First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 3. INTERPOLATION 120
Exercise 3.3-1: The following table gives the values of y = f(x) and y ′ = f ′ (x) where
f(x) =e x +sin10x. Compute the Hermite interpolating polynomial and the polynomial
interpolant for the data in the table. Plot the two interpolating polynomials together with
f(x) =e x +sin10x on (0, 3).
x 0 0.4 1 2 2.6 3
y 1 0.735 2.17 8.30 14.2 19.1
y ′ 11 -5.04 -5.67 11.5 19.9 21.6
3.4 Piecewise polynomials: spline interpolation
As we observed in Section 3.2, a polynomial interpolant of high degree can have large oscillations,
and thus provide an overall poor approximation to the underlying function. Recall
that the degree of the interpolating polynomial is directly linked to the number of data
points: we do not have the freedom to choose the degree of the polynomial.
In spline interpolation, we take a very different approach: instead of finding a single
polynomial that fits the given data, we find one low-degree polynomial that fits every pair
of data. This results in several polynomial pieces joined together, and we typically impose
some smoothness conditions on different pieces. The term spline function means a function
that consists of polynomial pieces joined together with some smoothness conditions.