On 12/24/2012 6:18 AM, Rupert Swarbrick wrote:
> Richard Fateman <fateman at eecs.berkeley.edu> writes:
>> oops. here's the link
>> RJF
>>
>> http://www.eecs.berkeley.edu/~wkahan/Math128/SOLVEkey.pdf
> I was thinking about this a bit after reading the linked article. I
> agree with the (clever and short) proof that a "black box" equation
> solver can't possibly always work.
>
> However, programs like Maxima are often given formulas. Is there
> research about numerical root finding for, say, rational expressions
Certainly there is a huge literature on the roots of polynomials ( and
hence the poles of
rational functions).
> 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...
I think Google may find papers on these topics too.
For a sufficiently complicated expression it has been proven that no uniform
algorithm can determine if it is identically zero. (Daniel Richardson).
>
> 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."
Not for all possible expressions, but certainly for some.
>
> Have people worked on this?
Not exactly that problem, but related ones.
RJF
>
> Rupert
>
>
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