Izpeljava metod za iskanje ničel polinomov z uporabo optimizacije : delo diplomskega seminarja

Abstract
V tem diplomskem seminarju obravnavamo naslednje metode za iskanje ničel polinomov: Newtonovo metodo, metodo Ostrovskega in Laguerrovo metodo. Izpeljali jih bomo s pomočjo vezanega optimizacijskega problema, potem pa na podoben način izpeljali še dve naprednejši metodi: izboljšano Newtonovo metodo in diskretno Laguerrovo metodo. Dokazali bomo nekaj izrekov, ki nam povedo, za koliko lahko povečan korak posamezne metode preseže najmanjšo ničlo. Metode bomo še numerično testirali in jih med seboj primerjali na problemu iskanja najmanjše lastne vrednosti simetrične tridiagonalne matrike. Primerjali bomo število potrebnih korakov za dovolj dober približek, njihovo časovno zahtevnost in red konvergence.

Language: Slovenian Newtonova metoda, metoda Ostrovskega, Laguerrova metoda, izboljšana Newtonova metoda, diskretna Laguerrova metoda Final seminar paper (mb14) FMF - Faculty of Mathematics and Physics 2019 519.6 18711641 72 23 (0 votes) Voting is allowed only to logged in users. AddThis uses cookies that require your consent. Edit consent...

## Secondary language

Language: English Derivation of polynomial zerofinders via optimization problems In this diploma seminar we study the following polynomial zerofinders: Newton's method, Ostrowski's method and Laguerre's method. We will derive them via a constrained optimization problem and will also derive some new methods using the same approach: the improved Newton's method and the discrete Laguerre's method. We will prove some theorems that give us a bound on how far a magnified step for a given method can overshoot the smallest zero of a polynomial. We will test the methods numerically and compare them to one another. We will do that for the problem of finding the smallest eigenvalue of a symmetric tridiagonal matrix. We will compare the number of steps we need for a good approximation of the smallest eigenvalue, the time complexity of the methods and rate of convergence. Newton's method, Ostrowski's method, Laguerre's method, improved Newton's method, discrete Laguerre's method