Cubic Spline

Posted by: aimeemarie88 on: December 9, 2008

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Cubic Spline

DEFINITION: Suppose that ${(x_k,y_k)}_{k=0}^n$ are n+1 points, where $a=x_0<x_1 \ldots <x_n=b$ . The function S(x) is called a cubic spline if there exists $n$ cubic polynomials $S_k(x)$ with coefficients $S_{k,0},S_{k,1}, S_{k,2}, S_{k,3}$ that satisfy the properties:

(i) $S(x)=S_k(x)=S_{k,0}+S_{k,1}(x-x_k)+S_{k,2}(x-x_k)^2+S_{k,3}(x-x_k)^3$ for $x\in[x_k,x_{k+1}]$ and $k=0,1, \ldots ,n-1$

(ii) The spline passes through each data point: $S(x_k)=Y_k$ for $k=0,1, \ldots n$

(iii) The spline forms a continuous function over [a,b]: $S_k(x_{k+1})=s_{k+1}(s_{k+1})$ for $k=0,1. \ldots ,n-2$

(iv) The spline forms a smooth function: $S'_k(x_{k+1})=S'{k+1}(x_{k+1})$ for $k=0,1, \ldots ,n-2$

(v) The second derivative is continous: $S''_k(x_{k+1})=S''{k+1}(x_{k+1})$

Natural Spline: There exists a unique cubic spline with the free boundary conditions $S''(a)=0$ and $S''(b)=0$

Clamped Spline: There exists a unique cubic spline with the first derivative boundary conditions $S'(a)=d_0$ and $S'(b)=d_n$

Theorem: Minimum Property of Clamped cubic splines: Assume $f\in C^2[a,b]$ and $S(x)$ is the unique clamped spline interpolant for $f(x)$ which passes through ${[x_k,f(x_k))}_{k=0}^n$ and satisfies the clamped end conditions $S'(a)=f(a)$ and $S'(b)=f(b)$ . Then $\int_{a}^{b}(S''(x))^2dx\le \int_{a}^{b}(f''(x))^2dx$

Numerical Analysis (aimee)

Cubic Spline

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