Geodesic
Solution of an Algebraic Equation Using an Irrational Iteration Function
Matematičeskie zametki, Tome 92 (2012) no. 5, pp. 778-785.

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It is proved that, for the choice $z_{n}^{[0]}=-a_{1}$ of the initial approximation, the sequence of approximations $z_{n}^{[i+1]}=\varphi_{n}(z_{n}^{[i]})$, $[i]=0,1,2,\dots$, of a solution of every canonical algebraic equation with real positive roots which is of the form $$ P_{n}(z)=z^{n}+a_{1}z^{n-1}+a_{2}z^{n-2}+\cdots+a_{n}=0,\qquad n=1,2,\dots, $$ where the sequence is generated by the irrational iteration function $\varphi_{n}(z)=(z^{n}-P_{n}(z))^{1/n}$, 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 in a molecule and in a crystal are presented. The possibility of constructing similar irrational iteration functions in order to solve an algebraic equation of general form is considered.
Keywords: canonical algebraic equation, largest root, irrational iteration, electron system in molecules and crystals, method of divided differences. 
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L. S. Chkhartishvili. Solution of an Algebraic Equation Using an Irrational Iteration Function. Matematičeskie zametki, Tome 92 (2012) no. 5, pp. 778-785. http://geodesic.mathdoc.fr/item/MZM_2012_92_5_a11/

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