MATH 640 – Numerical Analysis
Section 8.2: Orthogonal Polynomials and Least Squares Approximation
Let f(x)∈C[a, b]. We are trying to find the nth order least squares approximating polynomial Pn(x) =
anxn+an−1xn−1+···+a1x+a0. In other words we are tying to determine the coefficients a0, a1,· · · , an
that minimize the new error
E(a0, a1,· · · , an) = Zb
a
[f(x)−Pn(x)]2dx
NOTE: this is a bit different than our work in Section 8.1, where we were given specific data point,
and not the original function f(x).
Normal Equations Associated with E(a0, a1,·, an=Zb
a
[f(x)−Pn(x)]2dx
n
X
k=0
akZb
a
xk+jdx =Zb
a
f(x)xjdx
or expanded to be
a0Rb
ax0dx +a1Rb
ax1dx +··· +anRb
axndx =Rb
af(x)x0dx
a0Rb
ax1dx +a1Rb
ax2dx +··· +anRb
axn+1 dx =Rb
af(x)x1dx
.
.
..
.
..
.
..
.
.
a0Rb
axndx +a1Rb
axn+1 dx +··· +anRb
ax2ndx =Rb
af(x)xndx
Definition 1 (Linearly Independent Functions)
A set of functions {φ0, φ1,· · · , φn}is linearly independent on [a, b]if whenever
c0φ0(x) + c1φx(x) + · · · +cnφn(x) = 0 for all x∈[a, b]
it must be that c0=c1=· · · =cn= 0. Otherwise the set is linearly dependent.
Theorem 1
If φj(x)is a polynomial of degree j, for each j= 0,1,· · · , n then {φ0, φ1,· · · , φn}is linearly independent
on any interval [a, b].
Definition 2 (Πn)
The set of all polynomials of degree at most n, is denoted by Πn.
Theorem 2
If {φ0, φ1,· · · , φn}is a collection of linearly independent polynomials in Πn, then any polynomial in Πn
can be written uniquely as a linear combination of φ0, φ1,· · · , φn.
Definition 3 (Weight Function)
An integrable function wis called a weight function on the interval Iif w(x)≥0for all xin Ibut
w(x)6≡ 0on any subinterval of I.
Definition 4 (Another Error Function, E, and Normal Equations)
E=Zb
a
w(x)[f(x)−Pn(x)]2dx
