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Ayan ShafqatIf you've ever wrestled with fixed-point math, scaling, and rounding, this blog post is for you!I go deeper into binary arithmetic, division tricks, rounding, and what it really takes to build fixed-point systems that work.As always, I appreciate all of your feedback and corrections.<a href="https://shafq.at/fixed-point-tutorial-2.html" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://shafq.at/fixed-point-tutorial-2.html</a><a href="https://hachyderm.io/tags/EmbeddedSystems" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#EmbeddedSystems</a> <a href="https://hachyderm.io/tags/DSP" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DSP</a> <a href="https://hachyderm.io/tags/FixedPoint" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#FixedPoint</a> <a href="https://hachyderm.io/tags/CProgramming" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#CProgramming</a> <a href="https://hachyderm.io/tags/SignalProcessing" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#SignalProcessing</a> <a href="https://hachyderm.io/tags/Microcontrollers" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Microcontrollers</a> <a href="https://hachyderm.io/tags/C" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#C</a> <a href="https://hachyderm.io/tags/programming" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#programming</a>

AnthonyAI models fed AI-generated data quickly spew nonsense <blockquote>Researchers gave successive versions of a large language model information produced by previous generations of the AI — and observed rapid collapse. </blockquote>From: <a href="https://www.nature.com/articles/d41586-024-02420-7?WT.ec_id=NATURE-20240801" rel="nofollow noopener noreferrer" target="_blank">https://www.nature.com/articles/d41586-024-02420-7?WT.ec_id=NATURE-20240801</a> Not to dunk on this research, which I think is interesting and important, but if you've ever explored iterated function systems, discrete dynamical systems, fractals or the like, this is a wholly unsurprising observation. A general class of observations is that repeatedly iterating a function on a given input will diverge from that input and start assuming qualities reflective of the function itself. For instance, watch some of the videos on this page: <a href="https://www.algorithm-archive.org/contents/barnsley/barnsley.html" rel="nofollow noopener noreferrer" target="_blank">https://www.algorithm-archive.org/contents/barnsley/barnsley.html</a> . In one set, you'll see a square with randomly-placed dots being squished down into various shapes. In another set, you'll see the Barnsley fern itself run through the same functions being squished down to roughly the same shapes. This is a general fixed-point result of this (and all contractive affine) systems: any input set of points will be squished into the same shapes, and precisely the same fern image will emerge no matter what (non-empty) input you start with when you iterate these processes often enough (by iterate I mean feeding the output of the functions back in as input, as in the linked paper). This is an instance of the Banach fixed-point theorem applied to the Hausdorff metric on images; the theorem states that any self-map that's contractive in the metric has a unique fixed point. In this case, the unique fixed point is the fern image; the map being iterated is a bit complicated but detailed on that linked page about the fern. The theorem tells us this unique fixed point is dependent only on the self-map, not on what input is put in. Naturally <a href="https://buc.ci?t=generativeai" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#GenerativeAI</a> training and input-output procedures are considerably more complicated than affine functions, but the same class of fixed point phenomena are almost surely at play, especially for the image-generating ones. Personally I'd find it surprising and interesting if there weren't fixed point theorems like this for <a href="https://buc.ci?t=generativeai" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#GenerativeAI</a> systems trained on their own outputs. <a href="https://buc.ci?t=contractionmap" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ContractionMap</a> <a href="https://buc.ci?t=fixedpoint" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#FixedPoint</a> <a href="https://buc.ci?t=fractals" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Fractals</a> <a href="https://buc.ci?t=ai" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#AI</a> <a href="https://buc.ci?t=genai" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#GenAI</a>

IT NewsFixed Point Math Exposed - If you are used to writing software for modern machines, you probably don’t think ... - <a href="https://hackaday.com/2024/06/23/fixed-point-math-exposed/" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://hackaday.com/2024/06/23/fixed-point-math-exposed/</a> <a href="https://schleuss.online/tags/softwaredevelopment" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#softwaredevelopment</a> <a href="https://schleuss.online/tags/microcontrollers" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#microcontrollers</a> <a href="https://schleuss.online/tags/floatingpoint" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#floatingpoint</a> <a href="https://schleuss.online/tags/fixedpoint" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#fixedpoint</a>

Nafnlaus 🇮🇸 🇺🇦Now, rise-over-run, that's a <a href="https://fosstodon.org/tags/FloatingPoint" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#FloatingPoint</a> operation. But *of course* you can't do that. You're running on a single-threaded <a href="https://fosstodon.org/tags/CPU" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#CPU</a> (no <a href="https://fosstodon.org/tags/GPU" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#GPU</a>) with no lookahead; everything blocks, and floating point ops block for a LONG time. So instead you're going to have integers mimick floating point (<a href="https://fosstodon.org/tags/FixedPoint" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#FixedPoint</a>)Okay, so you check to see when they're going over a certain remainder value and you should move up or down one row of <a href="https://fosstodon.org/tags/pixels" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#pixels</a>? OF COURSE you don't have time for that.

Ted LochHello! My brief <a href="https://mathstodon.xyz/tags/introduction" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#introduction</a>: I'm a professor of <a href="https://mathstodon.xyz/tags/economics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#economics</a> and a <a href="https://mathstodon.xyz/tags/publicpolicy" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#publicpolicy</a> scholar at Rice U. I use <a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#math</a> modeling to inform science-based policy. I work on the economics of emerging infectious diseases, climate and the energy transition, and using <a href="https://mathstodon.xyz/tags/gametheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#gametheory</a> to study symbiotic relationships in <a href="https://mathstodon.xyz/tags/ecology" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ecology</a>. In <a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#math</a> I am interested in <a href="https://mathstodon.xyz/tags/topology" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#topology</a> and have studied <a href="https://mathstodon.xyz/tags/fixedpoint" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#fixedpoint</a> theorems for locally contractive set-valued maps. I enjoy travel, kayaking, reading, and dreaming of classic cars.

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