On 06/15/2013 04:37 PM, Rupert Swarbrick wrote:
> Incidentally, plotting eq with
>
> plot2d(lhs(eq)-rhs(eq), [x,0,2*%pi]);
>
> convinced me that there are six roots. A relief, since each of the three
> roots of eq5 is the square of two roots of eq4. Numerically, you can do:
>
> (%i45) zroots: allroots(eq5), numer;
> (%o45) [z = .7071067811865129 %i + .7071067811865259,
> z = - .7071067811865503 %i - .7071067811865429,
> z = .5000000000000374 %i + .8660254037844556]
>
> (the numer flag is to put in a numerical value for the sqrt(3)).
>
> AHAH! If you recognise the decimal expansion of 1/sqrt(2), this looks
> very familiar! Now, there are two things we can do. Firstly, we can
> continue with the numerical calculations. I confess that I didn't spot
> the magic numbers the first time around, so that's what I did
> first. Secondly, you can factor the cubic by eye.
For the record,
to help identify such floating point numbers one can try
http://oldweb.cecm.sfu.ca/projects/ISC/ISCmain.html
Andre