First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 3. INTERPOLATION 121
In linear spline interpolation, we simply join data points (the nodes), by line segments,
that is, linear polynomials. For example, consider the following figure that plots three data
points (x i−1 ,y i−1 ), (x i ,y i ), (x i+1 ,y i+1 ). We fit a linear polynomial P (x) to the first pair of
data points (x i−1 ,y i−1 ), (x i ,y i ), and another linear polynomial Q(x) to the second pair of
data points (x i ,y i ), (x i+1 ,y i+1 ).
yi
yi+1
P(x)
Q(x)
yi-1
xi-1 xi xi+1
Figure 3.4: Linear spline
Let P (x) =ax + b and Q(x) =cx + d. We find the coefficients a, b, c, d by solving
P (x i−1 )=y i−1
P (x i )=y i
Q(x i )=y i
Q(x i+1 )=y i+1
which is a system of four equations and four unknowns. We then repeat this procedure for
all data points, (x 0 ,y 0 ), (x 1 ,y 1 ), ..., (x n ,y n ), to determine all of the linear polynomials.
One disadvantage of linear spline interpolation is the lack of smoothness. The first
derivative of the spline is not continuous at the nodes (unless the data fall on a line). We
can obtain better smoothness by increasing the degree of the piecewise polynomials. In
quadratic spline interpolation, we connect the nodes via second degree polynomials.