Back to Search
Start Over
Newton’s method with fractional derivatives and various iteration processes via visual analysis
- Source :
- Numerical Algorithms. 86:953-1010
- Publication Year :
- 2020
- Publisher :
- Springer Science and Business Media LLC, 2020.
-
Abstract
- The aim of this paper is to visually investigate the dynamics and stability of the process in which the classic derivative is replaced by the fractional Riemann–Liouville or Caputo derivatives in the standard Newton root-finding method. Additionally, instead of the standard Picard iteration, the Mann, Khan, Ishikawa and S iterations are used. This process when applied to polynomials on complex plane produces images showing basins of attractions for polynomial zeros or images representing the number of iterations required to achieve any polynomial root. The images are called polynomiographs. In this paper, we use the colouring according to the number of iterations which reveals the speed of convergence and dynamic properties of processes visualised by polynomiographs. Moreover, to investigate the stability of the methods, we use basins of attraction. To compare numerically the modified root-finding methods among them, we demonstrate their action for polynomialz3− 1 on a complex plane.
- Subjects :
- Polynomial
Applied Mathematics
Numerical analysis
Stability (learning theory)
Fractional derivative
01 natural sciences
Fractional calculus
010101 applied mathematics
symbols.namesake
Newton method
Fixed-point iteration
ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION
0103 physical sciences
Convergence (routing)
symbols
Polynomiography
Applied mathematics
Iterations
0101 mathematics
010301 acoustics
Complex plane
Newton's method
Mathematics
Subjects
Details
- ISSN :
- 15729265 and 10171398
- Volume :
- 86
- Database :
- OpenAIRE
- Journal :
- Numerical Algorithms
- Accession number :
- edsair.doi.dedup.....1b3d957108bf476396499953f3df73c2