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''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

$$ \tag{* }
\int\limits _ { a } ^ { b } 
\rho (x) f (x)  dx  = \ 
\sum _ {k = 1 } ^ { n } 
\sum _ {i = 0 } ^ { m } 
p _ {ki} f ^ { (i) } (x _ {k} ) + R (f),
$$

where  $  \rho (x) $
is a weight function. The remainder (error) term  $  R (f) = R (f, X _ {n} , P _ {nm} ) $
depends both on the function  $  f (x) $, 
and on the vector  $  (X _ {n} , P _ {nm} ) $
consisting of the interpolation nodes  $  x _ {k} $(
it is usually assumed that  $  x _ {k} \in [a, b] $) 
and the coefficients  $  p _ {ki} $, 
$  k = 1 \dots n $; 
$  i = 0 \dots m $. 
Fixing  $  n \geq  1 $
and  $  m \geq  0 $, 
let  $  A $
denote some set of vectors  $  (X _ {n} , P _ {nm} ) $(
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  $  A = A ( \overline{X}\; _ {n} ) $
of coefficients  $  p _ {ki} $
for a fixed node vector  $  \overline{X}\; _ {n} $). 
Let  $  \mathfrak M $
be some class of functions  $  f (x) $, 
it being assumed that the integral and the sum in (*) exist. The best quadrature formula of type (*) for the class  $  \mathfrak M $
relative to the set  $  A $
is defined by a vector  $  (X _ {n}  ^ {*} , P _ {nm}  ^ {*} ) $
for which

$$ 
\sup _ {f \in \mathfrak M }  |
R (f, X _ {n}  ^ {*} , P _ {nm}  ^ {*} ) | =
$$

$$ 
= \ 
\inf _ {(X _ {n} , P _ {nm} ) \in A }  \sup _ {f
\in \mathfrak M }  | R (f, X _ {n} , P _ {nm} ) | .
$$

The construction of best quadrature formulas is intimately connected with certain problems in [[Spline approximation|spline approximation]]; in many cases it reduces to minimizing the norm of a monospline (see [[#References|[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  $  \mathfrak M $
may be viewed as the problem of optimal recovery of a functional

$$ 
J (f)  =  \int\limits _ { a } ^ { b }  \rho (x) f (x)  dx,
$$

where  $  f \in \mathfrak M $, 
on the basis of the information  $  \{ f ^ { (i) } (x _ {k} ) \} $, 
$  k = 1 \dots n $; 
$  i = 0 \dots m $. 
The concept of a best quadrature formula generalizes in a natural way to functions of several variables (cubature 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.M. Krylov,   "Approximate calculation of integrals" , Macmillan  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.J. Laurent,   "Approximation et optimisation" , Hermann  (1972)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.A. Zhensykbaev,   "Monosplines of minimal norm and quadrature formulas"  ''Uspekhi Mat. Nauk'' , '''36''' :  4  (1981)  pp. 107–159  (In Russian)</TD></TR></table>

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

A few recent textbooks are listed below.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Brass,   "Quadraturverfahren" , Vandenhoeck &amp; Ruprecht  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.J. Davis,   P. Rabinowitz,   "Methods of numerical integration" , Acad. Press  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Engels,   "Numerical quadrature and cubature" , Acad. Press  (1980)</TD></TR></table>