[fricas-devel] Denesting roots

[nob...@nowhere.invalid]

Waldek Hebisch wrote Mon, 14 Aug 2023 11:53:03 -0700:
>
> I have now a prototype denesting code. It is somewhat long for
> an email, so I put it at:
>
> http://fricas.math.uni.wroc.pl/~hebisch/rsimp2.input
>
> So try it first do
>
> )read rsimp2.input
>
> ATM there is one user oriented function called 'rsimp'. Argument
> to 'rsimp' should be a single root, it tries to produce
> denested version of the root, like:
>
> rsimp(sqrt(12 + 2*sqrt(6) + 2*sqrt(14) + sqrt(6)*sqrt(14)))
>
> +-+ +--+ +-+
> (\|6 + 3)\|14 + 7 \|6
> (275) ------------------------
> +--+
> \|42

Derive 6.10 simplifies this example of nested square roots to a much
simpler:

SQRT(12 + 2*SQRT(6) + 2*SQRT(14) + SQRT(6)*SQRT(14))
= SQRT(7) + SQRT(3) + SQRT(2)

Is something wrong with this?

>
> or
>
> rsimp(sqrt(x + sqrt(x^2 - 1)))
>
> +-------+ +------+ +-------+
> | 1 | 2 | 1
> (276) |------- \|x - 1 + (x + 1) |-------
> \|2 x + 2 \|2 x + 2
>
> or
>
> rsimp(sqrt(13^(1/3)+(108)^(1/3)))
>
> 3+--+2 3+-+3+--+ 3+-+2
> 2 \|13 + 2 \|4 \|13 - \|4
> (277) ------------------------------
> 6
>
>
> Due to interpreter limitation argument must be of type Expression(Integer).
> In principle the code should be able to denest roots of degree up
> to 12 when nested roots are all square roots, and roots of degree up
> to 4 when nested roots involve cube roots. But there are shortcuts
> in implementation which may lead to loss of some denstings. ATM
> code does not handle dependent roots, nested roots must be
> independent. And computational complexity is quite high, so
> complicated examples may take very long time or run out of
> memory. Still, I think that even in its current state this is
> quite powerful denester.
>
> --
> Waldek Hebisch
>

This was posted in <fricas-devel> on Google Groups.

Martin.