[Maxima] realonly in algsys.lisp

Alexey Beshenov al at beshenov.ru
Sun Apr 26 05:05:35 CDT 2009


The option variable realonly is quite confusing since it does not find
real solutions, but only solutions which are free of %i (purely using
freeof).

For example, when realonly is false,

  algsys ([x^4 + 1], [x]);

returns

  [[x = (-1)^(1/4)],  [x = -(-1)^(1/4)*%i],
   [x = -(-1)^(1/4)], [x = (-1)^(1/4)*%i]]

But when realonly is true, it returns

  [[x = (-1)^(1/4)], [x = -(-1)^(1/4)]]

while it is natural to expect [].


Maybe it is better to modify the behavior and filter roots by
checking something like

  is_real(x) := is(trigsimp(imagpart(x)) = 0);

and not just

  freeof(%i, x)


With the freeof approach we may also omit real roots of irreducible
polynomials:

  sol : map ('rhs, solve (3*x^3  - 3*x + 1));
  [(sqrt(3)*%i/2-1/2)/(3*(3^-(3/2)*%i/2-1/6)^(1/3))
    +(3^-(3/2)*%i/2-1/6)^(1/3)*(-sqrt(3)*%i/2-1/2),
   (3^-(3/2)*%i/2-1/6)^(1/3)*(sqrt(3)*%i/2-1/2)
    +(-sqrt(3)*%i/2-1/2)/(3*(3^-(3/2)*%i/2-1/6)^(1/3)),
   (3^-(3/2)*%i/2-1/6)^(1/3)+1/(3*(3^-(3/2)*%i/2-1/6)^(1/3))]

  map (lambda([x], freeof(%i, x)), sol);
  [false,false,false]

  map ('is_real, sol);
  [true,true,true]

So when realonly and algexact are set to true,

  algsys ([3*x^3  - 3*x + 1], [x])

just returns [].


What do you think?

-- 
Yours sincerely,
Alexey Beshenov <al at cadadr.org>


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