Theorem 20 (Gauss quadratures). Let be a weight function on the interval , as introduced in Eq. (9.17), and let be the associated orthogonal polynomial of degree . Then, the quadrature rule (9.63) with the choice (9.65) for the discrete weights, and as nodal points the roots of is exact for all polynomials of degree at most .

The following remarks are in order:

- The roots of are referred to as Gauss points or nodes.
- Suppose that . Then the Gauss points, i.e., the roots of the Chebyshev polynomial [see Eq. (9.68)], are exactly the points that minimize the infinity norm of the nodal polynomial in the interpolation problem, as discussed in Section 9.3.3.

One can see that the Gauss points actually lie inside the interval , and do not contain the endpoints or . Now suppose that for some reason we want the nodes to include the end points of integration,

One reason for including the end points of the interval in the set of nodes is when applying boundary conditions in the collocation approach, as discussed in Section 10. Then we are left with two less degrees of freedom compared to Gauss quadratures and therefore expect to be able to make the quadrature exact for polynomials of order up to . This leads to:Theorem 21 (Gauss–Lobatto quadratures). If we choose the discrete weights according to Eq. (9.65) as before but as nodal points, the Gauss–Lobatto ones, i.e., the roots of the polynomial

with and chosen so that , then the quadrature rule (9.63) is exact for all polynomials of degree at most .Note that the coefficients and in the previous equations are obtained by solving the simple system

One can similarly enforce that only one of the end points coincides with a quadrature one, leading to Gauss–Radau quadratures. The proofs of Theorems 20 and 21 can be found in most numerical analysis books, in particular [242].

For Chebyshev polynomials there are closed form expressions for the nodes and weights in Eqs. (9.63) and (9.65):

Since both products and are polynomials of degree , their quadratures are exact (in fact, the equality holds for each term separately):

where we have introduced the discrete counterpart of the weighted scalar product (9.17), with the nodes and discrete weights those of the corresponding quadrature.On the other hand, in the Legendre case,

and therefore Property (9.75) will be used in Section 9.7 when discussing stability through the energy method, much as in the FD case.
Living Rev. Relativity 15, (2012), 9
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