Square roots (Part VI)

February 20, 2018

I’ve discussed algorithms for computing square roots a couple of times already, and then some. While sorting notes, I’ve came across something interesting: Archytas’ method for computing square roots.

Archytas’ method is basically the old Babylonian method, where you first set

a=1,

b=n,

and iterate

\displaystyle a'=\frac{a+b}{2},

\displaystyle b'=\frac{n}{a'}=\frac{2n}{a+b},

until desired precision is achieved (or the scribe is exhausted).

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Unary numbers.

February 13, 2018

A positional number system needs a base that is either greater than one, or smaller than minus one—yes, we can have a negative base for a number system. The system, however, seems to break down if the base we chose is base 1.

If the base is 1, then there are no permissible digits since the digits d, in a base b system, must be 0\leqslant{d}<b. But we can still represent numbers using just 1s. That's the unary numeral system, and numbers are just represented as repeated 1s. 15? Fifteen ones: 111111111111111. Operations? Not very complicated, just… laborious.

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Elementary Automata (Generating Random Sequences XI)

February 6, 2018

So 3-cells context elementary automata seem too “self-correcting” to be useful pseudo-random generators. What if we fixed that boundary problem and have the automaton run on a cylinder (with both end joined)? What if we augment the context from 3 to 5 cells?

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Elementary Automata (Generating Random Sequences X)

January 30, 2018

I was rather discontented with last week’s post results. Most automata seemed to produce self-correcting patterns, even when seeded randomly—one could argue that rand() isn’t the strongest random generator, but that wasn’t the problem. No, indeed, most automata exhibit self-correcting behavior, forming the same self-similar pattern, or worse, the same periodic pattern.

So I made a few more experiments with random seeds and larger images. The code isn’t very complicated and isn’t of interest in itself, but it reveals a couple of interesting things.

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Elementary Automata (Generating Random Sequences IX)

January 23, 2018

A while ago, Wolfram (re)introduced elementary cellular automata, a special case of cellular automata that lives in one dimension. One cutesy aspect of these automata is that the rules are easy to describe and number. As any automata, it is given a context, a region of interest, and gives an output depending on the context. For example, we can have the rules:

Context output
000 1
001 0
010 1
011 0
100 1
101 0
110 1
111 1

The output column can be viewed as a number, in this case 11010101, 0xd5, or 213. This is rule 213.

It may be tempting to use those to generate chaos, noise, and random numbers. But is it that a good idea?

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Woe is coding.

January 16, 2018

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#include <the_usual>

January 9, 2018

Recently on Freenode channel ##cpp, I saw some code using an include-all-you-can header. The idea was to help beginners to the language, help them start programming without having to remember which header was which, and which headers are needed.

Is that really a good idea?

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