HP calcs are really not that accurate..

[Paul Dale]

(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

[pier4r]

(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)

Thus, apparently, you only needed 2 digits to measure your plot in square planks.

Finally someone noticing it.

[DA74254]

(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.

Finally someone noticing it.

Ok, then..

5.5239270495615041346253643432252334828540904763434087809926093365504231\
804817592757697276389655441007940097088632679222802317397420429653488735108049\
431522283055306025755503308162972536920967090801455644232660234663533541930694\
817334452387637360300940472628380640156082840510901474744449309617737322204628\
587857751372514370713807275201882769301371525005286841080723750834087309062435\
786357479267274998838754526213738836620730505474652837615093166252526834316754\
046480271470039410437409593057379145134933345546416041604225885527104187396816\
222529775627122498896098356291836153104596440768891614076851135059293733838206\
324717202805921738614097762147662658563248581721585263902179318564270997228015\
355171051818799271443241875167662659776580186704352442235721802530022563019264\
476709810881299909889223152355355454806291706563321136110714040551396736772884\
815235066188761998588231596924493632289988054229591203322875289740833929857937\
532990991958317041160217176167458475693651120361566485792429849638653120217264\
03334754862313137E+72

[Claudio L.]

(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.
Not wanting to "do something" with my precious 50G, I'd ask here; can it be run on the 49G+ as well?
I'm more inclined to use that calc as an experimental thingy than my 50G.

I don't mean to pollute this thread with any more ads, but yes, 39gs/40gs/49g+/50g for now. Thanks for the interest.

[Claudio L.]

(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:

1 - a number is totally meaningless if you don't specify its units
2 - a number is also totally meaningless if you don't specify its error margin

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.

And then threads like this would not even exist...

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].

[SlideRule]

"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)

[Thomas Okken]

(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...

[AlexFekken]

(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.

[AlexFekken]

(12-03-2017 08:05 PM)Claudio L. Wrote:  Totally meaningless without units?
That's not the physics they taught me.

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...

[AlexFekken]

(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?

[Paul Dale]

(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

[AlexFekken]

(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.

[ttw]

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.

[AlexFekken]

(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...

[ttw]

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.

[AlexFekken]

(12-04-2017 01:22 AM)ttw Wrote:  Point-valued Newton does not guarantee convergence compared to interval-valued Newton which does.

Interesting observation....

[Thomas Okken]

(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?

[AlexFekken]

(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...e_analysis

[Thomas Okken]

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?

[AlexFekken]

(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.