Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

70 CHAPTER 5. INTERPOLATION
roots. Then apply Rolle’s Theorem again to find φ ′′ (t) has n roots. In this way we see that φ (n+1) (t)
has a root, call it ξ. That is
0 = φ (n+1) (ξ) = f (n+1) (ξ) − p (n+1) (ξ) − cw (n+1) (ξ).
But p(t) is a polynomial of degree ≤ n, so p (n+1) is identically zero. And w(t) is a polynomial
of degree n + 1 in t, so its n + 1 th derivative is easily seen to be (n + 1)! Thus
which is what was to be proven.
0 = f (n+1) (ξ) − c(n + 1)!
c(n + 1)! = f (n+1) (ξ)
f(x) − p(x) 1
=
w(x) (n + 1)! f (n+1) (ξ)
1
f(x) − p(x) =
(n + 1)! f (n+1) (ξ)w(x),
Thus the error in a polynomial interpolation is given as
1
n∏
f(x) − p(x) =
(n + 1)! f (n+1) (ξ) (x − x i ) .
We have no control over the function f(x) or its derivatives, and once the nodes and f are fixed, p
is determined; thus the only way we can make the error |f(x) − p(x)| small is by judicious choice
of the nodes x i .
The Chebyshev nodes on [−1, 1] have the remarkable property that
n∏
(t − x
∣
i )
∣ ≤ 2−n
i=0
for any t ∈ [−1, 1] . Moreover, it can be shown that for any choice of nodes x i that
n∏
max
(t − x
∣
i )
∣ ≥ 2−n .
t∈[−1,1]
i=0
Thus the Chebyshev nodes are considered the best for polynomial interpolation.
Merging this result with Theorem 5.8, the error for polynomial interpolants defined on Chebyshev
nodes can be bounded as
|f(x) − p(x)| ≤
1
2 n (n + 1)!
max
ξ∈[−1,1]
i=0
∣
∣f (n+1) (ξ)
The Chebyshev nodes can be rescaled and shifted for use on the general interval [α, β]. In this
case they take the form
x i = β − α [( ) ] 2i + 1
cos
π + α + β , 0 ≤ i ≤ n.
2 2n + 2 2
In this case, the rescaling of the nodes changes the bound on ∏ (t − x i ) so the overall error bound
becomes
(β − α)n+1 ∣
|f(x) − p(x)| ≤
2 2n+1 max ∣f (n+1) (ξ) ∣ ,
(n + 1)! ξ∈[α,β]
for x ∈ [α, β] .
∣ .