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

V. I. Shmoylov^a, M. V. Hisamutdinov^a, G. A. Kirichenko<sup>b

a</sup> Scientific-Research Institute Multiprocessing Computing Systems after Kalyaev of South Federal University
^b Institute of Computer Technology and Information Security, Engineering and Technological Academy, Southern Federal University

Abstract: 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.

Keywords: algebraic equations, infinite Toeplitz determinants, divergent continuous fractions, $r/\varphi$-algorithm.

UDC: 517.524

Received: 01.04.2014