Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

Chapter 6
Spline Interpolation
Splines are used to approximate complex functions and shapes. A spline is a function consisting of
simple functions glued together. In this way a spline is different from a polynomial interpolation,
which consists of a single well defined function that approximates a given shape; splines are normally
piecewise polynomial.
6.1 First and Second Degree Splines
Splines make use of partitions, which are a way of cutting an interval into a number of subintervals.
Definition 6.1 (Partition). A partition of the interval [a, b] is an ordered sequence {t i } n i=0 such
that
a = t 0 < t 1 < · · · < t n−1 < t n = b
The numbers t i are known as knots.
A spline of degree 1, also known as a linear spline, is a function which is linear on each subinterval
defined by a partition:
Definition 6.2 (Linear Splines). A function S is a spline of degree 1 on [a, b] if
1. The domain of S is [a, b].
2. S is continuous on [a, b].
3. There is a partition {t i } n i=0 of [a, b] such that on each [t i, t i+1 ], S is a linear polynomial.
A linear spline is defined entirely by its value at the knots. That is, given
t t 0 t 1 . . . t n
y y 0 y 1 . . . y n
there is only one linear spline with these values at the knots and linear on each given subinterval.
For a spline with this data, the linear polynomial on each subinterval is defined as
S i (x) = y i + y i+1 − y i
t i+1 − t i
(x − t i ) .
Note that if x ∈ [t i , t i+1 ] , then x − t i > 0, but x − t i−1 ≤ 0. Thus if we wish to evaluate S(x), we
search for the largest i such that x − t i > 0, then evaluate S i (x).
Example 6.3. The linear spline for the following data
is shown in Figure 6.1.
t 0.0 0.1 0.4 0.5 0.75 1.0
y 1.3 4.5 2.0 2.1 5.0 3
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