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.
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/
@article{MZM_2012_92_5_a11,
author = {L. S. Chkhartishvili},
title = {Solution of an {Algebraic} {Equation} {Using} an {Irrational} {Iteration} {Function}},
journal = {Matemati\v{c}eskie zametki},
pages = {778--785},
year = {2012},
volume = {92},
number = {5},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/MZM_2012_92_5_a11/}
}
TY - JOUR
AU - L. S. Chkhartishvili
TI - Solution of an Algebraic Equation Using an Irrational Iteration Function
JO - Matematičeskie zametki
PY - 2012
SP - 778
EP - 785
VL - 92
IS - 5
UR - http://geodesic.mathdoc.fr/item/MZM_2012_92_5_a11/
LA - ru
ID - MZM_2012_92_5_a11
ER -
%0 Journal Article
%A L. S. Chkhartishvili
%T Solution of an Algebraic Equation Using an Irrational Iteration Function
%J Matematičeskie zametki
%D 2012
%P 778-785
%V 92
%N 5
%U http://geodesic.mathdoc.fr/item/MZM_2012_92_5_a11/
%G ru
%F MZM_2012_92_5_a11
[6] L. S. Chkhartishvili, “Kvaziklassicheskie otsenki postoyannoi reshetki i shiriny zapreschennoi zony kristalla: dvumernyi nitrid bora”, FTT, 46:11 (2004), 2056–2063
[7] L. Chkhartishvili, “Quasi-classical approach: Electronic structure of cubic boron nitride crystals”, J. Solid State Chem., 177:2 (2004), 395–399 | DOI
