Rich,
Okay, yes. It will not always work. f(x) = sin(x)^2 is an example, it has
roots but never changes sign. I think the general conclusion that it is
impossible to find an algorithm that always works does not mean you should
give up. You can find some that usually work. For f(x) you can use
Newton's method on some good initial guess. Mix up the point choices with
some random choices too to add some chance (think entropy) at plain good
luck to finding the roots. The Falsi method combined with Newton's method
and random guessing may work most of the time. You can always find the
derivative, so Newton's method could be applied to each f(x). You can also
try squaring some g(x) if the function g(x) has a point of inflection at
g(x) = 0.
I think this is already well researched. I think probably there are really
good methods out there so there is no point in reinventing the wheel. I
would not want to put too much work into coming up with my own recipe, it
would be unlikely to be better than ideas already proposed.
Richard Hennessy
-----Original Message-----
From: Richard Fateman
Sent: Sunday, December 23, 2012 6:52 PM
To: Richard Hennessy
Cc: maxima at math.utexas.edu ; Barton Willis
Subject: Re: [Maxima] strange quad_qags behavior?
oops. here's the link
RJF
http://www.eecs.berkeley.edu/~wkahan/Math128/SOLVEkey.pdf
On 12/23/2012 3:51 PM, Richard Fateman wrote:
> If you haven't seen this explanation of how the HP-34C calculation
> implements "SOLVE", you might enjoy it.
>
> On 12/23/2012 3:42 PM, Richard Hennessy wrote:
>> . My TI-89 calculator can find a and b in most cases. It has a nsolve()
>> command
>