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## [Axiom-developer] Re: Provisos generalize intervals.

 From: G. William Walster Subject: [Axiom-developer] Re: Provisos generalize intervals. Date: Tue, 14 Mar 2006 12:25:18 -0800 User-agent: Mozilla/5.0 (Macintosh; U; PPC Mac OS X Mach-O; en-US; rv:1.7) Gecko/20040616

```
Tim,

There are lots of possible generalizations of a real or extended
real interval.  For example: a vector of intervals, or a "box";
a complex interval consisting of a real and imaginary interval pair,
or an interval magnitude and phase; or a set of intervals, sometimes
called a list.  Another important interval construct is a Taylor Model
consisting of selected terms in a Taylor series and an interval error
bound.

In thinking about your question, it seems as if there is only one
important requirements that *must* be satisfied:  That is, the set of
possible results (what I call the containment set, or cset) must be
```
defined for all permitted operations on or functions of some interval construct. When some operation or function is operationalized, the resulting "interval" must enclose the cset for the given operation or function. We can never permit a containment failure.
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Usually, but not always, one wants a closed system in which operations
```
or functions of an interval construct result in new members of the same interval construct. For example basic arithmetic operations on
```extended real intervals produce extended real intervals.  No exceptions.
However, intersections of exterior intervals can result in two disjoint
```
intervals. This is a set of intervals.
```The pragmatic question to answer is whether the utility value of adding
a new interval construct and operations on it will add sufficient value
to justify the effort.  Value can be measured in terms of parsimonious
and transparent mathematics and code and overall resulting algorithm
speed.  I assume that narrow width contributes to speed.

So, the bottom line is that any operation or function of an interval
```
entity must have a defined cset. Evaluation must never produce a containment failure.
```
Hope this helps to clarify what you need to do.

Cheers,

Bill

root wrote:

```
```Bill,

```
I've taken a step back and am reviewing Moore's book again in preparation for a deeper reading of your book.
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It seems that the fundamental difference between the work that you and
Moore have pursued and the work I've been doing lies in the definition
of the endpoints of the intervals. Correct me if I'm wrong but it appears
that the endpoints of your intervals are all ordered numbers.

The endpoints of the intervals I'm looking at for provisos are not
numbers and, in most cases, are ordered by explicit constraints. My
work assumes that the endpoints can be pretty much anything, complex
numbers, vectors, symbols, polynomials, etc.

```
Thus I raise the question of intervals whose endpoints specify the inner and outer radius of two bounding surfaces. For example, the
```inner and outer radius of two spheres or the upper and lower bounding
surfaces of a function during integration. This is useful symbolically
since if I can show that the two bounding surfaces tends to zero I can
conclude that the integral tends to zero even if the actual function
is too complex to compute.

Provisos generalize intervals.

t
```
```

```