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Re: "Assuming"

  • To: mathgroup at smc.vnet.net
  • Subject: [mg85649] Re: "Assuming"
  • From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
  • Date: Sun, 17 Feb 2008 07:21:25 -0500 (EST)
  • References: <fp0m11$5u7$1@smc.vnet.net> <fp192b$gju$1@smc.vnet.net> <fp2khg$pka$1@smc.vnet.net> <fp3ui0$9jt$1@smc.vnet.net>

dh <dh at metrohm.ch> wrote:
> Hi David,
>
> we all learned not to divide by zero,
> but things are a bit more subtle than this.

Of course.

> You claim that (a^2 - 1)/(a - 1) should simplify to If[a == 1,
> Indeterminate, a + 1] and not to (a+1),

I said that it should simplify to _something equivalent to_
If[a == 1, Indeterminate, a + 1]. Furthermore, when I said "should", I
should have made it clear that I was talking about "in an ideal sense",
rather than practicality. Sorry that I failed to make that clear.

I do not claim that Mathematica should be rewritten so that
Simplify[(a^2 - 1)/(a - 1)] yields anything other than what it currently
does. Perhaps you noticed what I said at the end of my response: "But such
a result [like If[a == 1, Indeterminate, a + 1] here] is normally
considered to be too cumbersome to be practical; the 'misdemeanor'
simplification is considered preferable. All CASs known to me make such
simplifications." Thus, if the developers consider
If[a == 1, Indeterminate, a + 1] to be too cumbersome to be practical, I
accept their judgement, despite the fact that (a + 1) is not equivalent to
(a^2 - 1)/(a - 1).

Also, note that "simplify to something equivalent to
If[a == 1, Indeterminate, a + 1]" leaves open the possibility that
Simplify[(a^2 - 1)/(a - 1)] would yield (a^2 - 1)/(a - 1) unchanged.

> but you do not give a good reason for this.

[P.S. Thanks to Mariano for having already addressed this point.]

I didn't mention a reason; I thought it was obvious. Namely,
(a^2 - 1)/(a - 1) and (a + 1) are not equivalent. In Mathematica, when
a is 1, the former expression is Indeterminate while the latter is 2.

> Obviously (a^2 - 1)/(a - 1) is not defined for a==1,

Well, in Mathematica, it's not defined as anything I'd normally call a
"number", but it is defined to be Indeterminate. [You may consider that
comment to be mere pedantry. However, Indeterminate in Mathematica
corresponds fairly well to NaN in standard floating-point arithmetic. IIRC,
Kahan is quite adamant that NaN is not the same as "undefined".]

> but note that the left and right limit exist and are identical.

Sure, the singularity is removable.

> Now what harm is done if
> we replace the first function by a second one (a+1) that agrees with
> the first one where it is defined, but that is also defined for a==1?

A singularity has <poof!> vanished all of a sudden. Although that required
us to invoke Simplify in this case, there are simpler cases when this can
happen without even asking for any simplifcation, such as

In[7]:= x/x
Out[7]= 1

Whatever potential harm this "misdemeanor" might do is outweighed, in the
minds of most (perhaps, all) CAS developers, by the utility of having a
simpler result.

For all I know, some future version of Mathematica might return 0 and
Sinc[x] for FullSimplify[0^Abs[z]] and FullSimplify[Sin[x]/x], resp.; in
the current version, FullSimplify leaves those arguments unchanged.

> This seems to me a better aproach than your proposal, especially if
> we consider the implications on practical calculations.

As I said above, I am not proposing that Mathematica should be
rewritten so that Simplify[(a^2 - 1)/(a - 1)] yields anything other
than (a + 1), despite the fact that (a^2 - 1)/(a - 1) should not, in
an ideal sense, simplify to (a + 1).

This response should also have addressed a point raised by David Bailey.

David W. Cantrell


> David W.Cantrell wrote:
[snip]
> > Here's a portion of a response of mine which appeared some years
> > ago in this newsgroup. It's relevant to what Markus was asking about.
> >
> > -------------------------------------------------------
> >
> > In a sense, there is something wrong. These simplifications are
> > what David Stoutemyer called "misdemeanors" in his article "Crimes and
> > Misdemeanors in the Computer Algebra Trade", _Notices of the American
> > Mathematical Society_ 38:7 (1991) 778-785.
> >
> > In case anyone doesn't see why simplifying, say, x^0 to 1 is a
> > misdemeanor: In Mathematica, 0^0 is regarded as being Indeterminate.
> > Thus, without knowing that x is nonzero, Mathematica should not bluntly
> > simplify x^0 to 1. Rather, it would seem that, in Mathematica, x^0
> > should simplify to, say, If[x == 0, Indeterminate, 1]. But such a
> > result is normally considered to be too cumbersome to be practical; the
> > "misdemeanor" simplification is considered preferable. All CASs known
> > to me make such simplifications.
> >
> > -------------------------------------------------------


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