Numerical Methods

What’s the idea behind piecewise polynomial interpolation?

Break domain into subdomains, apply polynomial interpolation on each subdomain (interpolation points are now “knots”)

What do we avoid with piecewise polynomials and what problem is introduced?

Avoid higher order polynomials and therefore avoid “blow-up”, however we lose smoothness if the interpolant.

Name a popular type of piecewise polynomial interpolant

A spline

What is a spline of degree k?

A piecewise polynomial that is continuously differentiable k-1 times

What do splines solve?

To some extent, the “loss of smoothness” issue since they have continuous derivatives

How many conditions are we short by in a cubic spline and why?

We are missing 2 conditons since cubic splines are constructed to be continuous, first-derivative and second derivative continuous but ONLY on the interior points.

What are the two ways to make up the last two conditions of a cubic spline?

Natural cubic spline: right at the end points, the curved curvature is zero (second derivatives equal zero)

Knot-a-not: the first two and last two intervals share the same cubic so that the edges aren’t really breaks