Ted Kosan wrote:
> Ondrej wrote:
>
>>> Could you please clarify, what exact functionality in solve you expect
>>> in order for 1235 to be solved?
>>>
>>> Should it just run the iterative numerical solver if it cannot find
>>> the solution analytically?
>
>
> And William wrote:
>
>> I don't know. However, Ted, what do you think of the following, i.e.,
>> it is a way in Sage to solve your problem which is probably pretty
>> clean and flexible, and could certainly made a little more student
>> friendly?
>>
>> sage: var('t')
>> sage: a = .004*(8*e^(-(300*t)) - 8*e^(-(1200*t)))*(720000*e^(-(300*t))
>> - 11520000*e^(-(1200*t))) +.004*(9600*e^(-(1200*t)) -
>> 2400*e^(-(300*t)))^2
>> sage: from scipy.optimize import brentq
>> sage: # Given two points x, y such that a(x) and a(y) have different sign,
>> this
>> sage: # brentq uses "inverse quadratic extrapolation" to find a root of a in
>> the
>> sage: # interval [x,y]. It has lots of extra tolerance and other options.
>> sage: brentq(a, 0, 0.002)
>> 0.00041105140493493411
>> sage: show(plot(a,0,.002),xmin=0, xmax=.002)
>>
>> I.e., what we provide an numerical_root method so that
>> a.numerical_root(x,y)
>> would fine a numerical root of a in the interval [x,y], if it exists?
>> It could be built on brentq. The main thing we would have to add
>> is some sort of analysis to find x', y' in the interval so that a(x')
>> has different sign from a(y'), i.e., decide if there is a sign switch,
>> which could be doable for many analytically defined functions at least.
>
> Here is an excerpt from a Mathematica FAQ that I located on the Internet:
> -----
> 3.2 I've properly entered a Solve command but all Mathematica returns
> is an empty list!
> What's going on:
>
> You've asked Mathematica to solve an equation it can't solve
> analytically. So instead
> of a list of solutions, it gives you an empty list. The same thing can
> happen, incidentally, with NSolve.
>
> How to fix the problem: Try using FindRoot to solve the equation.
> First write the equation in the form
> expression = 0.
> (1)
> Then use Plot to graph the expression. Use Mathematica's coordinate
> locator to determine roughly where
> the zeros of the expression are. Feed these to FindRoot as initial guesses.
> -----
>
> It appears that Mathematica uses the same technique that you describe
> using brentq to solve this problem.
>
> Also, this recent discussion on sage-developer seems to be related to
> this issue:
>
> http://www.mail-archive.com/[EMAIL PROTECTED]/msg06571.html
>
>
> For the engineering oriented problems like the two I originally
> submitted, we are usually interested in numeric results. I am now
> thinking that having functions like nsolve() and find_root() in SAGE
> would serve our needs better than enhancing the solve() function.
>
> What is coming to mind is that nsolve() would work like Mathematica's
> NSolve function
> (http://documents.wolfram.com/mathematica/functions/NSolve) and
> find_root() would be a wrapper around brentq.
A sidenote: the current Mathematica 6 documentation (which seems to be
much more comprehensive than the 5.2 documentation noted above) is here:
http://reference.wolfram.com/mathematica/ref/NSolve.html
and here:
http://reference.wolfram.com/mathematica/ref/FindRoot.html
-Jason
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