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.
Does this seem reasonable?
Ted
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