Summary of interpolation

We have seen now that polynomial interpolation, splines, and RBFs can all be expressed in terms of a linear function basis with the coordinates given by a set of weights, $\vec{\omega}$ which is determined by the data:

This is excellent since we can use linear system tools to solve for $\vec{\omega}$ and therefore build our function. Note that the functional dependence of $\vec{x}$ is within $A$.

Pragmatic Interpolators¶

Let’s recap and generalize:

• For any $n$ points there is a polynomial that fits it, but because of Runge’s phenomenon you don’t want to use that!

• Piecewise polynomials are stiffer and avoid Runge’s phenomenon, but smoothness causes issues for N-D.

• RBFs are great and flexible but prone to numerical error due to solving a poorly conditioned matrix.

One pragmatic possibility is to use local interpolator methods, but these have the drawback of having to find (triangulate) the nearest neighbours before being able to use them:

• Nearest ND interpolator: Find the nearest data point and use that.

• Linear ND interpolators: For each input, a triangulation finds the nearest data points and a linear barycentric Lagrange interpolation is performed.

Moving to Curve-Fitting¶

Let’s now consider moving to curve-fitting in which the function doesn’t have to pass through every data point. Through a nifty tool called the pseudoinverse we’ll see that the methods of interpolation can be applied almost unchanged!