HP calcs are really not that accurate.. - Printable Version +- HP Forums ( https://www.hpmuseum.org/forum)+-- Forum: HP Calculators (and very old HP Computers) ( /forum-3.html)+--- Forum: General Forum ( /forum-4.html)+--- Thread: HP calcs are really not that accurate.. ( /thread-9610.html) |

RE: HP calcs are really not that accurate.. - Paul Dale - 12-03-2017 11:40 AM
(12-03-2017 10:03 AM)emece67 Wrote: 256 digits may be an overkill when dealing with physics problems, but there are other problems where they may be not enough. Digits of π calculations would have been my example There are others. Pauli RE: HP calcs are really not that accurate.. - pier4r - 12-03-2017 01:45 PM
(12-03-2017 10:03 AM)emece67 Wrote:(12-02-2017 07:42 AM)DA74254 Wrote: And yes, I know exactly how big my land plot is in square plancks (just over 5,5x10^72) Finally someone noticing it. RE: HP calcs are really not that accurate.. - DA74254 - 12-03-2017 02:15 PM
(12-03-2017 01:45 PM)pier4r Wrote:(12-03-2017 10:03 AM)emece67 Wrote: Thus, apparently, you only needed 2 digits to measure your plot in square planks. Ok, then.. 5.5239270495615041346253643432252334828540904763434087809926093365504231\ 804817592757697276389655441007940097088632679222802317397420429653488735108049\ 431522283055306025755503308162972536920967090801455644232660234663533541930694\ 817334452387637360300940472628380640156082840510901474744449309617737322204628\ 587857751372514370713807275201882769301371525005286841080723750834087309062435\ 786357479267274998838754526213738836620730505474652837615093166252526834316754\ 046480271470039410437409593057379145134933345546416041604225885527104187396816\ 222529775627122498896098356291836153104596440768891614076851135059293733838206\ 324717202805921738614097762147662658563248581721585263902179318564270997228015\ 355171051818799271443241875167662659776580186704352442235721802530022563019264\ 476709810881299909889223152355355454806291706563321136110714040551396736772884\ 815235066188761998588231596924493632289988054229591203322875289740833929857937\ 532990991958317041160217176167458475693651120361566485792429849638653120217264\ 03334754862313137E+72 RE: HP calcs are really not that accurate.. - Claudio L. - 12-03-2017 07:33 PM
(12-03-2017 08:54 AM)DA74254 Wrote:(12-02-2017 11:51 PM)Claudio L. Wrote: newRPL commercial..I've been looking at that site every now and then. I don't mean to pollute this thread with any more ads, but yes, 39gs/40gs/49g+/50g for now. Thanks for the interest. RE: HP calcs are really not that accurate.. - Claudio L. - 12-03-2017 08:05 PM
(12-03-2017 02:31 AM)AlexFekken Wrote: My previoius post reminded me of literally the first two basic principles that I learned when I started studying physics (and mathematics) at university in 1976: Totally meaningless without units? That's not the physics they taught me. The error margin I agree, if you stick to physics. When you get to engineering, it's combined with many other uncertainty factors into a global safety factor, so it's not necessary to repeat and analyze each result's error margin, we know roughly what it is, and combined with all the other things that can go wrong, we establish what works or not. No need for a lot of digits precision though, we are talking large percentage factors... (12-03-2017 02:31 AM)AlexFekken Wrote: But still, should we not have an abundance of (freely available) tools now to do, for example, interval arithmetic? And should these not be the standard in scientific education by now. Clearly, this is much more fundamental and important than e.g. CAS or graphing capabilties. Sounds like a nice challenge... I'll start thinking about this, you might see it implemented in 6 months or so in... [that project I mentioned a million times]. RE: HP calcs are really not that accurate.. - SlideRule - 12-03-2017 08:15 PM
