Talk:Interval arithmetic

Interval Comparison
The article on interval arithmetic is quite long. But I didn't find a mention of interval comparison. It seems to me that for practical use interval arithmetic needs to be complemented by interval comparison.

Take for example <, we have 3 different outcomes:

[a,b] < [c,d] == true,    b < c  [a,b] < [c,d] == false,    d < a  [a,b] < [c,d] == maybe,    otherwise

The "maybe" turns an original deterministic pointwise program into a non-deterministic intervalwise program.

Jan Burse (talk) 15:17, 3 July 2012 (UTC)


 * There is some discussion here (under "Comparisons")--Rumping (talk) 23:38, 4 July 2012 (UTC)


 * I see, assertions or mapping of maybe to either true or false.
 * BTW, constraint logic programming usually can easily deal with this
 * tri-logic. In the case of true, the constraint is dropped, in the
 * following example X #< Y disappears from the constraint store:

?- X in 0..3, Y in 4..7, X #< Y. X in 0..3 Y in 4..7


 * In case of false, the constraint solving fails, in the
 * following example unsatisfiability is detected and false is
 * returned:

?- X in 4..7, Y in 0..3, X #< Y. false


 * In case of maybe, the constraint simply stays in the constraint
 * store until it either turns into true or false, when the variables
 * become more restricted:

?- X in 2..7, Y in 0..5, X #< Y. X in 2..7 Y in 0..5 X #< Y ?- X in 2..7, Y in 0..5, X #< Y, label([X,Y]). X = 2, Y = 3 ; X = 2, Y = 4 ; Etc...


 * The above should work with http://www.swi-prolog.org/man/clpfd.html,
 * How much true/false is detected early on depends a little bit on
 * the system, there are stronger and weaker ones. The systems might
 * also return smaller intervals or smaller sets in the course of
 * solving a problem, and/or can non-deterministically show multiple
 * solutions. Jan Burse (talk) 11:55, 7 July 2012 (UTC)

Dependency problem
Isn't it kind of trivial to determine the correct resulting value of a complicated function if it is continuous and differentiable along the entire interval's domain? For any function of any number of variables, you just have to get the (partial) derivatives in order to get the maxima and minima inside the interval's region, get the function's values at the bounds, and then the resulting interval would consist of the minimum and maximum value in this set. Am I missing or overlooking something here? Rinku*gt (talk) 14:10, 9 February 2013 (UTC)


 * Well, let's take a relatively simple case where that's not possible. Suppose you have a sixth-order polynomial in one variable.  It's clearly continuous and differentiable, and if you differentiate it, you get a quintic.  To find the maxima and minima, you have to find the zeroes of that quintic, which may indeed be feasible or even easy, and it is certainly possible to approximate reasonably well with sufficient calculation, but by the Abel-Ruffini theorem, it's not possible to do algebraically in general.  You can do it with ultraradicals, which I didn't know about until now, but that article says, "the resulting expressions can be enormous,...taking many megabytes of storage for a general quintic with symbolic coefficients."


 * Which is to say, the approach you describe is not necessarily trivial or possible, and using a more conservative approximation may be a lot easier. But in the cases where you can find the zeroes of the derivative exactly, yes, you can do that.  If you can only find them approximately, you need to worry whether the second or higher derivative is big enough that when you compute the value of your complicated function at your approximate critical point, it may not be approximately the value of your complicated function at the actual critical point: $$x_0 \approx x_1 \not\Rightarrow f(x_0) \approx f(x_1)$$, even though $$f$$ is continuous and differentiable, because $$f''(x_0)$$ might be $$10^{100}$$ or $$10^{10^{100}}$$.


 * This is actually the reason I came to this page: I had a problem I was trying to solve with derivatives, and I realized that interval arithmetic bisection would be a much simpler approach! In fact, you can solve the above approximation problem with interval bisection; if instead of an approximate root, you get an interval which is guaranteed to contain the root, you can algebraically compute the second derivative (a fourth-order polynomial) and use simple interval arithmetic on it to verify that it doesn't get so large that your approximate answer could be badly wrong.  Even a quite crude conservative approximation of the bounds may be good enough to give you your answer.  (Interval arithmetic is also one of the simplest ways to approximate the roots of an arbitrary polynomial.)


 * There are also other interesting differentiable functions whose derivatives aren't even expressible in closed form, let alone solvable algebraically. Kragen Javier Sitaker (talk) 00:45, 4 June 2014 (UTC)

Formulas unnecessarily complex
In the second paragraph of the article, the middle parts of the formulas for addition and subtraction (highlighted below in bold)


 * [a, b] + [c, d] = [min (a + c, a + d, b + c, b + d), max (a + c, a + d, b + c, b + d)] = [a + c, b + d]
 * [a, b] &minus; [c, d] = [min (a &minus; c, a &minus; d, b &minus; c, b &minus; d), max (a &minus; c, a &minus; d, b &minus; c, b &minus; d)] = [a &minus; d, b &minus; c]

are pedagogically confusing and mathematically misleading. They are pedagogically confusing because they make the lines unnecessarily long and complex. One never needs to compute the cross results a + d, b + c, a - c, or b - d. They are mathematically misleading because they suggest that the definition of the interval sum and difference involves only the endpoints of the input intervals, whereas a more appropriate suggestion would refer to all possible values within the respective intervals, such as perhaps


 * [a, b] + [c, d] = {x = y + z | a &le; y &le; b, c &le; z &le; d} = [a + c, b + d]

I suggest that the bolded parts of the formulas be omitted for clarity and simplicity. I have made this change in the article.

Best regards, Scwarebang (talk) 23:22, 11 September 2013 (UTC)

Patents
The section Patents describes an alleged controversy over patents in interval arithmetic, sourced only to the website of a product that disputes the validity of the patents. This cannot be considered an independent reliable source. Even more, the person named, G. William Walster, appears to still be alive. I am removing the entire section as ontentious material about living persons that is unsourced. Deltahedron (talk) 16:06, 11 February 2014 (UTC)

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 * the https://p2k.stiealrifaie.ac.id/IT/2-2741-2638/Interval_2522_p2k-stiealrifaie.html is has or no copy right(only link to this www.)and what is thing i should to edit in this acticle. พนรนร (talk) 08:27, 11 September 2022 (UTC)