Best way to compute powers of complex rational functions

[Stephen Crowley]

Dear flint-devel,

I'm currently working on a project that involves computations with complex rational functions. I've implemented these functions using a pair of `fmpz_poly_q` to represent the real and imaginary parts.

I need to compute the power of these complex rational functions, with both positive and negative integer exponents. I'm considering two approaches:

1. **Direct computation using the binomial theorem:**  I've implemented a function that expands the power using the binomial theorem and performs the necessary arithmetic operations on the `fmpz_poly_q` components.
2. **Matrix representation and `fmpz_poly_mat_q`:**  I could represent the complex rational function as a 2x2 matrix of `fmpz_poly_q` and use the `fmpz_poly_mat_q_pow` function to compute the power.

I'm wondering which approach is generally more efficient and recommended in FLINT. Are there any potential pitfalls or considerations I should be aware of for either method?

Any insights or suggestions would be greatly appreciated.

Thanks in advance,

Stephen

**Regarding which method is better**

* The matrix approach, leveraging `fmpz_poly_mat_q_pow`, is likely to be more efficient, especially for larger exponents. FLINT's matrix functions are generally well-optimized.
* The direct computation approach might be simpler to implement and understand, but could be slower for higher powers due to the nature of the binomial expansion.

[Stephen Crowley]

Well, now we know what a complete and utter piece of worthless junk google gemini is. It was hallucinating, there is no such fmpz_poly_mat_q .

That being said, any thoughts on the most efficient way to accomplish this given a complex rational function as a pair of real and imaginary rational function parts?

[Oscar Benjamin]

If you factor out the denominator then you can use fmpz_poly_mat.

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[Stephen Crowley]

Thank you I may try that if I need to I won't be computing powers more than a hundred I would be suspect

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