The solution of algebraic equations using Rutishauser-Nikiportsa

Presents analytical expressions representing all the roots of a random algebraic equation of the n-th degree through the ratio of the original equation. These formulas consist of two relations infinite Toeplitz determinants, diagonal elements of which are the coefficients of the algebraic equation. Such structures were called continuous fractions of Nikiportsa. For efficient calculation of the values of continued fractions of Nikiportsa used recurrent algorithm of Rutishauser. As an example, consider the solution of the algebraic equation 41-th degree.