"There are four properties of computers that are relevant to their use in the numerical solution of problems of algebra and analysis. These properties are causes of many pitfalls: (i) Computers use not the real number system, but instead a simulation of it called a "floating-point number system." This introduces the problem of round-off. (ii) The speed of computer processing permits the solution of very large problems. And frequently (but not always) large problems have answers that are much more sensitive to perturbations of the data than small problems are. (iii) The speed of computer processing permits many more operations to be carried out for a reasonable price than were possible in the pre-computer era. As a result, the instability of many processes is conspicuously revealed. (iv) Normally the intermediate results of a computer computation are hidden in the store of the machine, and never known to the programmer. Consequently the programmer must be able to detect errors in his process without seeing the warning signals of possible error that occur in desk computation, where all intermediate results are in front of the problem solver. Or, conversely, he must be able to prove that his process cannot fail in any way". Pitfalls in Computation, or Why a Math Book Isn't Enough by George E. Forsythe (Stanford University) RE: HP calcs are really not that accurate.. - Thomas Okken - 12-03-2017 09:53 PM
(12-03-2017 02:31 AM)AlexFekken Wrote: But still, should we not have an abundance of (freely available) tools now to do, for example, interval arithmetic? How would you automate calculation with intervals? Without knowing the actual distribution of each variable, and without knowing which ones are independent and which ones are not, those calculations would be meaningless. I think we're still stuck with leaving error analysis to humans... RE: HP calcs are really not that accurate.. - AlexFekken - 12-03-2017 10:41 PM
(12-03-2017 09:53 PM)Thomas Okken Wrote: How would you automate calculation with intervals? Without knowing the actual distribution of each variable, and without knowing which ones are independent and which ones are not, those calculations would be meaningless.You are right, it is not that straightforward. Which is why I (sneakily) switched from talking about error margins to interval arithmetic. Interval arithmetic should not be a problem (technically) and it should at least give you a rough idea of how fast errors could be propagating. Better than lying with either hidden digits or unknown rounding errors. RE: HP calcs are really not that accurate.. - AlexFekken - 12-03-2017 10:42 PM
(12-03-2017 08:05 PM)Claudio L. Wrote: Totally meaningless without units?OK I admit: not all of them, just the vast majority... (12-03-2017 08:05 PM)Claudio L. Wrote: Sounds like a nice challenge... I'll start thinking about this, you might see it implemented in 6 months or so in... [that project I mentioned a million times].Looking forward to it. Please keep Thomas's caveat in mind... RE: HP calcs are really not that accurate.. - AlexFekken - 12-03-2017 11:18 PM
(12-03-2017 08:05 PM)Claudio L. Wrote: Sounds like a nice challenge... I'll start thinking about this, you might see it implemented in 6 months or so in... [that project I mentioned a million times]. By the way, I am serious enough to buy an HP 50g (while they are still reasonably priced) if you are serious about this. I assume you are talking about something like interval arithmetic in newRPL...? So are you serious? RE: HP calcs are really not that accurate.. - Paul Dale - 12-03-2017 11:18 PM
(12-03-2017 09:53 PM)Thomas Okken Wrote: How would you automate calculation with intervals? Without knowing the actual distribution of each variable, and without knowing which ones are independent and which ones are not, those calculations would be meaningless. I think we're still stuck with leaving error analysis to humans... Be very conservative and watch the intervals blow out to infinity. There are interval libraries. MPFI based on MPFR seemed good when I looked some time back. Pauli RE: HP calcs are really not that accurate.. - AlexFekken - 12-03-2017 11:29 PM
(12-03-2017 11:18 PM)Paul Dale Wrote: Be very conservative and watch the intervals blow out to infinity.At least then you have some confirmation then that your answer is probably meaningless. Of course for purely mathematical calculations you could start with a very large number of digits, e.g. more than 256 :-), as it usually takes a while before you actually get to infinity. EDIT: (12-03-2017 11:18 PM)Paul Dale Wrote: There are interval libraries. MPFI based on MPFR seemed good when I looked some time back.Thanks, now let's see if there will be a calculator version soon. RE: HP calcs are really not that accurate.. - ttw - 12-03-2017 11:39 PM
