View source for Optimal quadrature

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A [[Quadrature formula|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 [[#References|[1]]]–[[#References|[3]]]). Optimal quadratures are also called best quadrature formulas or extremal quadrature formulas.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.M. Nikol'skii,   "Quadrature formulae" , H.M. Stationary Office , London  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.L. Sobolev,   "Introduction to the theory of cubature formulas" , Moscow  (1974)  (In Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Engels,   "Numerical quadrature and cubature" , Acad. Press  (1980)</TD></TR></table>