Steffensenova metoda a její zobecnění

Bc. Vendula Švendová

Master's thesis

Steffensen's method and its generalization

Anotácia:

Predlozena prace popisuje Steffensenovu metodu jako variantu Newtonovy metody. Uvodni kapitola nastinuje problematiku reseni soustav nelinearnich rovnic. Druha kapitola predstavuje obe metody v oboru realnych cisel. Steffensenova metoda je navic odvozena z Aitkenovy metody pro urychleni konvergence jinych iteracnich metod. Treti kapitola se zabyva teorii majorizace v Banachove prostoru. Ve ctvrte kapitole je pomoci teto teorie dokazana konvergence tzv. metod "Newtonova typu". Pata kapitola zobecnuje Steffensenovu metodu, jako konkretni pripad metody Newtonova typu. Metoda je implementovana v programu Matlab.

Abstract:

Presented work describes the study of Steffensen's method as a variant of Newton's method. Introductory chapter foreshadows the problem of solving nonlinear equations. The second chapter introduces both methods in the field of real numbers. Additionally, Steffensen's method is derived from Aitken method for the acceleration of convergence of other iterative methods. The third chapter covers the majorization theory in Banach spaces. In the fourth chapter, the convergence of the so-called "Newton's type" method is proven using this theory. The fifth chapter generalizes Steffensen's method as a specific case of the Newton's type method. The method is implemented in Matlab.

Kľúčové slová

Steffensenova metoda, Newtonova metoda, Aitkenova metoda, Banachuv prostor Steffensen's method, Newton's method, Aitken's method, Banach space

Jazyk práce: Czech

Datum vytvoření / odevzdání či podání práce: 18. 1. 2010

Obhajoba závěrečné práce

• Obhajoba proběhla 14. 6. 2010
• Vedúci: prof. RNDr. Ivanka Horová, CSc.
• Oponent: Mgr. Jiří Zelinka, Dr.

Masaryk University

Faculty of Science

Master programme / odbor:
Mathematics / Mathematical Modelling and Numeric Methods