First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 3. INTERPOLATION 123
Si-1
Si
S0
Sn-1
x0 x1 xi-1 xi xi+1
xn-1 xn
Figure 3.6: Cubic spline
The polynomial S i interpolates the nodes (x i ,y i ), (x i+1 ,y i+1 ). Let
S i (x) =a i + b i x + c i x 2 + d i x 3
for i =0, 1, ..., n − 1. There are 4n unknowns to determine: a i ,b i ,c i ,d i ,asi takes on values
from 0 to n − 1. Let’s describe the equations S i must satisfy. First, the interpolation
conditions, that is, the requirement that S i passes through the nodes (x i ,y i ), (x i+1 ,y i+1 ):
S i (x i )=y i
S i (x i+1 )=y i+1
for i =0, 1, ..., n − 1, which gives 2n equations.
smoothness:
The next group of equations are about
S ′ i−1(x i )=S ′ i(x i )
S ′′
i−1(x i )=S ′′
i (x i )
for i =1, 2, ..., n − 1, which gives 2(n − 1) = 2n − 2 equations. Last two equations are called
the boundary conditions. There are two choices:
• Free or natural boundary: S ′′
0 (x 0 )=S ′′
n−1(x n )=0