First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 3. INTERPOLATION 89
We want f to pass through the data points, that is, f(x i )=y i . Then determine a k so that:
f(x i )=
n∑
a k φ k (x i )=y i ,i=0, 1, ..., n,
k=0
which is a system of n +1equations with n +1unknowns. Using matrices, the problem is
to solve the matrix equation
Aa = y
for a, where
⎡
⎤ ⎡ ⎤ ⎡ ⎤
φ 0 (x 0 ) ... φ n (x 0 ) a 0 y 0
φ
A =
0 (x 1 ) ... φ n (x 1 )
⎢
⎥
⎣ .
⎦ ,a= a 1
⎢ ⎥
⎣ . ⎦ ,y = y 1
⎢ ⎥
⎣ . ⎦ .
φ 0 (x n ) ... φ n (x n ) a n y n
3.1 Polynomial interpolation
In polynomial interpolation, we pick polynomials as the family of functions in the interpolation
problem.
• Data: (x i ,y i ),i=0, 1, ..., n
• Family: Polynomials
The space of polynomials up to degree n is a vector space. We will consider three choices
for the basis for this vector space:
• Basis:
– Monomial basis: φ k (x) =x k
– Lagrange basis: φ k (x) = ∏ n
j=0,j≠k
– Newton basis: φ k (x) = ∏ k−1
j=0 (x − x j)
where k =0, 1, ..., n.
( )
x−xj
x k −x j
Once we decide on the basis, the interpolating polynomial can be written as a linear combination
of the basis functions:
n∑
p n (x) = a k φ k (x)
where p n (x i )=y i ,i=0, 1, ..., n.
k=0