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Two New Iterated Maps for Numerical Nth Root Evaluation
Department of Electrical Engineering, Universidade do Estado de Santa Catarina (UDESC), Joinville, Brazil
Department of Physics, Universidade do Estado de Santa Catarina (UDESC), Joinville, Brazil
Department of Physics, Universidade do Estado de Santa Catarina (UDESC), Joinville, Brazil
- 1 Department of Electrical Engineering, Universidade do Estado de Santa Catarina (UDESC), Joinville, Brazil
- 2 Department of Physics, Universidade do Estado de Santa Catarina (UDESC), Joinville, Brazil
- 3 Department of Physics, Universidade do Estado de Santa Catarina (UDESC), Joinville, Brazil
Applied Mathematics·Volume 05 (2014)·Pages 2974–2981·Published 4 November 2014·DOI10.4236/am.2014.519283
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Abstract
In this paper we propose two original iterated maps to numerically approximate the nth root of a real number. Comparisons between the new maps and the famous Newton-Raphson method are carried out, including fixed point determination, stability analysis and measure of the mean convergence time, which is confirmed by our analytical convergence time model. Stability of solutions is confirmed by measuring the Lyapunov exponent over the parameter space of each map. A generalization of the second map is proposed, giving rise to a family of new maps to address the same problem. This work is developed within the language of discrete dynamical systems.
KeywordsIterated MapNth Root of a Real NumberNumerical MethodNewton-Raphson MethodDynamical System
- Faber, X. and Voloch, J.F. (2011) On the Number of Places of Convergence for Newton’s Method over Number Fields. Journal de Theorie des Nombres de Bordeaux, 23, 387-401.
- Grau-Sánchez, M. and Daz-Barrero, J.L. (2011) A Technique to Composite a Modified Newton’s Method for Solving Nonlinear Equations. ArXiv e-prints.
- Pan, B., Cheng, P. and Xu, B. (2005) In-Plane Displacements Measurement by Gradient-Based Digital Image Correlation. SPIE Proceedings, 5852, 544-551.
- Amin, A.M., Thakur, R., Madren, S., Chuang, H.-S., Thottethodi, M., Vijaykumar, T., Wereley, S.T. and Jacobson, S.C. (2013) Software-Programmable Continuous-Flow Multi-Purpose Lab-on-a-Chip. Microfluidics and Nanofluidics, 15, 647-659. http://dx.doi.org/10.1007/s10404-013-1180-2
- Mungan, C.E. and Lipscombe, T.C. (2012) Babylonian Resistor Networks. European Journal of Physics, 33, 531. http://dx.doi.org/10.1088/0143-0807/33/3/531
- Senthilpari, C., Mohamad, Z.I. and Kavitha, S. (2011) Proposed Low Power, High Speed Adder-Based 65-nm Square Root Circuit. Microelectronics Journal, 42, 445-451. http://dx.doi.org/10.1016/j.mejo.2010.10.015
- Sun, T., Tsuda, S., Zauner, K.-P. and Morgan, H. (2010) On-Chip Electrical Impedance Tomography for Imaging Biological Cells. Biosensors and Bioelectronics, 25, 1109-1115. http://dx.doi.org/10.1016/j.bios.2009.09.036
- Ausloos, M. and Dirickx, M. (2005) The Logistic Map and the Route to Chaos: From the Beginnings to Modern Applications. Springer, New York.
- Eve, J. (1963) Starting Approximations for the Iterative Calculation of Square Roots. The Computer Journal, 6, 274-276. http://dx.doi.org/10.1093/comjnl/6.3.274
- Dellajustina, F.J. and Martins, L.C. (2014) The Hidden Geometry of the Babylonian Square Root Method. Accepted by Applied Mathematics, August.
- Lyapunov, A.M. (1992) The General Problem of the Stability of Motion. International Journal of Control, 55, 531-534. http://dx.doi.org/10.1080/00207179208934253
- Schuster, H.G. and Just, W. (2005) Deterministic Chaos: An Introduction. 4th Edition, John Wiley & Sons, New York. http://dx.doi.org/10.1002/3527604804