One of the most general solvable classes of such problems is "Real Closed Fields", which basically talks about roots of polynomial equations.
http://en.wikipedia.org/wiki/Real_closed_field
I believe that this solvable class has been extended with exponentials (due to an off-hand remark in a video lecture I saw within the past year), but I haven't been able to confirm this.
At 06:18 AM 12/24/2012, Rupert Swarbrick wrote:
>However, programs like Maxima are often given formulas. Is there
>research about numerical root finding for, say, rational expressions or
>maybe ones containing trig functions, logarithms etc? Then your
>functions are analytic and you've got a hope of spotting icky points in
>the denominator etc. etc...
>
>Presumably an ideal computer root finder would be able to say something like:
>
> "I have found <n> roots, which lie in the following (tiny)
> intervals. There are possibly up to <m> other roots that I haven't found."
>
>Have people worked on this?