Page 179 - Numerical methods for chemical engineering

Gaussian quadrature 165

Orthogonal polynomials

We now identify some useful properties of orthogonal polynomials.
TheoremGQ1 For the interval [a, b] and a nonnegative weight function w(x), there exists
a set of orthogonal polynomials of leading coefficient one, p j (x) ∈ 1 j , j = 0, 1, 2,...,
such that p j |p k = 0 for j = k. These polynomials are defined uniquely by the recursion
formula

p j+1 (x) = (x − δ j+1 )p j (x) − γ 2 p j−1 (x) (4.68)
j+1
where
p −1 (x) = 0 p 0 (x) = 1
δ j+1 = xp j |p j / p j |p j j = 1, 2,. . .

γ 2 = 0, j = 0 (4.69)
j+1 p j |p j / p j−1 |p j−1 , j = 1, 2,...
A proof of this theorem is provided in the supplemental material.
TheoremGQ2 The N roots {x 1 , x 2 ..., x N } of the orthogonal polynomial p N (x) are real,
distinct and lie in (a, b), i.e. a < x 1 < x 2 ··· < x N < b. The roots of p N (x) may be computed
by the eigenvalue technique of Chapter 3.
A proof of this theorem is provided in the supplemental material.

Theorem GQ3 The N × N matrix
 
p 0 (x 1 ) p 0 (x 2 ) ... p 0 (x N )
p 1 (x 1 ) p 1 (x 2 ) ... p 1 (x N )
 
  (4.70)
. . . 
. . .
A = 
 . . . 
p N−1 (x 1 ) p N−1 (x 2 ) ... p N−1 (x N )
is nonsingular for any mutually distinct arguments {x 1 ,x 2 ,...,x N }.
A proof of this theorem is provided in the supplemental material.

Gaussian quadrature

We next find that the roots x 1 , x 2 ,..., x N of the orthogonal polynomial p N (x) provide a set
of support points that yield highly accurate estimates of weighted integrals on [a, b].

Theorem GQ4 Let {x 1 , x 2 ..., x N } be the N distinct roots of p N (x) in the open interval
(a, b). Let {w 1 , w 2 ,..., w N } be the solution of the linear system

N
p 0 |p 0 , if j = 0
p j (x k )w k = (4.71)
0, if j = 1, 2,..., N − 1
k=1
whose matrix (4.70) is nonsingular by Theorem GQ3. Then w j > 0 for all j = 1, 2,...,