View source for Optimal quadrature

A [[Quadrature formula|quadrature formula]] giving the best approximation to the integral

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for a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068470/o0684702.png" /> of integrands. If

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then

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

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is called the quadrature error in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068470/o0684706.png" />. If a quadrature formula exists such that for the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068470/o0684707.png" /> the equality

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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>