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 elastic 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 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 (under the constraint of passing through all knots) 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

(1)
where

(2)

(3)

(4)
As
one gets that:

(5)

(6)
Setting x = x_{1} and x = x_{2} respectively in equations (5) and (6) one gets from (2) that indeed first derivatives q′(x_{1}) = k_{1} and q′(x_{2}) = k_{2} and also second derivatives

(7)

(8)
If now (x_{i}, y_{i}), i = 0, 1, ..., n are n + 1 points and

(9)
where i = 1, 2, ..., n and are n third degree polynomials interpolating y in the interval x_{i−1} ≤ x ≤ x_{i} for i = 1, ..., n such that q′_{i} (x_{i}) = q′_{i+1}(x_{i}) for i = 1, ..., n−1 then the n polynomials together define a differentiable function in the interval x_{0} ≤ x ≤ x_{n} and

(10)

(11)
for i = 1, ..., n where

(12)

(13)

(14)
If the sequence k_{0}, k_{1}, ..., k_{n} is such that, in addition, q′′_{i}(x_{i}) = q′′_{i+1}(x_{i}) holds for i = 1, ..., n1, then the resulting function will even have a continuous second derivative.
From (7), (8), (10) and (11) follows that this is the case if and only if

(15)
for i = 1, ..., n1. The relations (15) are n − 1 linear equations for the n + 1 values k_{0}, k_{1}, ..., k_{n}.
For the elastic rulers being the model for the spline interpolation one has that to the left of the leftmost "knot" and to the right of the rightmost "knot" the ruler can move freely and will therefore take the form of a straight line with q′′ = 0. As q′′ should be a continuous function of x one gets that for "Natural Splines" one in addition to the n − 1 linear equations (15) should have that
i.e. that

(16)

(17)
Eventually, (15) together with (16) and (17) constitute n + 1 linear equations that uniquely define the n + 1 parameters k_{0}, k_{1}, ..., k_{n}.
There exist other end conditions: "Clamped spline", that specifies the slope at the ends of the spline, and the popular "notaknot spline", that requires that the third derivative is also continuous at the x_{1} and x_{N−1} points. For the "notaknot" 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
In Figure 2 the spline function consisting of the two cubic polynomials and given by (9) is displayed
See also
 Cubic Hermite spline
 Centripetal Catmull–Rom spline
 Discrete spline interpolation
 Monotone cubic interpolation
 NURBS
 Multivariate interpolation
 Polynomial interpolation
 Smoothing spline
 Spline wavelet
Computer code
TinySpline: Open source Clibrary for splines which implements cubic spline interpolation
External links
 Cubic Spline Interpolation Online Calculation and Visualization Tool (with JavaScript source code)
 Hazewinkel, Michiel, ed. (2001), "Spline interpolation", Encyclopedia of Mathematics, Springer, ISBN 9781556080104
 Dynamic cubic splines with JSXGraph
 Lectures on the theory and practice of spline interpolation
 Paper which explains step by step how cubic spline interpolation is done, but only for equidistant knots.
 Numerical Recipes in C, Go to Chapter 3 Section 33.
 A note on cubic splines