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Derivative Matrices 131
(7.2.7) (Jacobi) d (1)
ii
(7.2.8) (Laguerre) d (1)
ii
(α + β + 2)ξ(n) i
=
2 (1 − (ξ (n)
i ) 2 )
(7.2.9) (Hermite) d (1)
ii
ξ(n) i =
2ξ (n)
i
= ξ(n)
i .
+ α − β
, α > −1, β > −1,
− α − 1
, α > −1,
Higher order derivative matrices are obtained by multiplying Dn by itself a suitable
number of times. For example, an explicit expression of D 2 n in the Hermite case is given
in funaro and kavian(1988).
In general, we denote by d (k)
ij , 1 ≤ i ≤ n, 1 ≤ j ≤ n, the entries of the matrix Dk n,
k ≥ 1. Of course, when k ≥ n, we have d (k)
ij
= 0, 1 ≤ i ≤ n, 1 ≤ j ≤ n.
We can argue in the same way using the Gauss-Lobatto points. For any p ∈ Pn,
n ≥ 1, we have
(7.2.10) p ′ (η (n)
i ) =
where
(7.2.11)
˜ d (1)
ij :=
n
j=0
˜d (1)
ij p(η(n)
j ), 0 ≤ i ≤ n,
d
dx ˜l (n)
j (η (n)
i ), 0 ≤ i ≤ n, 0 ≤ j ≤ n.
Similarly, we define the (n + 1) × (n + 1) matrix ˜ Dn := { ˜ d (1)
ij
} 0≤i≤n . Again, we denote
0≤j≤n
by ˜ d (k)
ij , 0 ≤ i ≤ n, 0 ≤ j ≤ n, the entries of the (n + 1) × (n + 1) matrix ˜ D k n, k ≥ 1.
When k ≥ n + 1, ˜ D k n is the zero matrix.
The computation of the entries of ˜ Dn needs more perseverance, but it is not hard to
manage. From (3.2.8), we have