User talk:D.Lazard

Ok, let's try to discuss here: https://en.wikipedia.org/wiki/Talk:RSA_(cryptosystem)#Safe_Primes,_in_RSA_Key_Generation No "reliable sources" are needed here. Here you just need to think with your own head. This is math. In any case, I do not owe you anything, and I do not demand anything from you.

From any polynomial of one variable, divide all coefficients by the non null coefficient of highest degree term to reduce the original equation to a monic polynomial. Then, constructing the companion matrix and computing its eigenvalues provide directly all roots of the polynomial equation (real and complex). Hence, it is of great help for finding roots of polynomials of high degrees. Though eigenvalue computation has a significant cost, it is a rather simple algorithm to identify all roots which is not that straight forward with other methods like Newton's method. This method should be added in the paragraph "Computing roots" on page Zero of a function. Wathelem (talk) 09:45, 11 January 2025 (UTC)[reply]

Sure that the method you suggest for finding polynomial roots is theoretically nice. As far as I know, it does not works well in practice. If you have reliable sources that show the opposite, please provide them, and one could add section about that in the article Polynomial root-finding algorithms.

Even if it would exist an efficient root-finding algorithm based on companion matrice, it place would not be in Zero of a function § Computing roots. Note that I rewrote this section for making clear that it is not the place of a specific algorithm. D.Lazard (talk) 11:42, 11 January 2025 (UTC)[reply]

Great, having the link to Polynomial root-finding algorithms enlarges the subject. About companion matrix method, I have a bit of experience with it gained while writing this paper [1], section 3.6. I tested the method a on variety of polynomials up to degree 8. I did not notice any particular instability in the range I used but to be honest, I did not carry any complete review of this aspect. Wathelem (talk) 20:35, 11 January 2025 (UTC)[reply]

I'm sorry that photo was sitting around for so long waiting to be added. I found it by chance, when I read the article out of interest. I must say, battling with the system on Commons for specifying the copyright licence was an experience. Musiconeologist (talk) 18:11, 20 January 2025 (UTC)[reply]