Re: IEEE Std.1788, inner operations, MD-RDM-IA, the Laws of Motion

[Oliver Heimlich <oliver@xxxxxxxxxxxxxxxxxxx>]

On 19.10.2015 22:55, Mehran Mazandarani wrote:
> Dear Std.1788 members and IC members
> First of all let me thank John, Walter, Evgenija, and Svetoslav for kind
> attentions and advice.
> My workload was too high and I couldn't reply earlier, so sorry for that.
> John and others told that what I sent - *IC-2[corrected version].pdf
> <http://grouper.ieee.org/groups/1788/email/pdfgOOF1CFv5a.pdf>* -  for
> explaining differences between approaches was not clearly. That's why I
> prepared another one for clarifying the matter, and put on the table .
> Please bear in mind what Isaac Newton figured out more than 300 years
> ago - *The Laws of Motion*.
> I guess we should take a glance to the nature for almost whatever we do
> in mathematics.
> I mean the mathematics and its theories or discoveries should have
> emerged in real world or in the nature.
> In the new presentation, the file *IC-3.mp4* , the three approaches,
> *IEEE Std.1788 2015*, *Inner operations*, and *Multidimensional RDM
> interval arithmetic* are examined by the laws of motion. Simply put, the
> approaches are examined using the nature/real world. Please see the
> attachment.
> As is seen, the IEEE Std.1788 is in violation of the third law of
> motion, and Inner operations are in violation of second law of motion,
> whereas the Multidimensional RDM interval arithmetic preserves the Laws
> of motion.
> Indeed, the *cancelMinus/cancelPlus* approach could not either handle
> the limitations of Inner operations.
> 
> In the next step I am going to show that how capable MD-RDM-IA is of
> dealing with Newton's method for solving a 1-D nonlinear system.
> 
> I appreciate you kind consideration in advance.
> 
> Comments are welcome.

Mehran,

I haven't heard of RDM before. For others who are interested, paper [1]
gives a basic introduction. From what I have found out, RDM is just
another variant of affine arithmetic, but the errors are written as α ∈
[0, 1] instead of ε ∈ [-1, 1] and error variables are accumulated as
polynomials to some extend (I didn't look so much into detail).

Saying the set-based interval arithmetic *violates* Newton's laws is
wrong. The exact result is just over-estimated (which is correct and
should always be expected) because of the dependency problem. You can
reduce the dependency problem to some extend by carrying over symbolic
computations like you do in RDM, but I can certainly find formulas which
will bring RDM beyond its limits too (e.g. with exponential or
trigonometric functions).

Apropos symbolic computations: Since IEEE Std 1788-2015 distinguishes
between the point function and its interval extension, you can always
use some symbolic math on the point function before evaluation. To give
an example, I compute a2 from your slides in Octave:

>> pkg load symbolic interval
>> # Symbolic simplification of the point function
>> syms F_A F_B m
>> a2 = (F_B - F_A + F_A) ./ m;
>> a2 = simplify (a2);
>> # Interval extension of the simplification
>> Ia2 = function_handle (a2, 'vars', [F_A, F_B, m]) ...
         (infsupdec ('[1, 3]'), infsupdec ('[1, 3]'), 1)
Ia2 = [1, 3]_com

Of course, this also has its limits.

Best regards,
Oliver

[1] http://www.jtacs.org/archive/2012/2/3/JTACS_2012_02_03.pdf