The word quadrature has (at least) three incompatible meanings. Integration by quadrature either means solving an integral analytically (i.e., symbolically in terms of known functions), or solving of an integral numerically (e.g., Gaussian quadrature, Newton-Cotes formulas). Ueberhuber (1997, p. 71) uses the word "quadrature" to mean numerical computation of a univariate integral, and "cubature" to mean numerical computation of a multiple integral.
The word quadrature is also used to mean squaring: the construction of a square using only compass and straightedge which has the same area as a given geometric figure. If quadrature is possible for a plane figure, it is said to be quadrable.
For a function 
tabulated at given values 
(so the abscissas cannot be chosen at will), write the
function 
as a sum of orthonormal functions 
satisfying
 |
(1)
|
as
 |
(2)
|
and plug into the Lagrange interpolating polynomial of 
through the 
points (as is done in Gaussian quadrature)
 |  |  |
(3)
|
 |  |  |
(4)
|
where
 |
(5)
|
giving
![int_a^bsum_(j=0)^inftya_jp_j(x)W(x)dx=sum_(i=1)^nw_i[sum_(j=0)^inftya_jp_j(x_i)].](/images/equations/Quadrature/NumberedEquation4.svg) |
(6)
|
But we wish this to hold for all degrees of approximation, so
 |
(7)
|
 |
(8)
|
Setting 
in (◇) gives
 |
(9)
|
The zeroth order orthonormal function can always be taken as 
, so (9) becomes
 |  |  |
(10)
|
 |  |  |
(11)
|
where (◇) has been used in the last step. We therefore have the matrix equation
![[p_0(x_1) ... p_0(x_n); p_1(x_1) ... p_1(x_n); | ... |; p_(n-1)(x_1) ... p_(n-1)(x_n)][w_1; w_2; |; w_n]=[1; 0; |; 0],](/images/equations/Quadrature/NumberedEquation8.svg) |
(12)
|
which can be inverted to solve for the 
s (Press et al. 1992).
