3 ms·

The article comments: "In a lot of respects, it’s quite amazing how accurate many calculations can be made with floating numbers, like orbital mechanics and hea

by jamessb 11y ago

The article comments: "In a lot of respects, it’s quite amazing how accurate many calculations can be made with floating numbers, like orbital mechanics and heat equations" I'm not sure it is so surprising. In Nick Trefethen's Numerical Analysis entry in The Princeton Companion to Mathematics, he notes: "Thus, on a computer, the interval [1 , 2] , for example, is approximated by about 10^16 numbers. It is interesting to compare the fineness of this discretization with that of the discretizations of physics. In a handful of solid or liquid or a balloonful of gas, the number of atoms or molecules in a line from one point to another is on the order of 10^8 (the cube root of Avogadro’s number). Such a system behaves enough like a continuum to justify our definitions of physical quantities such as density, pressure, stress, strain, and temperature. Computer arithmetic, however, is more than a million times finer than this. Another comparison with physics concerns the precision to which fundamental constants are known, such as (roughly) 4 digits for the gravitational constant G, 7 digits for Planck’s constant h and the elementary charge e, and 12 digits for the ratio μ_e/μ_B of the magnetic moment of the electron to the Bohr magneton. At present, almost nothing in physics is known to more than 12 or 13 digits of accuracy. Thus IEEE numbers are orders of magnitude more precise than any number in science. (Of course, purely mathematical quantities like π are another matter.)" http://people.maths.ox.ac.uk/trefethen/NAessay.pdf http://people.maths.ox.ac.uk/trefethen/NAessay.pdf

eriktaubeneck 11y ago
That's an interesting point I had not considered. My view (mostly from the math side, my shallow dive into physical systems was just from a PDE class in grad school) is from the view of unstable differential equations where very small changes to initial conditions can cause massive changes to the output of the system. My uninformed instincts would tell me that even with orders of more digits than atoms in a physical system, changes smaller than the available accuracy to initial conditions could have significant impact on the eventual state of the system.
- replicant 11y ago
  Another point of view on why it is not striking that we can model heat conduction well. The heat equation is well-posed, which among other things, means that the solution depends continuously of the input data. And we observe this in many physical systems, that small changes in the cause produces small changes in the effect, therefore, we should expect from good mathematical models of reality to display this property. Now, orbital mechanics do display unstable behaviour. I don't dare to adventure on how people work around this. https://en.wikipedia.org/wiki/Well-posed_problem https://en.wikipedia.org/wiki/Well-posed_problem

by jamessb 11y ago

eriktaubeneck 11y ago

That's an interesting point I had not considered. My view (mostly from the math side, my shallow dive into physical systems was just from a PDE class in grad school) is from the view of unstable differential equations where very small changes to initial conditions can cause massive changes to the output of the system. My uninformed instincts would tell me that even with orders of more digits than atoms in a physical system, changes smaller than the available accuracy to initial conditions could have significant impact on the eventual state of the system.

replicant 11y ago
Another point of view on why it is not striking that we can model heat conduction well. The heat equation is well-posed, which among other things, means that the solution depends continuously of the input data. And we observe this in many physical systems, that small changes in the cause produces small changes in the effect, therefore, we should expect from good mathematical models of reality to display this property. Now, orbital mechanics do display unstable behaviour. I don't dare to adventure on how people work around this. https://en.wikipedia.org/wiki/Well-posed_problem https://en.wikipedia.org/wiki/Well-posed_problem