# Tag Info

4

If that is the case, what computational method can be used to confirm that the linked configuration is, in fact, a good spherical code? You can't, with the given data. With only 16 significant digits available, you will never know if that quantity is exactly 1 or not. Or even if you were given Float128s, or numbers with 2000 significant digits, for that ...

2

From the definition it seems that the optimization problem can be written as \begin{align} &\min_{\mathbf{x}_i}& &1\\ &\text{subject to } & &\mathbf{x}_i \cdot \mathbf{x}_j \geq t\, , i\neq j\\ & & &\Vert \mathbf{x}\Vert = 1 \end{align} I am not sure, though. Thompson problem In the case of the Thompson problem, you have \...

2

The program solves the problem Place n points on a sphere in d dimensions so as to maximize the minimal distance (or equivalently the minimal angle) between them. As an aside, let's take $n = 3$ and $d = 2$. Then the solution is three points on a circle 120° apart. But that means we have coordinates $(0,0)$,$(cos(2\pi/3),sin(2\pi/3))$ and $(cos(2\pi/3),-... 2 Either I am not understanding the issue, or you're making it out to be more difficult than it really is. You have a thing$A$that should ideally be equal to$I$. The norm$\|I-A\|_2$measures its distance from$I$; that's what norms do. 1 Here is a heuristic approach, suitable if you want an approximate solution, or a good starting point for the optimization: take$M \gg N$points uniformly at random on the$n$-sphere ($M=10N$should be enough), apply the$k$-means algorithm with$k=N$, and use the resulting$N\$ cluster centers.

1

There has been progress both in terms of the speed of computers (basically driven by Moore's law) and in the algorithms used to solve LP's and especially MILP's. Overall, improvements in algorithms have had at least as large an effect as improvements in hardware. How this will work out for your particular model is a different question. Especially for MILP'...

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