204
4. Series-Expansion Methods
distributed over the domain a :::: x :::: b. Then f(x) ean be expressed in the form
of a Lagrange interpolating polynomial
f(x) = L
m
f(xj)pj(x),
j=1
where
If
(4.59)
it follows immediately that
l
b
f(x) dx = AI f(x)) + A2f(X2) + ...+ Amf(x m),
(4.60)
and that (4.59) and (4.60) give the exaet integral of all polynomials f (x) of order
less than or equal to m - 1.
The preeeding fonnula aehieves exaet results for polynomials up to order m - 1
without imposing any eonstraint on the loeation ofthe x j within the interval [a, b].
Gaussian quadrature, on the other hand, obtains exaet results for polynomials up
to order 2m - 1 without adding more terms to the quadrature fonnula by ehoosing
the x j to be the zeros of the Legendre polynomial of order m. The role played
by Legendre polynomials in Gaussian quadrature is essentially independent of
their relation to the assoeiated Legendre funetions and spherieal hannonics. The
property of the Legendre polynomials that is important for Gaussian quadrature
is that these polynomials satisfy the orthogonality condition?
1
28m n
Pm (x)P n (x) dx = - - '-.
- I
2n + 1
In order to appreeiate how this judicious ehoice for the x j inereases the accuraey of (4.59) and (4.60), suppose that f (x) is a polynomial of order 2m - 1 and
let Pm(x) be the Legendre polynomial on [a , b] of order m. If q(x) and r(x) are,
respeetively, the quotient and the remainder obtained when dividing f by Pm,
then f = q Pm + r, where both q and r are polynomials of order less than or
90 ther commonly used sets of orthogonal polynomials are orthogonal with respect to a nonconstant
weight function. In the case of Chebyshev polynomials, for example,
1
1 Tm(x )Tn (x )(I -X 2)-1 / 2dx=O,
-I
1
unlessm = n .