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Derivative Matrices 145
The matrix is given by squaring the equivalent of the matrix in (7.4.11), for a general
n ≥ 2. In fact, (7.4.23) can be decoupled, with the substitution ˆp := p ′′ , into two
consecutive second order systems.
We finally conclude with some examples relative to unbounded domains. Let us
consider first the case I =]0,+∞[. Here, problems are formulated in the space of
Laguerre functions (see sections 6.7 and 9.5). For Q ∈ Sn−2, n ≥ 2, σ ∈ R and
µ > 0, we are concerned with finding the solution P ∈ Sn−1 of the problem
(7.4.25)
⎧
⎨
⎩
−P ′′ (η (n)
i ) + µP(η (n)
i ) = Q(η (n)
i ), 1 ≤ i ≤ n − 1,
P(η (n)
0 ) = σ,
where the η (n)
i ’s are the Laguerre Gauss-Radau nodes (see section 3.6). By setting
q(x) := Q(x)e x , p(x) := P(x)e x , ∀x ∈ [0,+∞[, we get an equivalent formulation in
the space of polynomials
(7.4.26)
⎧
⎨
⎩
−p ′′ (η (n)
i ) + 2p ′ (η (n)
i ) + (µ − 1)p(η (n)
i ) = q(η (n)
i ), 1 ≤ i ≤ n − 1,
p(η (n)
0 ) = σ.
This leads to a n×n linear system. The entries of the corresponding matrix are obtained
in the usual way. Similar considerations hold for other types of boundary conditions. We
remark that we only have one boundary point for a second-order equation. Nevertheless,
problem (7.4.25) admits a unique solution. Actually, another condition is implicitly
assumed by observing that P decays to zero at infinity exponentially fast. More
details are given in section 9.5.
For the Hermite case there are no boundary points. Let us discuss an example.
The generating problem will be given in section 10.2. Let Q ∈ Sn−1, n ≥ 1, be a
Hermite function and µ > 0. We want to find P ∈ Sn−1 such that
(7.4.27) −P ′′ (ξ (n)
i ) − 2ξ (n)
i P ′ (ξ (n)
i ) + µP(ξ (n)
i ) = Q(ξ (n)
i ), 1 ≤ i ≤ n,