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Numerical Integration 55
3.7 Clenshaw-Curtis integration formulas
The formulas analyzed in the previous sections are very accurate when the weighted
integral of a polynomial is needed. The weight function is strictly related to the set of
nodes in which the polynomial is known. Nevertheless, very often in applications, one
is concerned with evaluating the integral corresponding to a different weight function.
For example, we would like to compute 1
pdx, where p is determined at some Jacobi
−1
Gauss-Lobatto nodes. This time, after integrating (3.2.7), we end up with the formula
(3.7.1)
1
−1
p dx =
n
j=0
p(η (n)
j ) χ (n)
j ,
where χ (n)
j := 1
−1 ˜l (n)
j dx, 0 ≤ j ≤ n.
Of course, relation (3.7.1) is exact for all polynomials in the space Pn. Unfortunately,
if the nodes are not related to zeroes of derivatives of Legendre polynomials (in this
case we would obtain: χ (n)
j
= ˜w(n)
j , 0 ≤ j ≤ n), we cannot expect the formula to be
exact for polynomials of degree higher than n. On the other hand, (3.7.1) is possibly
interesting for future applications. For this reason, we give the expression of the weights
in the case of Chebyshev nodes (which are presented in (3.1.11)), when n ≥ 4 is even.
In practice, we have
(3.7.2) χ (n)
j =
⎧
⎪⎩
1
n 2 − 1
if j = 0,
⎪⎨
⎡
2
⎣1 −
n
(−1)j
n2 − 1 +
n/2−1
2 2kjπ
cos
1 − 4k2 n
k=1
⎤
⎦ if 1 ≤ j ≤ n
2 ,
χ (n)
n−j
if n
2
+ 1 ≤ j ≤ n.
With this choice of nodes and weights, relation (3.7.1) is known as Clenshaw-Curtis
formula (see davis and rabinowitz(1984)). This is exact for polynomials of degree
not larger than n.