Solution of an algebraic equation using an irrational iteration function

Abstract

It is proved that, for the choice z ^[n] _n = −a ₁ of the initial approximation, the sequence of approximations z ^[i+1] _n = φ _n (z ^[i] _n ), [i] = 0, 1, 2, ..., of a solution of every canonical algebraic equation with real positive roots which is of the form

$$P_n (z) = z^n + a_1 z^{n - 1} + a_2 z^{n - 2} + \ldots + a_n = 0, n = 1,2, \ldots ,$$

where the sequence is generated by the irrational iteration function φ _n (z) = (z ⁿ − P _n (z))^1/n, converges to the largest root z _n . Examples of numerical realization of the method for the problem of determining the energy levels of electron systems of a molecule or a crystal are presented. The possibility of constructing similar irrational iteration functions in order to solve an algebraic equation of general form is considered.