First Semester in Numerical Analysis with Julia, 2020a

Chapter 3
Interpolation
In this chapter, we will study the following problem: given data (x i ,y i ),i=0, 1, ..., n, find a
function f such that f(x i )=y i . This problem is called the interpolation problem, and f is
called the interpolating function, or interpolant, for the given data.
Interpolation is used, for example, when we use mathematical software to plot a smooth
curve through discrete data points, when we want to find the in-between values in a table,
or when we differentiate or integrate black-box type functions.
Howdowechoosef? Or, what kind of function do we want f to be? There are several
options. Examples of functions used in interpolation are polynomials, piecewise polynomials,
rational functions, trigonometric functions, and exponential functions. As we try to find a
good choice for f for our data, some questions to consider are whether we want f to inherit
the properties of the data (for example, if the data is periodic, should we use a trigonometric
function as f?), and how we want f behave between data points. In general f should be
easy to evaluate, and easy to integrate & differentiate.
Here is a general framework for the interpolation problem. We are given data, and we
pick a family of functions from which the interpolant f will be chosen:
• Data: (x i ,y i ),i=0, 1, ..., n
• Family: Polynomials, trigonometric functions, etc.
Suppose the family of functions selected forms a vector space. Pick a basis for the vector
space: φ 0 (x),φ 1 (x), ..., φ n (x). Then the interpolating function can be written as a linear
combination of the basis vectors (functions):
f(x) =
n∑
a k φ k (x).
k=0
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