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

80 CHAPTER 6. SPLINE INTERPOLATION
Thus there are 3n parameters to define Q(x).
For each of the n subintervals, the data
t t 0 t 1 . . . t n
y y 0 y 1 . . . y n
give two equations regarding Q i (x), namely that Q i (t i ) must equal y i and Q i (t i+1 ) must equal y i+1 .
This is 2n equations. The condition on continuity of Q ′ gives a single equation for each of the n − 1
internal nodes. This totals 3n − 1 equations, but 3n unknowns. This system is underdetermined.
Thus some additional user-chosen condition is required to determine the quadratic spline. One
might choose, for example, Q ′ (a) = 0, or Q ′′ (a) = 0, or some other condition.
6.1.3 Computing Second Degree Splines
Suppose the data
t t 0 t 1 . . . t n
y y 0 y 1 . . . y n
are given. Let z i = Q ′ i (t i), and suppose that the additional condition to define the quadratic spline
is given by specifying z 0 . We want to be able to compute the form of Q i (x).
Because Q i (t i ) = y i , Q ′ i (t i) = z i , Q ′ i (t i+1) = z i+1 , we see that we can define
Use this at t i+1 :
Q i (x) =
y i+1 = Q i (t i+1 ) =
z i+1 − z i
2 (t i+1 − t i ) (x − t i) 2 + z i (x − t i ) + y i .
y i+1 − y i = z i+1 − z i
2
y i+1 − y i = z i+1 + z i
2
z i+1 − z i
2 (t i+1 − t i ) (t i+1 − t i ) 2 + z i (t i+1 − t i ) + y i ,
(t i+1 − t i ) + z i (t i+1 − t i ) ,
(t i+1 − t i ) .
Thus we can determine, from the data alone, z i+1 from z i :
6.2 (Natural) Cubic Splines
z i+1 = 2 y i+1 − y i
t i+1 − t i
− z i .
If you recall the definition of the linear and quadratic splines, probably you can guess the definition
of the spline of degree k:
Definition 6.6 (Splines of Degree k). A function S is a spline of degree k on [a, b] if
1. The domain of S is [a, b].
2. S, S ′ , S ′′ , . . . , S (k−1) are continuous on (a, b).
3. There is a partition {t i } n i=0 of [a, b] such that on [t i, t i+1 ], S is a polynomial of degree ≤ k.
You would also expect that a spline of degree k has k − 1 “degrees of freedom,” as we show
here. If the partition has n + 1 knots, the spline of degree k is defined by n(k + 1) parameters. The
given data
t t 0 t 1 . . . t n
y y 0 y 1 . . . y n