# Algebarski pristup iterativnim metodama tangente i sekante

Sačić, Marko (2016) Algebarski pristup iterativnim metodama tangente i sekante. Diploma thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

In this thesis we present an algebraic approach to some well known iterative methods for calculating roots of real functions or in a different formulation, finding fixed points of functions. The use of the secant method is interpreted as a binary operation $\oplus$ on the extended set of real numbers $\overline{\mathbb{R}}$. We consider possible algebraic properties of operation $\oplus$ and try to find some classes of functions for which this operation is associative. Applying Pascal’s famous Hexagrammum Mysticum Theorem we prove the associativity of operation $\oplus$ for a class of rational functions and to obtain an analogue of the initial example of determing the value of the Golden Mean by various iterative methods: a sequence of iterations of function $m(x)=1+ \frac{1}{x}$ and sequences we get by using tangent method and secant method on polynomial $f(x) = x^2-x-1$. Furthermore, Möbius transformations also provide interesting generalization of the initial example where iterations converge to the root of the characteristic polynomial of the Möbius transformation. We list a variety of examples that illustrate how some well known operations such as the group law on elliptic curves, the velocity addition law of special relativity, the addition of electric resistances in parallel and in series and also the standard addition and multiplication of real numbers are special cases of binary operation $\oplus$ with an appropriate choice of initial $f$.