Re: The current proposal
Paul,
Your first example provides a motivation for including modal intervals in the standard.
As you mention, for the intervals
x \in [-3,2]
y \in [1,6]
z \in [-5,-4]
then x*(y+z) should be expected in [-8,12]. Yet it is the case that
x*y+x*z=[-28,27],
so here as you point out the product is no longer distributive over the sum.
However, the modal interval arithmetic:
dual(x)*y+x*z=[-8,12]
maintains the distributive property in this case.
This is one of the many rasons I wish the 1788 group will investigate the modal intervals and include them in the standard.
Sincerely,
Nate Hayes
Sunfish Studio, LCC
----- Original Message ----- From: "Paul Langevin" <kkwweett@xxxxxxxxxx>
To: <stds-1788@xxxxxxxx>
Sent: Tuesday, February 24, 2009 8:49 AM
Subject: Re: The current proposal
----- Original Message ----- From: "Nate Hayes" <nh@xxxxxxxxxxxxxxxxx>
To: <stds-1788@xxxxxxxxxxxxxxxxx>
Cc: "Kearfott Ralph B" <rbk5287@xxxxxxxxxxxxx>
Sent: Monday, February 23, 2009 9:34 AM
Subject: Re: The current proposal
>I think the point Siegfreid makes is that the "ideal" mathematical
>properties of the interval arithmetic should drive the implementation,
>not
>the other way around.
>
I agree.
Here is another example suggesting intervals of the form [a,b] should be treated differerently when sign(a) and sign(b) are different or when they're equal.
Let x in [a,b], y in [c,d]. The product xy can be expected in [min ((ac)+,(ad)-,(bc)-,(bd)+), max ((ac)+,(ad)-,(bc)-,(bd)+)] where z+ = max (0,z) and z-=min (0,z)
The sum, as you mentionned, can be expected in [a+c, b+d].
When the signs of some boundaries are not equal, the product ceased to be distributive over the sum. For instance, let x in [-3, 2], y in [1, 6] and z in [-5, -4],
x (y + z) should be expected in [-3, 2] ( [1-5, 6-4]) = [-3,2] [-4,2] =[-8, 12] whereas
xy + xz should be expected in [-18, 12] + [-10,15] = [-28, 27]
On the other hand, when 0 is on the same side for a,b,c,d,e and f, no matter what form we take the result is the same. For instance, let u in [2, 3], v in [1,6] and w in [4,5]
u (v + w) and uv + uw should both be expected in [10,33].
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