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Numerical Integration 47
This means that, by knowing the values p(ξ (n)
j ), 1 ≤ j ≤ n, we can for instance recover
the point values of Iw,np 2 . These are p 2 (ξ (n)
j ), 1 ≤ j ≤ n. Hence, the value in x of
the interpolant of p 2 is n
j=1 p2 (ξ (n)
j )l (n)
j (x).
This trivial remark is a key point in the treatment of nonlinear terms in the approxi-
mation of differential equations (see section 9.8).
d (α,β)
Another interpolant operator is defined in relation to the zeroes of dxP n . Thus
for n ≥ 1, let us consider Ĩw,n : C0 ([−1,1]) → Pn to be the operator associating to the
function f, the unique polynomial pn ∈ Pn satisfying pn(η (n)
j ) = f(η (n)
j ), 0 ≤ j ≤ n.
The operator is linear and we get
(3.3.3) Ĩw,np = p, ∀p ∈ Pn.
For Laguerre polynomials Ĩw,n : C0 (n)
([0,+∞[) → Pn−1 is characterized by ( Ĩw,nf)(η j )
= f(η (n)
j ), 0 ≤ j ≤ n − 1, where η (n)
0 = 0.
As already mentioned in section 2.4, the analysis when n tends to +∞, of the
operators introduced above, is one of the primary issues of chapter six.
3.4 Gauss integration formulas
We have reached the crucial subject of this chapter, that is the formulas for numerical
integration. Given a polynomial p ∈ Pn, one is concerned with finding the value of
I pwdx. If the Fourier expansion of p = n−1 k=0 ckuk is known, the answer has been
given in chapter two, i.e.,
pwdx = c0
I I u0wdx. On the other hand, when p is known
through its values at the zeroes of un, it is sufficient to integrate (3.2.2) obtaining
(3.4.1)
where w (n)
j :=
I l(n) j
I
pw dx =
n
j=1
p(ξ (n)
j ) w (n)
j ,
wdx, 1 ≤ j ≤ n, are the weights of the integration formula.