Tribonacci

[raoulb]

Carry over an old PR from Calcium to Flint

flintlib/calcium#42

[raoulb]

@fredrik-johansson Please consider this series of PRs for merging or let me know what is missing. May be this can even go into flint 3?

[videlec]

Wouldn't it be nicer with a generic k-bonacci x^k - x^(k-1) - x^(k-2) - ... - 1 that specializes to Goldenratio (Fibonacci), Tribonacci and Tetranacci?

[raoulb]

Hmm, I'm not sure I get your point. Which part of the code do you talk about?

Sure, there could be a function which constructs the general polynomial. But this series of PRs implements the constants, which are selected solutions of the corresponding polynomials. (And there even is no closed-form solution for degree 5 up to at least 11, according the the cited reference.)

Maybe we could write (for example) a single qqbar_kbonacci_constant(qqbar_t res, uint k) function, but I don't see the real benefit, because the main part of each individual function is constructing the expression for the explicit solution.

If this is about the example programs, then, yes, we could merge them or even just include a single one, if at all. They are my test cases used for development.

Maybe I miss the obvious point here. In that case, please give more details about what I should change in the code. Thanks.

[videlec]

Hmm, I'm not sure I get your point. Which part of the code do you talk about?

Sure, there could be a function which constructs the general polynomial. But this series of PRs implements the constants, which are selected solutions of the corresponding polynomials. (And there even is no closed-form solution for degree 5 up to at least 11, according the the cited reference.)

Maybe we could write (for example) a single qqbar_kbonacci_constant(qqbar_t res, uint k) function, but I don't see the real benefit, because the main part of each individual function is constructing the expression for the explicit solution.

If this is about the example programs, then, yes, we could merge them or even just include a single one, if at all. They are my test cases used for development.

Maybe I miss the obvious point here. In that case, please give more details about what I should change in the code. Thanks.

By constructing the expression for the explicit solution, you mean expressing the k-bonacci in terms of n-th roots? Isn't the function qqbar_get_fexpr_formula supposed to (partially) do that already? There is a general algorithm for writing such expressions for any algebraic number, see eg in sagemath

sage: x = polygen(QQ, 'x')
sage: tribonacci = QQbar.polynomial_root(x^3 - x^2 - x - 1, RIF(1, 2))
sage: tribonacci.radical_expression()
1/3*(3*sqrt(11)*sqrt(3) + 19)^(1/3) + 4/3/(3*sqrt(11)*sqrt(3) + 19)^(1/3) + 1/3

The above is, as far as I know, not fully implemented in flint but would be to my mind a better addition. While it is valuable to have concrete examples for illustration and testing purposes, I do not understand what would be the advantage of having these symbolic expressions hard coded for each single constant that someone care about.

Note that this is my personal opinion. It does not necessarily reflect the point of view of the of the flint maintainer team.