Article,
Solution of an algebraic equation using an irrational iteration function
Mathematical Notes, 92 (5-6): 714-719 (November 2012)
DOI: 10.1134/S0001434612110132
Abstract
It is proved that, for the choice z n n = −a 1 of the initial approximation, the sequence of approximations z n i+1 = φ n (z n i ), i = 0, 1, 2, ..., of a solution of every canonical algebraic equation with real positive roots which is of the form
Pn(z)=zn+a1zn−1+a2zn−2+…+an=0,n=1,2,…,
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.
