On Tue, Sep 8, 2009 at 5:21 PM, Richard Fateman<fateman at cs.berkeley.edu> wrote:
>... Now it is true that 0 is inside [-2,2], so you could maybe bend the
> situation and say that the answer is OK if we change the meaning of
> equality for intervals.
Rich, come on, this is pure sophistry. No one claimed that
lim(a)+lim(b)=lim(a+b). What I said was that
interval-plus(limitset(a), limitset(b)) is a superset of
limitset(a+b). This is as expected for interval arithmetic. After
all, in normal interval arithmetic, f(x,y):=x-y when applied to
z:[0,1] gives [-1,1] and not [0,0].
> This would be really convenient since you could re-implement the limit function to just return [-inf,inf] for every input and that would be OK too.:)
That would not be the only place in Maxima where we have a best-effort
superset result. I'm pretty sure we have cases where the *current*
limit (forget about limit sets) returns say IND where a finer analysis
would allow a constant result.
> ...I don't know if this extended notion of limit has other ramifications, like it breaks differentiation somehow.
Don't know what you mean here. If you mean the definition of
derivatives as limits, I suspect you get some nicely robust results,
though of course the derivative of a real-to-real function is no
longer a real-to-real function but a real-to-(set-of-reals) function.
Haven't worked through the derivations, but it would not surprise me
at all that this works out quite nicely.
-s