SoylentNews is people
SoylentNews
Arthur T Knackerbracket has found the following story:
Professor Peter Coveney, Director of the UCL Centre[*] for Computational Science and study co-author, said: "Our work shows that the behaviour of the chaotic dynamical systems is richer than any digital computer can capture. Chaos is more commonplace than many people may realise and even for very simple chaotic systems, numbers used by digital computers can lead to errors that are not obvious but can have a big impact. Ultimately, computers can't simulate everything."
The team investigated the impact of using floating-point arithmetic -- a method standardised by the IEEE and used since the 1950s to approximate real numbers on digital computers.
Digital computers use only rational numbers, ones that can be expressed as fractions. Moreover the denominator of these fractions must be a power of two, such as 2, 4, 8, 16, etc. There are infinitely more real numbers that cannot be expressed this way.
In the present work, the scientists used all four billion of these single-precision floating-point numbers that range from plus to minus infinity. The fact that the numbers are not distributed uniformly may also contribute to some of the inaccuracies.
First author, Professor Bruce Boghosian (Tufts University), said: "The four billion single-precision floating-point numbers that digital computers use are spread unevenly, so there are as many such numbers between 0.125 and 0.25, as there are between 0.25 and 0.5, as there are between 0.5 and 1.0. It is amazing that they are able to simulate real-world chaotic events as well as they do. But even so, we are now aware that this simplification does not accurately represent the complexity of chaotic dynamical systems, and this is a problem for such simulations on all current and future digital computers."
The study builds on the work of Edward Lorenz of MIT whose weather simulations using a simple computer model in the 1960s showed that tiny rounding errors in the numbers fed into his computer led to quite different forecasts, which is now known as the 'butterfly effect'.
[*] UCL: University College London
Journal Reference:
Bruce M. Boghosian, Peter V. Coveney, Hongyan Wang. A New Pathology in the Simulation of Chaotic Dynamical Systems on Digital Computers. Advanced Theory and Simulations, 2019; 1900125 DOI: 10.1002/adts.201900125
(Score: 2) by JoeMerchant on Thursday September 26 2019, @01:09PM
It all depends on the stability of your chaotic oscillator. For oscillators that diverge, or are very very close to diverging, any tiny difference may make a very significant change in the observed pattern - the difference between modelling with floats and doubles certainly can do that.
On the other hand, there are very richly patterned chaotic oscillators (or, as a UF professor of Chaos once chided me: quasi-periodic oscillators, though I think he's splitting hairs that are too fuzzy to classify), which deliver the same large scale, and mid scale patterns whether you compute them with floats, or doubles, or arbitrarily large precision. Of course, when you zoom down into these oscillators to reveal the fine structure at the limits of the computational precision, they all will eventually "chunk up" as you approach the limits of precision, whatever those limits are - this is easily seen with the classic Mandelbrot set zooming programs.
And, what is the difference between stability and instability in a chaotic oscillator? Usually just small changes in a single feedback coefficient can take you from an oscillator that converges to a single point, to bifurcation, to multiple bifurcations, sometimes to quasi-periodic oscillation, to chaos, to instability and divergence to infinity.
http://mangocats.com [mangocats.com]
🌻🌻 [google.com]