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Re: [mg 25674] Bugs in Abs etc.
Posted:
Oct 21, 2000 2:58 PM


Hi Group,
Many thanks to all of you who were kind enough to point out to me my blunder in assuming that numeric functions never evaluate nonnumeric symbolic arguments. On thinking over your comments, I would like to note some further peculiarities that are not necessarily serious, but do raise issues of consistency. I do my computations using ver 4.01 on a PowerMac and on a Sun. The results are the same, thank goodness!
(1) Look in the Help Browser for Abs. The second item down is this:"Abs[z] is left unevaluated if z is not a numeric quantity". Although this is not quite the same thing as saying Abs does not evaluate symbolic arguments, it is suggestive of this particularly because there is not the slightest hint that Abs[x^n]>Abs[x]^n anywhere in the Help Browser or the Mathematica Book references given by the Help Browser.
(2) Consider Sign[x]. When I entered ?Sign I get this "Sign[x] gives 1,0, or 1 depending on whether x is negative, zero, or positive. But Sign accepts complex entries! Sign[I] > I. You find this out when you use the Help Browser. Shouldn't this be part of ?.
(3) As I mention in an earlier post, Abs[(x1)^2]>Abs[x1]^2 but Abs[x^22x+1] returns unevaluated. There are significant problems here relative to pattern matching. This is obvious so I stop here.
Let me make one point very clear: None of these three items are "killers". As Alan Hayes points out, we can use Unevaluated and then Mathematica treats Abs[(x1)^2] and Abs[x^22x+1] the same way so we need not worry if Abs[u] is buried deep in some code we write  provided we remember to include a hold of some sort. But there are implications here.
What other numeric functions shares "evaluates (sometimes) its symbolic arguments" with Abs and Sign? Guess? Can't think of any? What should you do if some builtin, say BuiltIn[x_], does some simplification for symbolic arguments? How do you find out about it? Its not documented. Perhaps BuiltIn[x_a_]>BuiltIn[x]. How would you know to check this rather than say BuiltIn[x^2]>BuiltIn[x]^2? So, the only safe course is to use Unevaluated everywhere! But this is no good because Unevaluated stops evaluation of All arguments to BuiltIn including BuiltIn[11].
I don't recall whether Daniel Lichtblau (sp?) or some other kind responder pointed out that this troublesome feature (at least to me) was introduced in ver. 4. I for one believe Mathematica should, for the reasons I have outlined, go back to the way Abs and Sign were handled in earlier versions. Consistency and therefore predictability are second in importance only to correctness. You can quote me on this :)
Jack



