Re: [Maxima-discuss] probable bug in sqrt

[Oscar B. <osc...@gm...>]

On Sun, 9 Nov 2025 at 16:44, Igor Pesando <ipe...@gm...> wrote:
>
> Hi*,
>
> I have a linear system of 5 eqs in 7 unknows cA[1]... cA[7]
>
> The system depends also on a parameter d.
>
> I solve the system for 5 unknowns cA[1] .. cA[5].
>
> So the solution depends on cA[6], cA[7] and d.
>
> I substitute back and I do not find zero, one of the reason I cannot
> convince maxima to simplify expressions containing sqrt(7 - 4 *sqrt(3)).
>
> So I substitute some values of d and I still do not get zero.
>
> Since I know the solution of the system for d=26 and I noticed that
> substituting d=26 before solving or after solving yields some sign
> differences

The particular case d=26 is degenerate for this system. If you write
the linear equations as Ax = b with 5x5 matrix A then the determinant
of A is:

72*(45 - 26*sqrt(3))*(d - 26)^2*(d - 19 + 4*sqrt(3))^(3/2)

If either d = 26 or d = 19 - 4*sqrt(3) then the determinant is zero.
Otherwise for generic values of d the determinant is nonzero. I'm not
sure how linsolve handles that case in general but I assume that it
returns a result that gives the unique solution for generic values of
d but is invalid for the degenerate values i.e. not valid for d=26.

You can see a simpler example of this:

(%i8) linsolve([a*x - b], [x]);
(%o8) [x = b/a]

This is valid for a != 0 but invalid in the degenerate case that a = 0.

--
Oscar