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cubic spline interpolation
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(Definition)
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Suppose we are given  data points
 such that
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Then the function  is called a cubic spline interpolation if there exists  cubic polynomials  with coefficients
 such that the following hold.
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![$ S(x)=S_{k}(x)=\sum_{i=0}^{3}s_{k,i}(x-x_{k})^{i} \; \;\;\forall x\in [x_{k},x_{k+1}]\;\;\;0\leq k \leq N-1$](http://images.planetmath.org:8080/cache/objects/4339/l2h/img8.png "$ S(x)=S_{k}(x)=\sum_{i=0}^{3}s_{k,i}(x-x_{k})^{i} \; \;\;\forall x\in [x_{k},x_{k+1}]\;\;\;0\leq k \leq N-1$")
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The set of points
 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
 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
 , 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
 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)
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"cubic spline interpolation" is owned by yota. [ full author list (2) | owner history (7) ]
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(view preamble)
Cross-references: interpolation, curve, MATLAB, end point, endpoints, second derivatives, multiplication, scalar, addition, vector space, fixed set, knots, coefficients, polynomials, function, points
There is 1 reference to this entry.
This is version 4 of cubic spline interpolation, born on 2003-06-11, modified 2006-10-19.
Object id is 4339, canonical name is CubicSplinInterpolation.
Accessed 34005 times total.
Classification:
| AMS MSC: | 65-01 (Numerical analysis :: Instructional exposition ) |
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Pending Errata and Addenda
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