The solution of algebraic equations by the method of Rutishauser–Nieporte

Provides analytical expressions representing all the roots of a random algebraic equation of $n$-th degree through the coefficients of the initial equation. These formulas are based on the known ratio of Aitken and consist of two relations infinite Toeplitz determinants, the diagonal elements of which are the coefficients of algebraic equations. When calculating the relations of Toeplitz determinants algorithm is used, Rutishauser. For finding complex roots applies modification of the $r/\varphi$-algorithm developed for the summation of divergent continued fractions.