Računanje realnih ničel polinoma z izrezovanjem

delo diplomskega seminarja

Peter Jereb (Author), Gašper Jaklič (Mentor)

Abstract

V delu bom predstavil algoritem za računanje realnih ničel polinoma, imenovan kubično izrezovanje. Dan polinom $p$ najprej zapišemo v Bernsteinovi bazi in ga aproksimiramo s kubičnim polinomom $q$. Slednjega dobimo z nižanjem stopnje začetnega polinoma. Po Cardanovi formuli izračunamo ničle polinoma $q$, ki bodo oklepale ničle polinoma $p$ in bodo zmanjšale začetni interval. Iteracijo ponavljamo, dokler interval ni krajši od željene natančnosti. Dolžine intervalov z ničlami $p$ konvergirajo z redom 4 za enojne ničle, 2 za dvojne ničle in superlinearno 4/3 za ničle reda 3.

Keywords

matematika;polinomi;iskanje ničel;kubično izrezovanje;Bézierjeve krivulje;

Data

Language:	Slovenian
Year of publishing:	2019
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FGG - Faculty of Civil and Geodetic Engineering
Publisher:	[P. Jereb]
UDC:	519.6
COBISS:	18724185
Views:	1228
Downloads:	189
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Computing real roots of polynomial using cubic clipping
Secondary abstract:	In this work we present an algorithm for computing real zeros of a polynomial called cubic clipping. We write a given polynomial $p$ in Bernstein basis. Then we aproximate $p$ with a cubic polynomial $q$ using degree reduction on $p$. Using Cardano formula, we then compute the roots of $q$ which enclose zeros of $p$ and shorthen the length of the starting interval. Now we iterate this process, until we find zeros within the given accuracy. Lengths of the intervals containing zeros of $p$ have a convergence rate 4 for single roots, 2 for double roots and superlinear 4/3 for cubic roots.
Secondary keywords:	mathematics;polynomials;root finding;cubic clipping;Bézier curves;
Type (COBISS):	Final seminar paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	33 str.
ID:	11227541