Introduction to Splines

An alternative to Lagrange polynomials to interpolate throughpointswe can use piecewise polynomial functions called splines, i.e.where the low degree polynomialsare defined on the intervals

The simplest spline is composed of linear functions of the formfor(i.e. a straight line between the two successive pointsand).

The coefficientsandare determined by the conditions (i)and (ii)Thus,

We needequations to find thecoefficients

Each of (I) and (ii) gives rise toequations soequations in total.

The interpolating function is continuous but it is not differentiable, i.e. not smooth, at the

interior points:

To retain the smoothness of Lagrange interpolation without producing large oscillations, higher

order splines are needed. We can construct splines of any order, but the most common are maybe cubic splines. These are twice differentiable and are suitable for many purposes.

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