In general, an integration formula which is based on a Taylor series using \( N \) points, will integrate exactly a polynomial \( P \) of degree \( N-1 \). That is, the \( N \) weights \( \omega_n \) can be chosen to satisfy \( N \) linear equations, see chapter 3 of Ref.\ [3]. A greater precision for a given amount of numerical work can be achieved if we are willing to give up the requirement of equally spaced integration points. In Gaussian quadrature (hereafter GQ), both the mesh points and the weights are to be determined. The points will not be equally spaced.

The theory behind GQ is to obtain an arbitrary weight \( \omega \) through the use of so-called orthogonal polynomials. These polynomials are orthogonal in some interval say e.g., [-1,1]. Our points \( x_i \) are chosen in some optimal sense subject only to the constraint that they should lie in this interval. Together with the weights we have then \( 2N \) (\( N \) the number of points) parameters at our disposal.