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Numerical Integration 51
3.5 Gauss-Lobatto integration formulas
In this section, we study an integration formula based on the nodes obtained from
the derivatives of Jacobi polynomials. Integrating (3.2.6), we get for n ≥ 1 and any
polynomial p ∈ Pn, the so called Gauss-Lobatto formula
(3.5.1)
1
−1
pw dx =
n
j=0
p(η (n)
j ) ˜w (n)
j ,
where w is the Jacobi weight function and ˜w (n)
j := 1
Gauss-Lobatto weights.
−1 ˜l (n)
j
wdx, 0 ≤ j ≤ n, are the
The equation (3.5.1) represents a high degree integration formula by virtue of the fol-
lowing theorem.
Theorem 3.5.1 - Formula (3.5.1) is true for any polynomial p ∈ P2n−1.
Proof - Here we define q := Ĩw,np. Thus p − q = (1 − x 2 )u ′ nr, where r ∈ Pn−2.
Moreover, r is the derivative of some polynomial s ∈ Pn−1. Then, recalling (2.2.8) and
that a = (1 − x 2 )w, one finally obtains
1
−1
pw dx =
=
1
n
j=0
−1
qw dx +
q(η (n)
j ) ˜w (n)
j
1
−1
=
au ′ ns ′ dx =
n
j=0
Formula (3.5.1) does not hold in general when p ∈ P2n.
1
−1
p(η (n)
j ) ˜w (n)
j .
qw dx
One is now left with the determination of the weights. To this end we prove the following
proposition.