Optimal multistep Newton-type methods for finding multiple roots of nonlinear equation with known integer multiplicity

Abstract

<p>Ova disertacija se bavi problemom određivanja višestrukih rešenja realnih nelinearnih jednačina kada je višestrukost unapred poznati prirodan broj. Teorijski se analiziraju i numerički testiraju red konvergencije i optimalnost neki dobro poznatih metoda poput Liu-Čou metoda i Čou-Čen-Song metoda. Izvodi se i objašnjava zavisnost optimalnog reda konvergencije i parnosti/neparnosti višestrukosti rešenja. Takođe, konstruišu se dve nove familije postupaka osmog reda konvergecnije. Razmatraju se nove familije dvokoračnih postupaka namenjene za rešavanje problema koje klasični metodi NJutnovog tipa ne mogu da reše.</p>

<p>This thesis deals with the problem of determing multiple roots of real nonlinear equations where the multiplicity is some integer known in advance. The convergence order and optimal properties of some well-known methods such as Liu-Zhou method and Zhou-Chen-Song method are theoretically analyzed and numerically tested. The dependence of optimal convergence order on multiplicity has been derived and explained. Further, two new efficient families of methods with optimal eighth convergence order have been constructed. Furthermore, some new families of two-step methods are considered to solve certain problems where the classical Newton-type methods fail.</p>