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Manolo Sobrino · 04-14-2014, 11:32 PM

(04-14-2014 10:47 PM)Han Wrote: I was not debating what the answer for 12 significant digits was. I was debating on your suggestion any one among {4.79318*10^-12, 3.79309*10^-12, 2.793*10^-12, 1.79335*10^-12, 7.93266*10^-13, -2.06823*10^-13, -1.20691*10^-12, -2.207*10^-12, -3.20665*10^-12} would be fine. There is a difference between suggesting the correct answer is 0.000000000000 versus any one among this list. They are all "equal" when rounded to 12 sig. digits, but they are clearly not the same numbers. Surely you agree.

Oh Han... I proposed this, I took the time and pains to explain why it should be 0.000000000000. It should be that because if you consider that your result is accurate to more digits you are actually trespassing your own limits. But people just don't care, no one in this forum has ever mentioned the table maker's dilemma for instance, which is relevant for this kind of recurring discussions, it's all talk, talk, talk and vacuous claims. It's a kind of totemic attitude versus a device that I, due surely to cognitive biases having to do with my painful education can't understand.

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Quote:Well, in such a case that famous answer for sin (pi) is also wrong. You can't have your cake and eat it. Either you are consistent or you aren't.

In what sense are you suggesting it's wrong? Mathematically? or practically? Mathematically, no numerical calculations involving rational approximations of transcendental values will every be "exactly right", but they can be "right" in a practical sense once we account for roundoff errors. The Prime enables you to use two separate environments depending on your needs. If you only care to have approximate solutions, use the Home environment. If you need exact values, use the CAS environment.

First you like that your calc comes up with the exact result for Sin[3.14159265359000000000000000000000000000000000000...], which is not the true value of sin(Pi) yet is given with spurious precision to it's true value. Then you complain about them not being able to give the true answer of an arithmetic calculation when they're working out of their range. I just get lost with your point.

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Quote:That's what professional Mathematicians do, and HP has taken advantage of it in the past. Kahan might still be available. If not, I'd suggest Higham.

You know, this thing Maths is unluckily for some not a matter of opinion, either you get it right or you don't. In the real world of Logic, Math departments and peer reviewed journals your preferences don't matter. If you keep on being wrong for the sake of it you're out.

Mathematicians would not even consider using a calculator or CAS or any of the stable algorithms designed by Higham when it comes to "exact" mathematics for their proofs. The question of accuracy is moot in this case. That said, are we still discussing numerical algorithms applied to transcendental values -- on a calculator -- here? If you want the power of Higham or Kahan or (insert whatever numerical analyst you like here) then why are we even debating this in the context of a calculator? At some point one must concede that, given the constraints of a particular machine (RAM, CPU power, etc), some amount of accuracy is sufficient. Otherwise, we need a more powerful machine. As already mentioned, the calculator has a CAS for those who need "exact" answers.

Applied Mathematicians have been crunching numbers for a long time with less computing power than that which you have in your hands now. It can be done, and of course it has been done. We have digital computers since really the 50s, and people have been doing a great job with them since then, developing interval arithmetic and all kinds of nice stuff to provide meaningful answers with error bounds, which is the more "exact" mathematics you can do for this job, you're not proving the four colour theorem here. What would have given Feynman at Los Alamos for a dozen of these?... This is hopeless, we won't agree tonight (at Europe). Let's leave it for another day.