4 ms·
> Logarithmic growth asymptotically decreases Logarithmic growth slows down, but not asymptotically. Think about it: what would the asymptote be? (There is non
by sampo 8y ago
> Logarithmic growth asymptotically decreases
Logarithmic growth slows down, but not asymptotically. Think about it: what would the asymptote be? (There is none.)
- ThePhysicist 8y ago100 % accuracy is what the algorithm wants to achieve (in the most basic case), so that's your asymptote. If your precision increases logarithmically e.g. as 1-1/log(x) you get arbitrarily close to 1, which is the definition of an asymptote I think.
- sampo 8y ago> 100 % accuracy is what the algorithm wants to achieve Well, in that context every kind of (monotonic) growth is asymptotic, so the word has no meaning.
- air7 8y agoIt really slows down though. Log(n) indeed goes to infinity but if you numbered all the atoms in the universe and pass them through the function, the last one would be ~80.
- siekmanj 8y agoThat may be true, but that is a far cry from asymptotic
- air7 8y agoMathematically, yes. For all real-world intents and purposes, it's bounded. I sometimes use this very example to illustrate the profound difference between the two which can be easily under-estimated. (That or log(log(n)) which also goes to infinity but is real-world bounded at <3)
- sampo 8y agoThat is in log_10, in log_e it would be near 190.
- throwawaymath 8y agoOkay, yeah you make a good point. Strictly speaking a logarithmic function does not imply a convergent series. To be more precise, what I'm talking about are slowly varying functions. These are functions which don't converge but which have similar characteristics to convergent functions (namely, extremely slow rate of change). Since you made the correction you're probably already familiar with these, but for the benefit of others: https://en.wikipedia.org/wiki/Slowly_varying_function https://en.wikipedia.org/wiki/Slowly_varying_function