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Derivative Matrices 137
Next, consider second-order derivatives. Now, two additional conditions are re-
quired to select a unique primitive function. Since there are several possibilities, we
only consider one example. Let A, B, σ1, σ2, be real constants and q ∈ Pn−2, n ≥ 2.
In the ultraspherical case, one is concerned with finding p ∈ Pn such that
⎧
(7.3.5)
−p ′′ + Πw,n−2(Ap ′ + Bp) = q,
⎪⎨
p(−1) = σ1,
⎪⎩
p(1) = σ2.
For instance, when n = 5, problem (7.3.5) leads to a linear system of the form
(7.3.6)
⎡
B ∗ ∗ ∗ ∗
⎤⎡
∗
⎢ 0
⎢ 0
⎢ 0
⎣
∗
B
0
0
∗
∗
B
0
∗
∗
∗
B
∗
∗
∗
∗
∗
∗ ⎥⎢
⎥⎢
∗ ⎥⎢
⎥⎢
∗ ⎥⎢
⎦⎣
∗
∗ ∗ ∗ ∗ ∗ ∗
where the dk’s are the Fourier coefficients of q. Again, this can be solved with a direct
procedure with a cost proportional to n 2 .
From (1.3.2) and (1.3.3), in the ultraspherical case we have uk(1) = (−1) k uk(−1),
k ∈ N. This implies
(7.3.7) 2
n
k=0
k even
ck uk(1) = σ1 + σ2, 2
c0
c1
c2
c3
c4
c5
⎤
⎥
⎦
n
=
k=1
k odd
⎡
⎢
⎣
d0
d1
d2
d3
σ1
σ2
⎤
⎥
⎥,
⎥
⎦
ck uk(1) = σ1 − σ2.
Therefore, when A = 0, (7.3.6) can be decoupled into the subsystems
(7.3.8)
⎡
B
⎣ 0
∗
B
⎤⎡
∗
∗⎦
⎣
∗ ∗ ∗
c0
⎤ ⎡
c2 ⎦ = ⎣ d0
d2
⎤
⎦,
⎡
B
⎣ 0
∗
B
⎤⎡
∗
∗ ⎦⎣
∗ ∗ ∗
c1
⎤ ⎡
c3 ⎦ =
c4
σ1 + σ2
c5
⎣ d1
d3
σ1 − σ2
This is because c (2)
i , 0 ≤ i ≤ n − 2, is a linear combination of cj, i + 2 ≤ j ≤ n, where
i+j is even (see for instance (7.1.9) and (7.1.10)). Following this idea, a fast algorithm
for the solution of (7.3.5) is presented in clenshaw(1957), for the Chebyshev case.
Further hints are given in canuto, hussaini, quarteroni and zang(1988), p.129,
and boyd(1989), p.380.
⎤
⎦.