When a calculayions misleading?

Mrwhy · 2012-07-05 20:31:54

When I calculate a path, or movie, often each state is calculated from the previous state.
Thus the ERRORS may tend to build up - even to the "nonsense" level.

For example if I roll a marble down a child's playground slide it goes down wandering from side to side but usually NOT escaping over the side.

So how big, and what KIND, of errors give
(a) an unstable result - escape beyond the slide edges
(b) PERIODIC meanderings about the valley bottom as we roll down the slide
(c) NON-periodic but bounded wanderings down the valley
(d) Settling to a steady descent down the valley bottom

These questions are vital to the whole concept of computer calculation.
And even IF the slide has no bends, changes of gradient or changes in cross-sectional shape.

The question I am asking is WHEN do calculations of a "strange atteactor" give results "of the same structure" - and thus USEFULLY INFORMATIVE.
Beyond what point are they nonsense?
Can they ever recover and become "accurate" again?
How would we know?

I suspect the idea "of the same structure" is meaningless as "structure" is a variable in infinite-dimensional space.

So HOW CAN I TELL before I waste weeks calculating meaningless trajectories!
Only SOME results are stable enough to "forgive" small-enough errors.
SOME resuts - not only those that repeat steadily and periodically!

For HOW LONG do the calculations of planetery orbits stay accurate enough not to be totally invalid.
How do we know when?

All ideas welcome
Please help.

bobbym · 2012-07-05 21:42:17

Hi Mrwhy;

All the questions you are asking are covered in the field of numerical analysis. That is where the question of error propagation is examined in detail.

One could say in chaos theory the errors in calculations are magnified and studied, in numerical analysis the errors are minimized.

All calculations that are done in floating point arithmetic contain errors. How much error? That question was first asked by the great John Von Neuman. He developed the first bit of error analysis. It is important to remember that a total error analysis is sometimes not possible.

The following are some direct consequences of errors in how computers store and do arithmetic on numbers.

1)The number line has holes in it!

2)(a+b)(a-b) ≠ (a^2 - b^2)

3) The quadratic formula is garbage and should never be used.

4) Newton's iteration is numerically unstable and should be used rarely.

5) Adding a long list of numbers from left to right is often not the same as adding it right to left!

6) Computers can not subtract!

Mrwhy · 2012-07-05 23:14:24

Ok, thank you

So how do I know if the present calculated position of a satellite in a Trojan orbit between earth and moon is valid
What does "stable orbit" mean
Can it be closed BECAUSE OF computer error whereas really it is unstable around the Trojan point no matter how close we are to that point.
I think that some of the trojan points ARE unstable in that sense whilst others are stable within certain bounds.

bobbym · 2012-07-05 23:21:15

Hi;

The planets orbits are known to be chaotic. The term is defined as sensitivity to initial conditions. Meaning that small errors in the measured position of the planet will be greatly amplified by the equations that describe the motion.

But a physical problem of objects settling into Lagrangian points is not what your post asked about. For instance:

BECAUSE OF computer error

The error I am speaking of in post #2 is not a computer error. It is errors that are inherent in the way computers do arithmetic. In other words it is a property of computer arithmetic.

Mrwhy · 2012-07-05 23:53:14

Not real planets!

The "mathematically pure" set of point-but-massive bodies moving relatively by Newton's Inverse Square Gravity Rule-of-Thumb without collisions.

There are a few position-velocity states into which, if placed, a third body will seem to be stable rotationg "motionlessly" with respect to the other 2.
A very-slightly-disturbed such body will move in an orbit about this so-called Trojan Point.
Are the calculated "strange attractors" imaginary artefacts due to computer error
How would we know?

The general question "How do I tell when my calculations are rubbish (even when totally reproducible) is most interesting and the opposite of chaos!
With chaos as nextx=4x(1-x) there is chaos spread throughought 0<x<1.
But for less than 4 there are interesting BANDS of permitted x and some periodic results.
The results are NOT evenly-distributed: their structure SEEMS stable against very small errors, but of course if we look at detail sufficient to "show errors" we will surely find them.

