edwards@houxa.UUCP (D.LEWAN) (02/06/86)
I am currently considering the real numbers, some of their
constructive aspects and specific uses, and different
encodings for them for digital computers. This article is a
request for encodings that may not be well known as well
(but are indeed useful), as particular uses for those
encodings. Such a use is probably one for which
1. that encoding is convenient and
2. another encoding would be difficult to use.
A few philosophical notes should be made before I go too
far.
The act of encoding is that of assigning, to a *meaningful*
object, a name from which the object (or its meaning,
anyway) can be recreated. For example, to the object
"fifteen" we may assign the name "15". Now, from "15" we
can reconstruct "fifteen" by interpreting it as "10 + 5",
and the objects named by "10", "+" and "5" are all well
defined and understood. I cannot name all the real numbers
since there are only a countable number of names available,
and there are an uncountable number of reals. But I can
select subsets of the reals tailored to my particular needs
and for some of those subsets I can find worthwhile
encodings.
The standard exponent/base approach to real number
representation is a style that in some sense has the best
and worst of all worlds. Its precision is generally good
enough for most arithmetic applications, but absolute
precision waxes for large numbers, where the type integer
might in fact be more precise. It also doesn't guarantee,
for example, that n-th roots, when raised to the n-th power,
yield the original numbers, i.e. that
n ---- n
\/ q = q
in all cases. (Imagine moving your PC's best approximation
of the square root of 2 to a CRAY II and then performing
arithmetic.)
There are representations that do preserve such algebraic
properties, unfortunately they lose meaningful precision.
If one only knows that the square of a certain number always
yields 2 one does not necessarily know if it is greater than
or less than 0. (High school math conventions do no good
here.)
Now, I know that preservation of algebraic properties is
just one of many that might be of interest to a particular
application.
I am sending this plea to the net to find a wealth of
applications that might run better with less well known
encodings for specific real numbers.hofbauer@utcsri.UUCP (John Hofbauer) (02/09/86)
Mother of mercy, if you thought the debate on integer division got out of hand you ain't seen nothing yet! That is not the kind of constructive comment the poster was looking for but as a Numerical Analyst I'm not going to touch the representation of real numbers with a ten foot pole. Sorry.
aglew@ccvaxa.UUCP (02/12/86)
I saw a paper in the not-too-distant past about a real number representation that had three fields (four if you count the sign): fraction, exponent, and meta-exponent. I am not sure that I remember exactly what the meta-exponent was - it may have been something like the number of tens in 10**10**10**... exponent - but it was used to greatly increase range. I believe that you could express the loss in precision of large numbers as a fraction of the log of the number? Anybody remember this paper? Unfortunately, I have changed cities, and I remember where to find this paper only by the position in the stacks.
herndon@umn-cs.UUCP (02/13/86)
There've been some interesting representations suggested for "real" numbers (I shouldn't say floating point, but they aren't real either.) One article which appeared in Journal of the ACM a while back suggested using a very odd scheme involving a mantissa and another field which was (if I recall aright) the number of times e should be raised to itself before multiplying by the mantissa. Very useful for LARGE(!) and SMALL(!) numbers (we're talking numbers you couldn't even write down in scientific notation, they have so many zeroes). Mentioned lots of useful properties of these things, but computational efficiency wasn't one of them. This article appeared a year or two ago. There was an article in byte a while back suggesting an alternate representation for floating point numbers which might be useful on a micro... unfortunately I've forgotten all the details. I think the idea was to keep the logarithm of the number, and then do multiplies and divisions by adding and subtracting... mildly interesting, anyway. I heard something interesting from one of the hardware designers where I used to work (Phoenix, AZ) about a FP representation under consideration for large arrays of processing elements. I seem to recall that it involved non-unique representations for numbers, allowing arithmetic operations to be performed much faster since the representation put a finite bound on the number of carries which ever had to be performed (something like gray codes for FP numbers). Sorry I don't have more details, but maybe these notes will prompt somebody to mention more about them.
ken@rochester.UUCP (Ipse dixit) (02/16/86)
In article <17200002@umn-cs.UUCP> herndon@umn-cs.UUCP writes: > There was an article in byte a while back suggesting an alternate >representation for floating point numbers which might be useful >on a micro... unfortunately I've forgotten all the details. I >think the idea was to keep the logarithm of the number, and then >do multiplies and divisions by adding and subtracting... mildly >interesting, anyway. In 1979, Aug I think, an article appeared in CACM with the above scheme. It was called FOCUS. The rationale was that data from real world sensors, e.g. light, sound follow logarithmic scales so there were advantages to being able to multiply or divide quickly on cheap micros, and the lack of a true zero didn't matter much, the smallest number being essentially below thermal noise levels. I am curious to know if anybody actually used this scheme. Ken -- UUCP: ..!{allegra,decvax,seismo}!rochester!ken ARPA: ken@rochester.arpa Snail: Comp. of Disp. Sci., U. of Roch., NY 14627. Voice: Ken!