Spline interpolation

In the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. Spline interpolation is often preferred over polynomial interpolation because the interpolation error can be made small even when using low degree polynomials for the spline. Spline interpolation avoids the problem of Runge's phenomenon, in which oscillation can occur between points when interpolating using high degree polynomials.

Originally, spline was a term for rulers that were bent to pass through a number of predefined points ("knots"). These were used to make technical drawings for shipbuilding and construction by hand, as illustrated by Figure 1.

The approach to mathematically model the shape of such elastic rulers fixed by n + 1 knots ${\displaystyle \left\{(x_{i},y_{i}):i=0,1,\cdots ,n\right\}}$ is to interpolate between all the pairs of knots ${\displaystyle (x_{i-1},y_{i-1})}$ and ${\displaystyle (x_{i},y_{i})}$ with polynomials ${\displaystyle y=q_{i}(x),i=1,2,\cdots ,n}$.

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