First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 3. INTERPOLATION 112
there exists ξ ∈ (a, b) such that
f[x 0 , ..., x n ]= f (n) (ξ)
.
n!
To prove this theorem, we need the generalized Rolle’s theorem.
Theorem 59 (Rolle’s theorem). Suppose f is a differentiable function on (a, b). If f(a) =
f(b), then there exists c ∈ (a, b) such that f ′ (c) =0.
Theorem 60 (Generalized Rolle’s theorem). Suppose f has n derivatives on (a, b). If f(x) =
0 at (n +1) distinct numbers x 0 ,x 1 , ..., x n ∈ [a, b], then there exists c ∈ (a, b) such that
f (n) (c) =0.
Proof of Theorem 58 . Consider the function g(x) =p n (x)−f(x). Observe that g(x i )=0for
i =0, 1, ..., n. From generalized Rolle’s theorem, there exists ξ ∈ (a, b) such that g (n) (ξ) =0,
which implies
p n (n) (ξ) − f (n) (ξ) =0.
Since p n (x) =f[x 0 ]+f[x 0 ,x 1 ](x − x 0 )+... + f[x 0 , ..., x n ](x − x 0 ) ···(x − x n−1 ), p (n)
n (x) equals
n! times the leading coefficient f[x 0 , ..., x n ]. Therefore
f (n) (ξ) =n!f[x 0 , ..., x n ].
3.3 Hermite interpolation
In polynomial interpolation, our starting point has been the x and y-coordinates of some
data we want to interpolate. Suppose, in addition, we know the derivative of the underlying
function at these x-coordinates. Our new data set has the following form.
Data:
x 0 ,x 1 , ..., x n
y 0 ,y 1 , ..., y n ; y i = f(x i )
y ′ 0,y ′ 1, ..., y ′ n; y ′ i = f ′ (x i )