Editing Interpolation (section)

===Spline interpolation===
[[File:Interpolation example spline.svg|right|thumb|230px|Plot of the data with spline interpolation applied]]
{{Main|Spline interpolation}}

Linear interpolation uses a linear function for each of intervals [''x''<sub>''k''</sub>,''x''<sub>''k+1''</sub>]. Spline interpolation uses low-degree polynomials in each of the intervals, and chooses the polynomial pieces such that they fit smoothly together. The resulting function is called a spline.

For instance, the [[natural cubic spline]] is [[piecewise]] cubic and twice continuously differentiable. Furthermore, its second derivative is zero at the end points. The natural cubic spline interpolating the points in the table above is given by

: <math> f(x) = \begin{cases}
-0.1522 x^3 + 0.9937 x, & \text{if } x \in [0,1], \\
-0.01258 x^3 - 0.4189 x^2 + 1.4126 x - 0.1396, & \text{if } x \in [1,2], \\
0.1403 x^3 - 1.3359 x^2 + 3.2467 x - 1.3623, & \text{if } x \in [2,3], \\
0.1579 x^3 - 1.4945 x^2 + 3.7225 x - 1.8381, & \text{if } x \in [3,4], \\
0.05375 x^3 -0.2450 x^2 - 1.2756 x + 4.8259, & \text{if } x \in [4,5], \\
-0.1871 x^3 + 3.3673 x^2 - 19.3370 x + 34.9282, & \text{if } x \in [5,6].
\end{cases} </math>

In this case we get ''f''(2.5)&nbsp;=&nbsp;0.5972.

Like polynomial interpolation, spline interpolation incurs a smaller error than linear interpolation, while the interpolant is smoother and easier to evaluate than the high-degree polynomials used in polynomial interpolation. However, the global nature of the basis functions leads to ill-conditioning. This is completely mitigated by using splines of compact support, such as are implemented in Boost.Math and discussed in Kress.<ref>{{cite book|last1=Kress|first1=Rainer|title=Numerical Analysis|url=https://archive.org/details/springer_10.1007-978-1-4612-0599-9|year=1998|publisher=Springer |isbn=9781461205999}}</ref>
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