Cubic Spline Interpolation

Updated 01 Oct 2026

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What it is
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  fits piecewise cubic polynomials that interpolate the data and jion with continuous first and second derivatives
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Boundary conditions
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  Natural: second derivative is zero at both endpoints
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  Clamped: endpoint first derivatives are specified
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Example
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  a natural spline through $(0,1),(1,2),(2,9)$ is $S_0(x)=1-\frac12x+\frac32x^3$ on $0,1$ .
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  On $1,2$ , $S_1(x)=2+4(x-1)+\frac92(x-1)^2-\frac32(x-1)^3$ .
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Limitations
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  ordinary cubic splines can overshoot
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  their pieces are local polynomials, but their coefficients come from a coupled system; changing one data point cat affect multiple intervals
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Related
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  Thomas Algorithm
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  Polynomial Interpolation

Linked references 3