Untitled - Cdm.unimo.it

Numerical Integration 59
(3.8.14) pw =
p 2 w,n − 1
2π
2
[(p,Tn)w,n]
1
2
, ∀p ∈ Pn,
when specialized to the Legendre and Chebyshev cases respectively.
The result of theorem 3.8.3 can be generalized as follows:
(3.8.15) (p,q)w = (p,q)w,n − un 2 w,n − un 2 w
un 4 w,n
We leave the proof as exercise.
(p,un)w,n (q,un)w,n, ∀p,q ∈ Pn.
Another quantity we would like to compute is the integral 1
−1 p2 dx, in terms of
the values of p ∈ Pn at some nodes. When the nodes are the Legendre Gauss-Lobatto
points, we can use (3.8.13). On the contrary, we are in a situation similar to that of
section 3.7. We only solve the problem for Chebyshev Gauss-Lobatto nodes. Here we
have p 2 ∈ P2n, while (3.7.1) is satisfied only in Pn. The idea is to duplicate the number
of points and still use (3.7.1). Recalling (3.1.11) and (3.1.4), in the Chebyshev case we
have the remarkable result that
(3.8.16) {ξ (n)
i }1≤i≤n ∪ {η (n)
j }0≤j≤n = {η (2n)
k }0≤k≤2n.
Since p is known at the points (3.1.11) and p(η (n)
j ) = p(η (2n)
2j ), 0 ≤ j ≤ n, we can
extrapolate its value at the points (3.1.4) with the help of (3.2.7):
n
(3.8.17) p(η (2n)
2i−1 ) = p(ξ(n) i ) =
j=0
Now, we can use (3.7.1), obtaining ∀p ∈ Pn
(3.8.18)
=
n
j=0
1
−1
p 2 dx =
p 2 (η (n)
j ) χ (2n)
2j +
2n
m=0
n
n
i=1
p(η (n)
j ) ˜ l (n)
j (ξ (n)
i ), 1 ≤ i ≤ n.
k=0
p 2 (η (2n)
m ) χ (2n)
m
p(η (n)
k ) ˜ l (n)
k (ξ(n)
i )
2
χ (2n)
2i−1 .
Finally, the weights can be determined from (3.7.2), and the coefficients ˜ l (n)
k (ξ(n)
i )
from (3.2.10) and (3.1.16). This procedure has a computational cost proportional to n 2 ,
versus the cost of implementing (3.8.10), which is proportional to n.