# Mathematica

## Polynomial Interpolation

It is interesting to look at what happens when you try to “fit” the absolute value function using polynomials. Pick some number $$N$$ of points on the absolute value function, and you can find a polynomial of degree $$N - 1$$ that goes through those points. Here is the absolute value function, $$f(x) = |x|$$: For simplicity, I’ll use an odd number of evenly spaced points so that the resulting polynomial will be even, since $$|x|$$ is even.

## Another Parabola Construction

Building on my last post, here is another way to construct a parabola using a collection of straight lines. First, the description, taken from Lockwood’s A Book of Curves (page 7): Draw any two lines and mark on each a series of points at equal intervals. (The intervals on the second line need not be equal to those on the first.) Call the points on the first line $$A_1, A_2, A_3$$, etc.

## Constructing Parabolas with Mathematica

There are several ways to draw a parabola using straight lines. If you get a chance, you should try one sometime - it is always satisfying to see the outline of a curve slowly emerging from a collection of straight lines. One method uses a set-square. As described in A Book of Curves by E.H. Lockwood (page 3): Draw a fixed line $$AY$$ and mark a fixed point $$S$$. Place a set square $$UQV$$ (right-angled at $$Q$$) with the vertex $$Q$$ on $$AY$$ and the side $$QU$$ passing through $$S$$ (Fig.

## Bayesian Decision Boundaries

I started reading The Elements of Statistical Learning by Hastie, Tibshirani, and Friedman, and was curious about how to reproduce Figure 2.5. (The book is made available as a free and legal pdf here.) So I figured out how to produce similar figures using Mathematica. I assume that this is also fairly straightforward to do in R, but I don’t yet know enough R. The authors explain the sampling method on pages 16 and 17.