SPLINES
A cubic spline is a function defined piecewise with each piece being a cubic
polynomial. Let the break points (knots) be x1< x2< . . . < xn, and let
y1, y2,...,ynbe the data values at these points. We set
δk=yk+1 −yk
xk+1 −xk
.
There are n−1 subintervals xk≤x≤xk+1 and a cubic polynomial Pk(x)
defined on each subinterval. Using s=x−xk, and h=xk+1 −xk, we write the
polynomials in the form
Pk(x) = Pk(s) = 3hs2−2s3
h3yk+1 +h3−hs2+ 2s3
h3yk(1)
+s2(s−h)
h2dk+1 +s(s−h)2
h2dk.
With the polynomials written this way, it is easy to verify that Pk(xk) =
Pk(s= 0) = yk, and Pk(xk+1) = Pk(s=h) = yk+1 . Thus P=∪Pkinterpolates
the data points (xk, yk). The piecewise defined function P=∪Pkalso has a
continuous derivative. In fact you can verify that P′
k(xk) = P′
k(s= 0) = dk,
and P′
k(xk+1) = P′
k(s=h) = dk+1.
We still have the nconstants d1,...,dnto determine. If we knew a function
f(x) such that f(xk) = yk, we could just set dk=f′(xk). Another approach is
to determine the values for the dkusing the values of the divided differences δk.
This is the method used in constructing the Shape-Preserving Piecewise Cubic,
described in Moler, p100 and in the MATLAB code pchip.m.
In a spline, we impose other conditions on Pat the knots to determine the dk.
We shall require that P′′ be continuous at the interior knots xk, k = 2,...,n−1:
P′
k(x+
k) = P′′
k−1(x−
k), k = 2,...,n−1.
This yields the equations for the dk(k= 2,...,n−1):
hkdk−1+ 2(hk−1+hk)dk+hk−1dk+1 = 3(hkδk−1+hk−1δk).(2)
We now have n−2 equations for the nunknowns dk. To make a system
that has a unique solution for the dk, we must add two equations to the n−2
equations (2) or remove two unknowns. This can be done in several ways by
imposing extra conditions at the ends.
Complete Spline We assign values to d1and dnusing other outside in-
formation. For example, we can interpolate a parabola r(x) through the data
points (x1, y1),(x2, y2),(x3, y3), and take d1=r′(x1). Do the same thing at the
other end. Since d1and dnare considered known, we can put them to the other
side of equation in the first equation (k = 2) of (2) and in the last equation (k =
1
