cubic spline interpolation

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

(1)

Then the function $S(x)$ is called a cubic spline interpolation ifthere 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 $(\mathit{\text{<span class="ltx\_text ltx\_ref\_tag">1</span>}})$ are called the knots. The set ofcubic splines on a fixed set of knots, forms a vector space forcubic 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 derivativesat the knots. Notice, from the above definition, one is free tospecify constraints on the endpoints. One common end pointconstraint is $S^{\prime \prime }(a)=0S^{\prime \prime }(b)=0$,which is called the natural spline. Other popular choices are theclamped cubic spline, parabolically terminated spline andcurvature-adjusted spline. Cubic splines are frequently used innumerical analysis to fit data. Matlab uses the command spline tofind cubic spline interpolations with not-a-knot end pointconditions. For example, the following commands would find thecubic spline interpolation of the curve $4\cos (x)+1$ and plot thecurve 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)