I am currently writing a program in TI-BASIC to compute the eigenvalues and associated eigenvectors of a given matrix (I am working with a TI83 Premium CE that doesn't offer such a program). I have found and written an algorithm to compute the coefficients of the characteristic polynomial of the given matrix (using Le Verrier algorithm). So I want to implement a root-finding algorithm to compute the matrix's eigenvalues. I found several methods in this article : https://en.wikipedia.org/wiki/Root-finding_algorithms (especially in the polynomial section). I found here the source code for plysmlt2, the program implemented by default to find root of polynomial but I can't open it because of the format. It was written in assembly code.
Is there a simple (= efficent, not too memory-consuming) root finding method to find all root of a given polynomial ?
I am searching for a method that converges almost certainly (the Newton-method is good but not robust in my case).
Edit: I'm interested in real AND complex roots