Cubic Spline Interpolation

Consider points . A cubic spline is a function defined piecewise by  cubic polynomials such that:

Where for for

We want for and we’d like ,, and to be continuous.

The Coefficient Matrix for this system is:

This gives us unknowns to come up with enough equations we use the following:

  1. Interpolation at Left Endpoints: ( equations)
    for
    for
  2. Interpolation at Right Endpoints: ( equations)
    for
    for
  3. is continuous: ( equations)
    for
  4. is continuous: ( equations)
    for

The last two equations are generally provided by the problem in the case of the natural cubic spline the equations are:

Example

Go to Find Cubic Spline Interpolation Coefficients to see a worked example of finding cubic spline coefficients given an incomplete coefficient matrix.