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 It might be outdated or ideologically biased.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, …, xn are the nodes of the quadrature formula, A1, A2 . . . ,An are its coefficients, and Rn 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.


Krylov, V. I. Priblizhennoe vychislenie integralov, 2nd ed. Moscow, 1967.
The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
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.
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.
To integrate an arbitrary function h(x) over the element [[OMEGA].sup.e], the symmetric quadrature formulas, described by Zhang et al.
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.
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.
Now if f is as above, then we have the one-point and two-points open Newton-Cotes quadrature formulas
Quadrature formulas based on rational interpolation.
USPENSKY, On the convergence of quadrature formulas related to an infinite interval, Trans.
Stancu, "Quadrature formulas with multiple Gaussian nodes," Journal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis, vol.

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