It is this ability to "accept and forgive" tiny errors while preserving the structural integrity for me to SEE - that is what fascinates me.

bobbym · 2012-07-06 00:02:56

Hi;

Are the calculated "strange attractors" imaginary artefacts due to computer error
How would we know?

Not quite following what you are asking. If you are looking for an explanation of the why of chaos then that can not be done. Years ago when the computer jockeys were looking at it they were making progress. Then the topologists took over and since then zilch!

Can you describe what you mean by computer error?

Mrwhy · 2012-07-06 06:22:42

Does the iteration nextx=4x(1-x) repeat or does it not?

On any MACHINE eventually it gets an x that it cannot tell different to an earlier x.
As nextx depends ONLY on x, from then on the whole list of xs repeats.

So HOW CAN WE TELL if the x-sequence "really" does nover repeat or not?

bobbym · 2012-07-06 06:50:18

Hi Mrwhy;

That equation and all the rest that chaoticians went wild with about 20 years ago are considered to be numerical anomalies. In numerical analysis we rearrange that equation or abandon it entirely.

If you want to study Julia sets then you do chaos and work with equations where sensitivity to initial conditions is welcomed. When you are working with a real calculation like where is Mars right now you certainly do not want chaotic effects to ruin the calculation. Just a different perspective.

Remember #2 in post #2. If you interested in allowing round off errors to magnify then you use the LHS. If you are interested in keeping error as small as possible then you use the RHS.

When a calculayions misleading?

Your original question is a straight numerical analysis question. Chaos is our enemy there and is to be reduced if not eliminated.

Mrwhy · 2012-07-06 20:40:20

Thanks for your comments

So you are saying that "Truth" cannot be found by calculations.
You are saying it depends what sort of answer you want!

The sort of answer I'd find helpful is this:-
For the value of k that gives a stable 3-cycle from the iteration nextx=kx(1-x) do YOU say this is "sensitive to intital conditions" or not?
For the STRUCTURE is the same (a stable 3-cycle and an attractor) whatever tiny changes you make or suffer, for 0<x<1. Start in this range and you stay in this pattern for ever - no escape! If you start outside the 3-cycle range (but with 0<x<1) you soon get attracted into it

bobbym · 2012-07-06 21:59:43

Hi;

So you are saying that "Truth" cannot be found by calculations.
You are saying it depends what sort of answer you want!

I personally disagree with that conclusion. I think the only way to truth is through calculations.

nextx=kx(1-x) do YOU say this is "sensitive to intital conditions" or not?

To know the answer to that we would need to compute the Lyapunov exponent, which I have forgotten how to do.

If you were discussing error propagation, measurement errors, loss of significant digits or how a get a number then I believe that I am the best bet. If we are going to talk about chaos, attractors and cycles then I am not an expert and can only be someone to talk to. A sounding board for your own ideas as I have very few of my own.

Mrwhy · 2012-07-07 07:26:36

Bobby, thanks for being all three
Help
Sounding Board
Someone to talk to

bobbym · 2012-07-07 10:24:54

Hi Mrwhy;

Your welcome.

Math Is Fun Forum

#1 2012-07-05 20:31:54

When a calculayions misleading?

#2 2012-07-05 21:42:17

Re: When a calculayions misleading?

#3 2012-07-05 23:14:24

Re: When a calculayions misleading?

#4 2012-07-05 23:21:15

Re: When a calculayions misleading?

#5 2012-07-05 23:53:14

Re: When a calculayions misleading?

#6 2012-07-06 00:02:56

Re: When a calculayions misleading?

#7 2012-07-06 06:22:42

Re: When a calculayions misleading?

#8 2012-07-06 06:50:18

Re: When a calculayions misleading?

#9 2012-07-06 20:40:20

Re: When a calculayions misleading?

#10 2012-07-06 21:59:43

Re: When a calculayions misleading?

#11 2012-07-07 07:26:36

Re: When a calculayions misleading?

#12 2012-07-07 10:24:54

Re: When a calculayions misleading?

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