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A method for approximating an [[Integral operator|integral operator]] by constructing numerical methods for the solution of integral equations. The simplest version of a quadrature-sum method consists in replacing an integral operator, for instance of the form

$$ 
\int\limits _ { a } ^ { b }  K ( x , s ) \phi ( s)  d s ,
$$

in an [[Integral equation|integral equation]]

$$ 
\lambda \phi ( x) +
\int\limits _ { a } ^ { b }  K ( x , s ) \phi ( s)  d s  =  f ( s)
$$

by an operator with finite-dimensional range, according to the rule

$$ \tag{1 }
\int\limits _ { a } ^ { b }  K ( x , s ) \phi ( s)  d s  \approx \ 
\sum _ { i= } 1 ^ { N }  a _ {i}  ^ {(} N) K ( x , s _ {i} ) \phi ( s _ {i} ) .
$$

The integral equation, in turn, is approximated by the linear algebraic equation

$$ 
\lambda \widetilde \phi   ( s _ {j} ) +
\sum _ { i= } 1 ^ { N } 
a _ {i}  ^ {(} N) K ( s _ {j} , s _ {i} )
\widetilde \phi   ( s _ {i} )  =  f ( s _ {j} ) ,\ \ 
j = 1 \dots N .
$$

On the right-hand side of the approximate equation (1) is a [[Quadrature formula|quadrature formula]] for the integral with respect to  $  s $. 
Various generalizations of (1) are possible:

$$ \tag{2 }
\int\limits _ { a } ^ { b }  K ( x , s ) \phi ( s)  d s  \approx \ 
\sum _ { i= } 1 ^ { N }  a _ {i}  ^ {(} N) ( x) \phi ( s _ {i} ) ,
$$

where the  $  a _ {i}  ^ {(} N) ( x) $
are certain functions constructed from the kernel  $  K ( x , s ) $. 
The quadrature-sum method as generalized in the form (2) can be applied for the approximation of integral operators with singularities in the kernel and even of singular integral operators.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.V. Kantorovich, V.I. Krylov, "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian) {{MR|0106537}} {{ZBL|0083.35301}} </TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C.T.H. Baker, "The numerical treatment of integral equations" , Clarendon Press (1977) pp. Chapt. 4 {{MR|0467215}} {{ZBL|0373.65060}} </TD></TR></table>