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48 Polynomial Approximation of Differential Equations
Expression (3.4.1) is known as Gauss quadrature formula. We will also refer to the ξ (n)
j ’s
as the Gauss integration nodes. Of course, once the weights are evaluated, (3.4.1) holds
for any polynomial p ∈ Pn−1.
Until now, we have not made use of the property that the nodes are related to
zeroes of orthogonal polynomials. This assumption is fundamental in proving the next
remarkable result.
Theorem 3.4.1 - Formula (3.4.1) is true for any polynomial p of degree less or equal
2n − 1.
Proof - We take q := Iw,np. Hence, q ∈ Pn−1 and p − q vanishes at the points
ξ (n)
j , 1 ≤ j ≤ n. Then p − q = unr, where r is a polynomial of degree at most n − 1.
Finally, by orthogonality and by using the exactness of (3.4.1) for polynomials of degree
n − 1, we get
I
pw dx =
=
n
j=1
I
qw dx +
q(ξ (n)
j ) w (n)
j
I
unrw dx =
=
n
j=1
I
qw dx
p(ξ (n)
j ) w (n)
j .
We leave as exercise the proof of the theorem’s converse, i.e., if formula (3.4.1) holds
for any p ∈ P2n−1 then the nodes are the zeroes of un. The degree of the integration
formula cannot be improved further. In fact, if one takes p = u 2 n ∈ P2n the right-hand
side of (3.4.1) vanishes. On the contrary, the left-hand side is different from zero.
Explicit expressions are known for the quantities w (n)
j , 1 ≤ j ≤ n, in the various
cases. We summarize the formulas.
Jacobi case - For α > −1, β > −1 and n ≥ 1, we have