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3
NUMERICAL INTEGRATION
As we shall see in this chapter, orthogonal polynomials allow us to generate in a very
elegant way high order integration formulas, known as Gauss formulas. Since the theory
is based on the knowledge of the zeroes of the polynomials, the first part is devoted to
the characterization of their main properties.
3.1 Zeroes of orthogonal polynomials
It is well-known that a polynomial of degree n has at most n distinct complex zeroes.
In general, very little can be said about real zeroes. An amazing theorem gives a precise
and complete answer in the case of orthogonal polynomials.
Theorem 3.1.1 - Let {un} n∈N be a sequence of solutions of (1.1.1), where un is a
polynomial of degree n, satisfying the orthogonal relation (2.2.1). Then, for any n ≥ 1,
un has exactly n real distinct zeroes in I.
Proof - We first note that
I un(x)w(x)dx = 0. Therefore un changes sign in I, hence
it has at least one real zero ξ1 ∈ I. If n = 1 the proof is ended. If n > 1, then we can
find another zero ξ2 ∈ I of un, with ξ2 = ξ1, since if un vanished only at ξ1, then the
polynomial (x − ξ1)un would not change sign in I, which is in contradiction with the
relation
I (x − ξ1)un(x)w(x)dx = 0, obtained by orthogonality. In a similar fashion,