Hadamard and Vandermonde determinants and Bernoulli-Euler-Lagrange-Aitken-Nikiporets type numerical method for roots of polynomials

In the article we develop Euler-Lagrange method and calculate all the roots of an arbitrary complex polynomial $P(z)$ on the base of calculation (similar to the Bernoulli-Aitken-Nikiporets methods) of the limits of ratios of Hadamard determinants built by means of coefficients of expansions into Taylor and Laurent series of the function~$\frac{P^{\mathrm{\prime }}(z)}{P(z)}$.