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Numerical Integration 53
The proof is completed using (2.2.10). For the weights relative to the endpoints, we
write
(3.5.5) 0 =
1
−1
= 2un(−1)u ′ n−1(−1) ˜w (n)
0
unu ′ n−1(1 − x)w dx =
n
m=0
(unu ′ n−1)(η (n)
m )(1 − η (n)
m ) ˜w (n)
m
− 2α+β (2n + α + β)Γ(n + α)Γ(n + β)
(n + α + β + 1) n! Γ(n + α + β)
n−1
m=1
(1 − η (n)
m ).
The first equality in (3.5.5) is due to orthogonality, while in the last equality we used
the expression of the weights obtained above for 1 ≤ m ≤ n − 1. Then, by (1.3.3) and
(3.1.18) we deduce the value of ˜w (n)
0 . In the same way one gets the value of ˜w (n)
n .
In the ultraspherical case, since the polynomials are even or odd, the nodes are
symmetrically distributed in [−1,1]. Therefore, when α = β we have n−1
j=m η(n)
m = 0.
Hence, in the Legendre case we obtain
(3.5.6) ˜w (n)
j
=
⎧
2
⎪⎨ n(n + 1)
⎪⎩
−2
Pn(η
n + 1
(n)
j )P ′ n−1(η (n)
−1 j )
if j = 0 or j = n,
if 1 ≤ j ≤ n − 1.
For the Chebyshev case, relations (3.5.2) are drastically simplified and give
(3.5.7) ˜w (n)
j
=
⎧
π
⎪⎨ 2n
⎪⎩
π
n
if j = 0 or j = n,
if 1 ≤ j ≤ n − 1.
This is a consequence of (3.1.15) and the relation T ′ n−1(η (n)
j ) = −(n − 1)(−1) j+n ,
1 ≤ j ≤ n − 1.
Also for Gauss-Lobatto formulas it is possible to prove that the weights are positive.
More about the subject can be found in engels(1980) or davis and rabinowitz(1984).
Integration formulas of Gauss-Lobatto type allow the determination of the weighted