Best quadrature formula

optimal quadrature formula

An approximate integration formula that guarantees the minimum error for a given class of functions, relative to all formulas of a specified type. As an example, consider the quadrature formula

(*)

where is a weight function. The remainder (error) term depends both on the function , and on the vector consisting of the interpolation nodes (it is usually assumed that ) and the coefficients , ; . Fixing and , let denote some set of vectors (and hence also some set of quadrature formulas), defined by some restrictions on the interpolation nodes and coefficients (in particular, one might consider the set of coefficients for a fixed node vector ). Let be some class of functions , it being assumed that the integral and the sum in (*) exist. The best quadrature formula of type (*) for the class relative to the set is defined by a vector for which

The construction of best quadrature formulas is intimately connected with certain problems in spline approximation; in many cases it reduces to minimizing the norm of a monospline (see [1]). Best quadrature formulas, together with sharp estimates for the remainder term, are known for many important classes of continuous and differentiable functions. From a more general point of view, the problem of finding best quadrature formulas and the corresponding errors for a class may be viewed as the problem of optimal recovery of a functional

where , on the basis of the information , ; . The concept of a best quadrature formula generalizes in a natural way to functions of several variables (cubature formulas).

References

[1]	S.M. Nikol'skii, "Quadrature formulae" , H.M. Stationary Office , London (1966) (Translated from Russian)
[2]	N.M. Krylov, "Approximate calculation of integrals" , Macmillan (1962) (Translated from Russian)
[3]	P.J. Laurent, "Approximation et optimisation" , Hermann (1972)
[4]	A.A. Zhensykbaev, "Monosplines of minimal norm and quadrature formulas" Uspekhi Mat. Nauk , 36 : 4 (1981) pp. 107–159 (In Russian)

Comments

The terminology "best formula" is often encountered in the literature on numerical analysis, but, as was observed in [a2], p. 75, it should be taken with a large dose of salt, because, after all, any quadrature formula, no matter how the weights and the nodes are chosen, will exactly integrate an infinite-dimensional family of functions.

A few recent textbooks are listed below.

References

[a1]	H. Brass, "Quadraturverfahren" , Vandenhoeck & Ruprecht (1977)
[a2]	P.J. Davis, P. Rabinowitz, "Methods of numerical integration" , Acad. Press (1984)
[a3]	H. Engels, "Numerical quadrature and cubature" , Acad. Press (1980)