cubic spline interpolation

Suppose we are given $N+1$ data points $\{(x_k,y_k)\}$ such that

(1)

Then the function $S(x)$ is called a cubic spline interpolation if there exists $N$ cubic polynomials $S_k(x)$ with coefficients $s_{k,i}\mathrm{\hspace{0.17em}\hspace{0.17em}0}\le i\le 3$ such that the following hold.

1. $S(x)=S_k(x)={\sum }_{i=0}^3{s}_{k,i}{(x-x_k)}^i\forall x\in [x_k,x_{k+1}]\mathrm{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}0}\le k\le N-1$

2. $S(x_k)=y_k\mathrm{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}0}\le k\le N$

3. $S_k(x_{k+1})=S_{k+1}(x_{k+1})\mathrm{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}0}\le k\le N-2$

4. $S_k^{\prime }(x_{k+1})=S_{k+1}^{\prime }(x_{k+1})\mathrm{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}0}\le k\le N-2$

5. $S_k^{\prime \prime }(x_{k+1})=S_{k+1}^{\prime \prime }(x_{k+1})\mathrm{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}0}\le k\le N-2$

The set of points $(\mathit{\text{<a href="\#S0.E1" title="(1) ‣ cubic spline interpolation" class="ltx\_ref ltx\_font\_italic"><span class="ltx\_text ltx\_ref\_tag">1</span></a>}})$ are called the knots. The set of cubic splines on a fixed set of knots, forms a vector space for cubic spline addition and scalar multiplication.

So we see that the cubic spline not only interpolates the data $\{(x_k,y_k)\}$ but matches the first and second derivatives at the knots. Notice, from the above definition, one is free to specify constraints on the endpoints. One common end point constraint is $S^{\prime \prime }(a)=0S^{\prime \prime }(b)=0$, which is called the natural spline. Other popular choices are the clamped cubic spline, parabolically terminated spline and curvature-adjusted spline. Cubic splines are frequently used in numerical analysis to fit data. Matlab uses the command spline to find cubic spline interpolations with not-a-knot end point conditions. For example, the following commands would find the cubic spline interpolation of the curve $4\cos (x)+1$ and plot the curve and the interpolation marked with o’s.

x = 0:2*pi;
y = 4*cos(x)+1;
 xx = 0:.001:2*pi;
yy = spline(x,y,xx);
plot(x,y,'o',xx,yy)

Title	cubic spline interpolation
Canonical name	CubicSplineInterpolation
Date of creation	2013-03-22 13:40:25
Last modified on	2013-03-22 13:40:25
Owner	yota (10184)
Last modified by	yota (10184)
Numerical id	7
Author	yota (10184)
Entry type	Definition
Classification	msc 65-01