There are lots of convergent interval algorithms. While intervals do use conservative rounding estimates, there are two common operations that shrink intervals. The first is the obvious division by a constant. (A,B)/2 becomes (A/2, B/2) with A/2 being rounded down and B/2 being rounded up. The new interval is smaller than the original (not twice as small though.) The second, and not as obvious shrinking algorithm is the interval Newton's method. One starts with an interval (A,B) which is guaranteed to cover the answer; next the interval Newton is applied (it's not just applying Newton's method with interval rounding to the endpoints; there is some other stuff that can be done but I don't remember all of it.) One gets a new interval (C,D) that also includes the solution. This interval may actually be larger than the input interval. (Magic Manipulation Alert) As both (A,B) and (C,D) include the desired point, the intersection of the intervals must also include the solution. So the interval (Max(A,C), Min(B,D)) (no rounding involved) must also contain the solution. In one dimension, this method always converges (unlike ordinary Newton) and provides a convergent sequence of intervals bracketing the solution. The convergence isn't guaranteed in several dimensions though. RE: HP calcs are really not that accurate.. - AlexFekken - 12-04-2017 12:00 AM
(12-03-2017 11:39 PM)ttw Wrote: there are two common operations that shrink intervals And any function with the absolute value of the derivative (uniformly) less than 1 should do that too. Chosing the right algorithms (e.g. forward vs backward iterations) will be important to keep the interval sizes under control. Of course this has always been the case but interval arithmetic would show the impact more clearly, i.e. it would help making the right choices and justifying them... RE: HP calcs are really not that accurate.. - ttw - 12-04-2017 01:22 AM
To me, one of the more interesting ideas in interval arithmetic is the ability to use intersections (and unions) of guaranteed solution-containing intervals. Point-valued Newton does not guarantee convergence compared to interval-valued Newton which does. RE: HP calcs are really not that accurate.. - AlexFekken - 12-04-2017 02:06 AM
(12-04-2017 01:22 AM)ttw Wrote: Point-valued Newton does not guarantee convergence compared to interval-valued Newton which does. Interesting observation.... RE: HP calcs are really not that accurate.. - Thomas Okken - 12-04-2017 02:16 AM
(12-04-2017 01:22 AM)ttw Wrote: Point-valued Newton does not guarantee convergence compared to interval-valued Newton which does. Newton's method not guaranteed to converge? That's news to me. Could you elaborate? RE: HP calcs are really not that accurate.. - AlexFekken - 12-04-2017 02:24 AM
(12-04-2017 02:16 AM)Thomas Okken Wrote: Newton's method not guaranteed to converge? That's news to me. Could you elaborate? As I remember it, it is only garanteed to converge when you start "close enough" (and the function smooth enough of course) because the function that you iterate is a (local) contraction with the zero as its fixed point. Global convergence is not garanteed unless you have additional (global) constraints. Will see if I can find an example. EDIT: https://en.wikipedia.org/wiki/Newton%27s_method#Failure_analysis RE: HP calcs are really not that accurate.. - Thomas Okken - 12-04-2017 02:38 AM
Yes, certain conditions have to be met in order for convergence to be guaranteed. That's high-school math. What changes when you calculate with intervals? RE: HP calcs are really not that accurate.. - AlexFekken - 12-04-2017 02:56 AM
(12-04-2017 02:38 AM)Thomas Okken Wrote: Yes, certain conditions have to be met in order for convergence to be guaranteed. That's high-school math. What changes when you calculate with intervals? You need to add an essential additional requirement: that the starting interval contains the solution/fixed point. Of course if you have such an interval then bisection or a combination of bisection and Newton would also garantee convergence to a solution. But I am not sure why the 'Starting point enters a cycle' situation could not occur and prevent convergence. Perhaps someone should try f(x)=x^3-2x+2 with an interval that includes 0 and 1... (Thinking a bit more) since 0 is mapped to 1 and 1 to 0 they will be in every iteration of the interval. I think that disproves it: interval-Newton does not necessarily converge either. |