<abstract xmlns="http://www.w3.org/1999/xhtml"><p>This paper is a small review of Chebyshev’s method. The geometric interpretation as a generalization of Newton’s method is derived. Using this interpretation its global convergence is proved. Some dynamical properties are studied. As a higher order method, they are more complicated than in Newton’s method. Finally, some applications are revisited pointing out the advantages of Chebyshev’s method with respect Newton’s method.</p></abstract>

This paper is a small review of Chebyshev’s method. The geometric interpretation as a generalization of Newton’s method is derived. Using this interpretation its global convergence is proved. Some dynamical properties are studied. As a higher order method, they are more complicated than in Newton’s method. Finally, some applications are revisited pointing out the advantages of Chebyshev’s method with respect Newton’s method.

After notes on Chebyshev’s iterative method

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"After notes on Chebyshev\u2019s iterative method"}

This paper is a small review of Chebyshev’s method. The geometric interpretation as a generalization of Newton’s method is derived. Using this interpretation...

p	${P_{i}}_{i = 3}^{2 p}$ $\begin{array}{} \displaystyle \{P_{i}\}_{i=3}^{2p} \end{array}$
2	0.375, 0.1406...
3	0.4814..., 0.2222..., 0.0493..., 0.0109...
5	0.56, 0.296, 0.1059..., 0.0355..., 0.0080..., 0.0017..., 2.0736... × 10⁻⁴, 2.4883... × 10⁻⁵
7	0.5918..., 0.3294..., 0.1345..., 0.0512..., 0.0151..., 0.0041..., 0.0008..., 0.0001..., 2.6281... × 10⁻⁵, 3.6251... × 10⁻⁶, 2.9592... × 10⁻⁷, 2.4157... × 10⁻⁸

Iteration	Cheby.
1	8.41e − 03
2	1.78e − 09
3	2.90e − 13
4	0.00e + 00

After notes on Chebyshev’s iterative method

Publicado en línea: 01 ene 2017

Páginas: 1 - 12

Recibido: 11 oct 2016

Aceptado: 01 ene 2017

DOI: https://doi.org/10.21042/AMNS.2017.1.00001

Palabras clave
Nonlinear equations, Chebyshev’s iterative method, geometric interpretation, global convergence, dynamics, applications

© 2017 S. Amat and S. Busquier, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Fig. 2

Sequence {Pi}i=32p$\begin{array}{} \displaystyle \{P.{i}\}.{i=3}^{2p} \end{array}$ for p = 2,3,5,7

Error max-norm, x0 such that ||x* − x0|| = 0.1

After notes on Chebyshev’s iterative method

Publicado en línea: 01 ene 2017

Páginas: 1 - 12

Recibido: 11 oct 2016

Aceptado: 01 ene 2017

DOI: https://doi.org/10.21042/AMNS.2017.1.00001

Palabras claveNonlinear equations, Chebyshev’s iterative method, geometric interpretation, global convergence, dynamics, applications

© 2017 S. Amat and S. Busquier, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Fig. 2

Sequence {Pi}i=32p$\begin{array}{} \displaystyle \{P.{i}\}.{i=3}^{2p} \end{array}$ for p = 2,3,5,7

Error max-norm, x0 such that ||x* − x0|| = 0.1

Palabras clave
Nonlinear equations, Chebyshev’s iterative method, geometric interpretation, global convergence, dynamics, applications