International Journal of Mathematics and Mathematical Sciences
Volume 28 (2001), Issue 9, Pages 545-548

Problems and solutions by the application of Julia set theory to one-dot and multi-dots numerical methods

Anna Tomova

Department of Mathematics, Physics and Informatics, Naval Academy, Varna, Bulgaria

Received 6 February 2001

Copyright © 2001 Anna Tomova. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In 1977 Hubbard developed the ideas of Cayley (1879) and solved in particular the Newton-Fourier imaginary problem. We solve the Newton-Fourier and the Chebyshev-Fourier imaginary problems completely. It is known that the application of Julia set theory is possible to the one-dot numerical method like the Newton's method for computing solution of the nonlinear equations. The secants method is the two-dots numerical method and the application of Julia set theory to it is not demonstrated. Previously we have defined two one-dot combinations: the Newton's-secants and the Chebyshev's-secants methods and have used the escape time algorithm to analyse the application of Julia set theory to these two combinations in some special cases. We consider and solve the Newton's-secants and Tchebicheff's-secants imaginary problems completely.