Over the last few months we have been taking a look at algorithms for interpolating over a set of points (x_i,y_i) in order to approximate values of y between the nodes x_i. We began with linear interpolation which connects the points with straight lines and is perhaps the simplest interpolation algorithm. Then we moved on to cubic spline interpolation which yields a smooth curve by specifying gradients at the nodes and fitting cubic polynomials between them that match both their values and their gradients. Next we saw how this could result in curves that change from increasing to decreasing, or vice versa, between the nodes and how we could fix this problem by adjusting those gradients.
I concluded by noting that, even with this improvement, the shape of a cubic spline interpolation is governed by choices that are not uniquely determined by the points themselves and that linear interpolation is consequently a more mathematically appropriate scheme, which is why I chose to generalise it to other arithmetic types for y, like complex numbers or matrices, but not to similarly generalise cubic spline interpolation.

The obvious next question is whether or not we can also generalise the nodes to other arithmetic types; in particular to vectors so that we can interpolate between nodes in more than one dimension.

Sir R-----! I must say that it is a relief to have the company of a fellow nobleman in these distressing times. That I have had to sell not one, but two of my several hundred antiquities to settle the burden of tax that this oppressive democracy has put upon me, simply to enrich slugabeds I might add, is quite intolerable!

Come, let us drown our sorrows whilst we still have the means to do so and engage in a little sport to raise our spirits.

I have a fancy for a game that I used to play when I was the Russian ambassador to the Rose Tree Valley commune. Founded by the philosopher queen Zway Remington as a haven for downtrodden wealthy industrialists, it was the purest of pure meritocracies; no handouts to the idle labouring classes there!

We have seen how it is possible to smoothly interpolate between a set of points (x_i, y_i), with the x_i known as nodes and the y_i as values, by specifying the gradients g_i at the nodes and calculating values between adjacent pairs using the uniquely defined cubic polynomials that match the values and gradients at them.
We have also seen how extrapolating such polynomials beyond the first and last nodes can yield less than satisfactory results, which we fixed by specifying the first and last gradients and then adding new first and last nodes to ensure that the first and last polynomials would represent straight lines.
Now we shall see how cubic spline interpolation can break down rather more dramatically and how we might fix it.

Recall that the Baron's game is comprised of taking turns to place dominoes on a six by six grid of squares with each domino covering a pair of squares. At no turn was a player allowed to place a domino such that it created an oddly-numbered region of empty squares and Sir R----- was to be victorious if, at the end of play, the lines running between the ranks and files of the board were each and every one straddled by at least one domino.

Last time we took a look at how we can use linear interpolation to approximate a function from a set of points on its graph by connecting them with straight lines. As a consequence the result isn't smooth, meaning that its derivative isn't continuous and is undefined at the x values of the points, known as the nodes of the interpolation.
In this post we shall see how we can define a smooth interpolation by connecting the points with curves rather than straight lines.

My fellow students and I have of late been thinking upon an equivalence between the roots of rational numbers and an infinite dimensional rational vector space, which we have named ℓ-space, that we discovered whilst defining analogues of logarithms that were expressed purely in terms of rationals.
We were particularly intrigued by the possibility of defining functions of such numbers by applying linear algebra operations to their associated vectors, which we began with a brief consideration of that given by their magnitudes. We have subsequently spent some time further exploring its properties and it is upon our findings that I shall now report.

Given a set of points (x_i,y_i), a common problem in numerical analysis is trying to estimate values of y for values of x that aren't in the set. The simplest scheme is linear interpolation, which connects points with consecutive values of x with straight lines and then uses them to calculate values of y for values of x that lie between those of their endpoints.
On the face of it implementing this would seem to be a pretty trivial business, but doing so both accurately and efficiently is a surprisingly tricky affair, as we shall see in this post.

Good heavens Sir R----- you look quite pallid! Come take a seat and let me fetch you a measure of rum to restore your humors.
To further improve your sanguinity might I suggest a small wager?

Splendid fellow!

I have in mind a game invented to commemorate my successfully quashing the Caribbean zombie uprising some few several years ago. Now, as I'm sure you well know, zombies have ever been a persistent, if sporadic, scourge of those islands. On that occasion, however, there arose a formidable leader from amongst their number; the zombie Lord J------ the Insensate.

In the last few posts we have implemented a type to represent Borel sets of the real numbers, which are the subsets of them that can be created with countable unions of intervals with closed or open lower and upper bounds. Whilst I would argue that doing so was a worthwhile exercise in its own right, you may be forgiven for wondering what Borel sets are actually for and so in this post I shall try to justify the effort that we have spent on them.

The Baron's latest wager set Sir R----- the task of rolling a higher score with two dice than the Baron should with one twelve sided die, giving him a prize of the difference between them should he have done so. Sir R-----'s first roll of the dice would cost him two coins and twelve cents and he could elect to roll them again as many times as he desired for a further cost of one coin and twelve cents each time, after which the Baron would roll his.
The simplest way to reckon the fairness of this wager is to re-frame its terms; to wit, that Sir R----- should pay the Baron one coin to play and thereafter one coin and twelve cents for each roll of his dice, including the first. The consequence of this is that before each roll of the dice Sir R----- could have expected to receive the same bounty, provided that he wrote off any losses that he had made beforehand.