This note is mainly based on the book Spectral Methods: Algorithms, Analysis and Applications (sec 3.1.4, 3.2.2, 3.3.1) by Jie Shen, Tao Tang and Li-Lian Wang.
The Gauss-type quadrature seek the best numerical approximation of an integral by selecting optimal nodes at which the integrand is evaluated. It belongs to the family of the numerical quadratures:
where are the quadrature nodes and weights, respectively, and is the error term. If , we say the quadrature is exact for . We assume the nodes are distinct. If , then the error term can be written as
where . The Lagrange basis polynomials asscoiated with the nodes are defined as
for . Taking in , the quadrature weights can be determined by
for .
The intergration formula has a degree of precision (DOP) , if there holds for all , but exists such that . In general, the DOP of a quadrature rule by any distinct nodes is between and . It’s remarkable that by choosing the nodes as the zeros of the orthogonal polynomials, the DOP can be maximized. Such quadrature rules are called Gauss quadratures.
Gauss Quadrature. Let be the set of zeros of the orthogonal polynomial . Then there exists a unique set of weights such that
where the quadrature weights are all positive and given by
for , where is the leading coefficient of and .
Note that all the nodes of the Gauss formula lie in the interior of the interval . This makes it difficult to impose boundary conditions. Next we consider the other types of quadratures that include the boundary points.
Gauss-Radau Quadrature. Assuming we would like to include the left boundary point in the quadrature nodes, we define
where . It’s obvious that and for any we have
Hence defines a sequence of orthogonal polynomials with respect to the weight function , and the leading coefficient of is . Let be the zeros of , then there exists a unique set of weights such that
Moreover, the quadrature weights are all positive and given by
and
for .
Similarly, we can define the Gauss-Lobatto quadrature by including the right endpoint instead of the left endpoint .
Gauss-Lobatto Quadrature. Now we turn to the Gauss-Lobatto quadrature whose nodes include both the left and right endpoints. In this case, we choose and such that
for and , and set
Also we have and for any we have
Therefore defines a sequence of orthogonal polynomials with respect to the weight function , and the leading coefficient of is . Let be the zeros of , then there exists a unique set of weights such that
where the quadrature weights are all positive and given by
and
for .
Let be a set of Gauss, Gauss-Radau or Gauss-Lobatto quadrature nodes and weights. We can define the following discrete inner product
and the corresponding norm
Note that the discrete inner product is an approximation of the continuous inner product , and the exactness of Gauss-type quadratures implies
where or for Gauss, Gauss-Radau or Gauss-Lobatto quadratures, respectively.
For any , we define the interpolation operator such that
where for Gauss, Gauss-Radau or Gauss-Lobatto quadratures, and are the corresponding quadrature nodes.
The above interpolation condition implies that for any . On the other hand, since , we have
By taking the discrete inner product with for , we obtain the forward discrete polynomial transform
for , where for and
Conversely, the backward discrete polynomial transform is given by
