Visual Analysis of the Newton's Method with Fractional Order Derivatives

被引：27

作者：

Gdawiec, Krzysztof ^{[1
]}

Kotarski, Wieslaw ^{[1
]}

Lisowska, Agnieszka ^{[1
]}

机构：

[1] Univ Silesia, Inst Comp Sci, Bedzinska 39, PL-41200 Sosnowiec, Poland

来源：

SYMMETRY-BASEL | 2019年 / 11卷 / 09期

关键词：

fractional derivative; Newton method; root-finding; polynomiography; POLYNOMIOGRAPHY;

D O I：

10.3390/sym11091143

中图分类号：

O [数理科学和化学]; P [天文学、地球科学]; Q [生物科学]; N [自然科学总论];

学科分类号：

07 ; 0710 ; 09 ;

摘要：

The aim of this paper is to investigate experimentally and to present visually the dynamics of the processes in which in the standard Newton's root-finding method the classic derivative is replaced by the fractional Riemann-Liouville or Caputo derivatives. These processes applied to polynomials on the complex plane produce images showing basins of attractions for polynomial zeros or images representing the number of iterations required to obtain polynomial roots. These latter images were called by Kalantari as polynomiographs. We use both: the colouring by roots to present basins of attractions, and the colouring by iterations that reveal the speed of convergence and dynamic properties of processes visualised by polynomiographs.

引用

页数：27