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.

Introduction

Originally, spline was a term for rulers that were bent to pass through a number of predefined points. These were used to make technical drawings for shipbuilding and construction by hand, as illustrated by Figure 1.
The approach to mathematically modelling the shape of such elastic rulers fixed by knots is to interpolate between all the pairs of knots and with polynomials.
The curvature of a curve is given by:
As the spline will take a shape that minimizes the bending both and will be continuous everywhere and at the knots. To achieve this one must have that
This can only be achieved if polynomials of degree 3 or higher are used. The classical approach is to use polynomials of degree 3 — the case of cubic splines.

Algorithm to find the interpolating cubic spline

A third-order polynomial for which
can be written in the symmetrical form
where
As
one gets that:
Setting and respectively in equations and one gets from that indeed first derivatives and and also second derivatives
If now are points and
where i = 1, 2,..., n and are n third degree polynomials interpolating in the interval for i = 1,..., n such that for i = 1,..., n−1 then the n polynomials together define a differentiable function in the interval and
for i = 1,..., n where
If the sequence is such that, in addition, holds for i = 1,..., n-1, then the resulting function will even have a continuous second derivative.
From,, and follows that this is the case if and only if
for i = 1,..., n-1. The relations are linear equations for the values.
For the elastic rulers being the model for the spline interpolation one has that to the left of the left-most "knot" and to the right of the right-most "knot" the ruler can move freely and will therefore take the form of a straight line with. As should be a continuous function of one gets that for "Natural Splines" one in addition to the linear equations should have that
i.e. that
Eventually, together with and constitute linear equations that uniquely define the parameters.
There exist other end conditions: "Clamped spline", that specifies the slope at the ends of the spline, and the popular "not-a-knot spline", that requires that the third derivative is also continuous at the and points.
For the "not-a-knot" spline, the additional equations will read:
where.

Example

In case of three points the values for are found by solving the tridiagonal linear equation system
with
For the three points
one gets that
and from and that
In Figure 2, the spline function consisting of the two cubic polynomials and given by is displayed.

Computer code