Re: [q-lang-users] Q 7.2 RC2

[John C. <co...@cc...>]

Albert Graef scripsit:

> Yes, I have to agree that 3/0 should now return inf instead of failing, 
> although that introduces a big incompatibility with earlier versions. :(

I think it's one that everyone can live with.

> I'm not so sure about 3%0, though. As I argue under item 6 below, for me 
> the right way to implement (%) on inexact numbers is the way it is done 
> now, i.e., to fall back to inexact division. But inexact division 
> returns (or rather will return when I fixed it) inf in this case. So 
> there is a case for having 3%0 evaluate to inf, too. X%0 should still 
> fail if X/0 does (the current definition isn't working correctly in that 
> case, but I've fixed that now). However, AFAICT this case wouldn't 
> actually arise any more.

I now agree with this:  % is not to be understood as exact division
(of the div-and-rem sort), but exactness-contagion division, just as +
- * are exactness-contagion operators: they return inexact results iff
at least one argument is inexact.  / on the other hand always produces
inexact results.  So both 3%0 and 3/0 should return inf.

> Therefore, if (%) is to work on all numbers (which is a good thing, 
> IMHO) then the right way to do this is to fall back to inexact division 
> on inexact operands.

This one was a consequence of my misunderstanding of %, and I agree it
should be as you say.  

> Besides, there's also the issue that there's no single "canonical" way 
> to do the rational approximation. So I think that it's much better to 
> leave that to a separate module (like Rob's ratutils.q) which can do a 
> much more comprehensive job and provide different kinds of specialized 
> approximation algorithms.

For the record, R5RS Scheme provides a "rationalize" procedure of two
arguments which returns the simplest rational number that differs from
X by no more than Y, and adds:

        A rational number r1 is simpler than another rational number r2
        if r1 = p1/q1 and r2 = p2/q2 (in lowest terms) and |p1| <= |p2|
        and |q1| <= |q2|. Thus 3/5 is simpler than 4/7. Although not all
        rationals are comparable in this ordering (consider 2/7 and 3/5)
        any interval contains a rational number that is simpler than
        every other rational number in that interval (the simpler 2/5
        lies between 2/7 and 3/5). Note that 0 = 0/1 is the simplest
        rational of all.

Rationalize, like almost all R5RS procedures, is exactness-contagious.

> > 7) This brings up the issue of coercions from exact to inexact (easy,
> > just need a special case for complex numbers) and for inexact to exact.
> > These should be in the library somewhere.
> 
> In fact they are. There's float to convert any Real number to a Float, 
> there are round, trunc, floor and ceil to go the opposite direction, and 
> there are Rob's Float -> Rational approximation algorithms. If needed, 
> these can be applied to the real and imaginary components of Complex. 
> What else do you need?
> 
> Granted, you can't just apply, say, round to Complex, but usually that 
> doesn't make mathematical sense anyway. The library cannot know whether 
> you'd prefer to round according to, e.g., the 1-, 2- or maybe the \infty 
> norm, that should be up to the application.

R5RS uses the phrase "{inexact,exact} [representable] number that is
numerically closest to the argument", presumably in the sense of
having the smallest difference from the argument.  Still, I agree
that flexibility probably beats simplicity here, especially in the
case of general complex numbers.

-- 
John Cowan      co...@cc...        http://www.ccil.org/~cowan
        Is it not written, "That which is written, is written"?