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62 Polynomial Approximation of Differential Equations
3.10 Scaled weights
For large n, the determination of the Laguerre weights can sometimes lead to severe
numerical problems. In fact, for the nodes largest in magnitude, the corresponding
weights are very small and decay quite fast to zero for increasing values of n. Following
the hints in section 1.6, it is advisable to introduce some scaled weights.
First, using (1.6.12) and (1.6.13), we note that, for n ≥ 1,
(3.10.1)
d
dx L(α) n (ξ (n)
j ) =
n + α
n
d
dx ˆ L (α)
n (ξ (n)
j )
n
k=1
1 + ξ(n) j
4k
Next, starting from (3.4.7), we define a new set of Gauss type weights
(3.10.2) ω (n)
j
= −
Γ(n + α + 1)
4n 2 n!
:= w (n)
j
(4n + ξ (n)
j )
S (α)
n (ξ (n)
−2 j )
, 1 ≤ j ≤ n.
ˆL (α)
n−1 (ξ(n) j ) d
dx ˆ L (α)
n (ξ (n)
−1 j ) , 1 ≤ j ≤ n.
The values of the scaled Laguerre functions at the nodes are evaluated by (1.6.14) and
(1.6.15). Recalling (3.4.1), we obtain the formula
(3.10.3)
where ˆp := pS (α)
n
+∞
p
0
2 (x) x α e −x dx =
n
j=1
ˆp 2 (ξ (n)
j ) ω (n)
j , ∀p ∈ Pn−1,
are used in place of p in numerical computations. The advan-
tages to this approach consist in a better numerical treatment. More details about the
implementation are examined later in section 7.5.
A similar definition is given for Gauss-Radau type weights (see (3.6.2)), i.e.,
(3.10.4) ˜ω (n)
0 := ˜w (n)
0
S (α)
−2 n (0)
= (α + 1) Γ(n + α + 1)
,
n n!