In the standard there are interval operations that
allow you to obtain x-x=0, these are the cancel plus/minus
operations. However, the cancel +/- are not "operations" in the
sense of algebra. I would suggest that the cancel plus/minus "operations"
are included in the basic standard.
S. Markov
PS. Algebraic operations have to be defined for all
intervals, whereas the cancel +/- are not. It is a pity that
the inner operations (proposed by me and Nate) were not
included in the standard. The inner +/- operations are
operations in algebraic sense.
On 26 Sep 2015 at 17:10, Kreinovich, Vladik wrote:
From: "Kreinovich, Vladik" <vladik@xxxxxxxx>
To: "Kreinovich, Vladik" <vladik@xxxxxxxx>,
Mehran Mazandarani <me.mazandarani@xxxxxxxxx>,
stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>,
"Andrzej Piegat" <apiegat@xxxxxxxxxxxxx>
Subject: RE: IEEE Standard 1788-2015
Date sent: Sat, 26 Sep 2015 17:10:56 +0000
On 1): we are talking about the problem that I mentioned in my first very email
reply: we have a function f(x1,…,xn) and n intervals, we need to find a range
of this function on these intervals.
If n is fixed, then we have a polynomial-time algorithm – but the power of the
corresponding polynomial grows with n.
From: stds-1788@xxxxxxxx [mailto:stds-1788@xxxxxxxx] On Behalf Of Kreinovich,
Vladik
Sent: Saturday, September 26, 2015 10:33 AM
To: Mehran Mazandarani <me.mazandarani@xxxxxxxxx>; stds-1788
<stds-1788@xxxxxxxxxxxxxxxxx>; Andrzej Piegat <apiegat@xxxxxxxxxxxxx>
Subject: RE: IEEE Standard 1788-2015
1) Yes, it would be desirable to compute the exact range all the time and
to avoid overestimation, but it is known theorem that computing the exact range
of a function under interval uncertainty is NP-hard, even for quadratic
functions, so if we want to get guaranteed bounds, excess width is inevitable,
i.e., if we want to always get an enclosure, an interval always containing the
exact range, it is inevitable that we have an enclosure which is sometimes
different from the exact range.
2) The original example of x-x when x is [1,3] (and a similar example of
x/x) are based on a major misunderstanding of interval computations which is,
unfortunately, rather common: that interval computations means replacing each
operation with numbers by a corresponding operation with intervals. This
so-called na"ive interval computations was never advocated by anyone, the very
first 1966 book by Moore has exactly this example of x-x to show that this is
NOT the way to go. A proper way – as shown in any textbook or survey on
interval computations and implemented in all the packages – is to first check
monotonicity and then use centered form if the function is not monotonic.
Monotonicity can be checked if we apply automatic differentiation to the
expression and then use na"ive interval computations to find the range of the
resulting derivative. If the resulting interval is always non-negative or
always non-positive, the function is monotonic and its range is easy to compute
by simply computing the values at endpoints. When we apply this to x-x,
interval computations lead to 0 range, NOT to [-2,2]. Same with x/x.
From:stds-1788@xxxxxxxx [mailto:stds-1788@xxxxxxxx] On Behalf Of Mehran
Mazandarani
Sent: Saturday, September 26, 2015 6:19 AM
To: stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>; Andrzej Piegat
<apiegat@xxxxxxxxxxxxx>
Subject: IEEE Standard 1788-2015
Dear John, Vladik, Michel, and IC members
Thank you for your kind reply.
I am writing to you regarding the previous emails about that section in Std.
1788 which deals with operations for interval arithmetic.
As I mentioned, based on the Std. 1788-2015 we may obtain results which lead to
values that are not possible actually.
Michel told that it is because of dependency issue, and that Moore's arithmetic
is indeed a worst-case arithmetic -- but this allows it to *guarantee* that the
computed result encloses any possible actual result.
Then, he told When an interval programmer writes a program to compute a
particular function, which could (in the point-function context) be written as
an
algebraic expression where certain variables may appear multiple times,
the programmer may take advantage of specific knowledge -- in particular,
monotonicity properties -- to compute a tighter enclosure than blindly
evaluating the expression using Moore arithmetic.
1. Using Moore's approach leads to obtain values which may not be possible,
i.e. impossible values. So, it makes us to analyze, decide, and design systems
and processes with too high costs and probably too complex. That all of these
are because of values which are impossible, i.e. they will not happen.
2. The standard should be based on an approach which makes us be assured in at
least some future development. This is while as you can see the attachment,
using advantage of specific knowledge we get to what I termed it Restoration
issue.
Additionally, we are just considering very simple cases, because the
cases themselves have their own issues of obtaining real possible
results/values, let alone problems related to differential equations and
defining derivatives.
I recommend to rethink about the section of Std. 1788-2015 which deals with
fourth basic operations in order to avoiding what will mislead us in so many
problems.
Comments are welcome.
Thank you so much for your kind attention and consideration.
Warmest regards,
Mehran Mazandarani
Department of Electrical Engineering
Ferdowsi University of Mashhad, Mashhad, Iran.
homepage:http://mehran.mazandarani.fumblog.um.ac.ir/
http://works.bepress.com/mehran_mazandarani
IEEE Member, me.mazandarani@xxxxxxxx
Prof. Svetoslav Markov, DSci, PhD,
Dept. "Math Modelling & Num Analysis", phone: +359-2-979-2876
Inst. of Mathematics and Informatics, fax: +359-2-971-3649
Bulgarian Academy of Sciences, e-mail: smarkov@xxxxxxxxxx
"Acad. G. Bonchev" st., block 8,
BG-1113 Sofia, BULGARIA mobile (gsm): 08888 24567
URL: http://www.math.bas.bg/~bio/
URL: http://www.biomath.bg/
www.biomath.bg/2015/
http://www.biomathforum.org/biomath/