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54 Polynomial Approximation of Differential Equations
integral of polynomials of degree 2n−1, knowing their values only at n+1 points. With
respect to Gauss formulas there is a slight loss in accuracy. Nevertheless, the fact that
the points x = −1 and x = 1 are included in the nodes is important when imposing
boundary conditions in the approximation of differential equations (see section 7.4).
3.6 Gauss-Radau integration formulas
Other integration formulas, known as Gauss-Radau formulas, are based on the nodes
given in (3.1.12), (3.1.13) and (3.1.14) respectively. The first case actually corresponds
to a Gauss type integration formula since, thanks to (1.3.6), the nodes given in (3.1.12)
are the zeroes of P (3/2,3/2)
n−1 . Therefore, the relative formula is exact for polynomials up
to degree 2n − 3. The other two sets of n nodes include the points x = 1 or x = −1
respectively. Appropriate weights can be defined so that the associated integration
formulas are valid for polynomials of degree up to 2n − 2. Some details are discussed
for instance in ralston(1965) or davis and rabinowitz(1984).
In this section we are concerned with an additional formula, based on the zeroes of
d
dx L(α)
n plus the point x = 0 (the only boundary point of the interval [0,+∞[). This is
(3.6.1)
+∞
0
pw dx =
n−1
j=0
p(η (n)
j ) ˜w (n)
j .
˜(n) l j wdx, where the Lagrange
Here w is the Laguerre weight function and ˜w (n)
j := +∞
0
polynomials are defined in (3.2.11). With a proof similar to that of theorem 3.5.1,
equation (3.6.1) turns out to be true for any polynomial p of degree at most 2n −2. For
the weights, we obtain
(3.6.2) ˜w (n)
j =
⎧
(α + 1) Γ
⎪⎨
⎪⎩
2 (α + 1) (n − 1)!
Γ(n + α + 1)
Γ(n + α)
n!
L (α)
n (η (n)
j ) d
dx L(α)
n−1 (η(n)
j )
−1
if j = 0,
if 1 ≤ j ≤ n − 1.
These formulas are proven with an argument similar to that used in theorem 3.5.2.