

Algorithm 793: GQRAT--Gauss Quadrature for Rational Functions

USPENSKY, On the convergence of quadrature formulas related to an infinite interval, Trans.

POLYNOMIAL APPROXIMATION WITH POLLACZECK-LAGUERRE WEIGHTS ON THE REAL SEMIAXIS. A SURVEY

Stancu, "Quadrature formulas with multiple Gaussian nodes," Journal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis, vol.

A New Sparse Gauss-Hermite Cubature Rule Based on Relative-Weight-Ratios for Bearing-Ranging Target Tracking