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Han · 04-14-2014, 10:47 PM

(04-14-2014 09:17 PM)Manolo Sobrino Wrote:
(04-14-2014 07:31 PM)Han Wrote: The calc picked the sixth case because it rounds \( \pi \) to 3.141592653590 and not any of the other values. And while all the other values are just as good, it only makes sense to map to one of them -- namely the one corresponding to an input of 3.141592653590.

No, it doesn't.

The consistent answer for 12 significant digits is 0.000000000000, and that is not zero, those zeroes have a meaning.

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.

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.

Quote:Maple is able to do arbitrary precision calculations, you should try that.

That is completely irrelevant. One could argue that the Prime has a CAS mode in which the value of \( \sin(\pi) \) is always 0 and that users "should try that."

Quote:
Quote:To suggest that there is one, right approach to handling numerical calculations is also going to result in awful consequences.

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.