[Maxima] bfloat roots of polynomials

[Raymond Toy (RT/EUS)]

Richard Fateman wrote:
> Rat() and ratcoef are so much faster, that I would suggest the user consider
> using rat() prior to calling this function, if the coefficients are in fact
> exact or floats and keepfloat:true is the case.  If they are bigfloats or
> who knows what else  (quad-double? ) that rat might not know about, maybe
> expand is better..
> list_of_poly_coeffs(p,x) := (if(ratp(p) then
> makelist(ratcoef(p,x,k),k,0,hipow(p,x)) else (p : expand(p),
> makelist(coeff(p,x,k),k,0,hipow(p,x)))$ 

With fixes and suggestions from Stavros, Barton, and Richard, I've fixed 
up the root finder, and you can find a copy at 
http://common-lisp.net/~rtoy/jtroot3.mac

polyroots(p,x) finds the roots of the polynomial p in the variable x.

poly_root_error(x, p, r) gives an estimate, if possible, of the 
approximate error in the roots.  r is a list of roots, p is the 
polynomial, and x is the variable.

I've added 5 tests of some special polynomials given in TOMS 419.  These 
work.  I also tested P(7), a 64-th order polynomial, from Adam Majewski. 
  This appears to work, and all roots have magnitude less than 2.  (I 
used fpprec = 64.  32 might work.)

I don't have any plans to really do anything else.  It was a fun 
project.  If someone one wants to add this to maxima in some contrib or 
share package, you have my permission.  The code is public domain.

Ray