Abstract
A first aim of the present work is the determination of the actual sources of the “finite precision error” generation and accumulation in two important algorithms: Bernoulli’s map and the folded Baker’s map. These two computational schemes attract the attention of a growing number of researchers, in connection with a wide range of applications. However, both Bernoulli’s and Baker’s maps, when implemented in a contemporary computing machine, suffer from a very serious numerical error due to the finite word length. This error, causally, causes a failure of these two algorithms after a relatively very small number of iterations. In the present manuscript, novel methods for eliminating this numerical error are presented. In fact, the introduced approach succeeds in executing the Bernoulli’s map and the folded Baker’s map in a computing machine for many hundreds of thousands of iterations, offering results practically free of finite precision error. These successful techniques are based on the determination and understanding of the substantial sources of finite precision (round-off) error, which is generated and accumulated in these two important chaotic maps.
Highlights
In recent years, a quite extensive research in connection with Bernoulli’s and Baker’s maps applications takes place
For example, this chaotic map is employed for image watermarking [2]; in this publication, the authors perform a statistical analysis of a watermarking system based on Bernoulli chaotic sequences
It has been established that the Bernoulli’s and Baker’s maps offer totally erroneous results, after an impressively small number of recursions; for example, when IEEE standard double precision is employed, both chaotic maps fail after few tens of iterations, frequently less than sixty
Summary
A quite extensive research in connection with Bernoulli’s and Baker’s maps applications takes place. The exact sources of finite precision error of this chaotic map are introduced for the first time It is, again, demonstrated that these sources make the results offered by Baker’s algorithm completely unreliable, after a relatively very limited number of recursions; this renders the classical execution of the Baker’s map, too, entirely inapplicable. Consider any computing machine, which uses a precision practically equivalent to n decimal digits in the mantissa; for an arbitrary floating-point number represented in this machine, it holds that:. Due to the previous finite word length computations, both quantities pn , qn have been evaluated with exactly λc correct decimal digits or equivalently with λ = n − λc erroneous c c d. Both multiplication and division may generate a large number of e. d. d., when they are repeatedly applied
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
