Optimal quadrature

A quadrature formula giving the best approximation to the integral

$$I(f)=\int\limits_\Omega f(P)\omega(P)dP$$

for a class $F$ of integrands. If

$$S_N(f)=\sum_{k=1}^Nc_kf(P_k),$$

then

$$R_N(f)=S_N(f)-I(f)$$

is called the quadrature error when calculating the integral of a given function, while

$$r_N(F)=\sup_{f\in F}|R_N(f)|$$

is called the quadrature error in the class $F$. If a quadrature formula exists such that for the corresponding $r_N(F)$ the equality

$$r_N(F)=\inf_{c_k,P_k}r_N(F)$$

holds, then this formula is called the optimal quadrature in this class.

Optimal quadratures have only been found for certain classes of functions which, basically, depend on one variable (see [1]–[3]). Optimal quadratures are also called best quadrature formulas or extremal quadrature formulas.

References

[1]	S.M. Nikol'skii, "Quadrature formulae" , H.M. Stationary Office , London (1966) (Translated from Russian)
[2]	N.S. Bakhvalov, "On optimal convergence estimates for quadrature processes and integration methods of Monte-Carlo type on function classes" , Numerical Methods for Solving Differential and Integral Equations and Quadrature Formulas , Moscow (1964) pp. 5–63 (In Russian)
[3]	S.L. Sobolev, "Introduction to the theory of cubature formulas" , Moscow (1974) (In Russian)

References

[a1]	H. Engels, "Numerical quadrature and cubature" , Acad. Press (1980)