ON THE HYPERBOLIC FIBONACCI MATRIX FUNCTIONS

Year 2018, Volume: 8 Issue: 2, 454 - 465, 01.12.2018

M. Bahşi S. Solak

Abstract

In this study, we will introduce a new class of hyperbolic matrix functions. By comparing Binet formulas for the Fibonacci and Lucas numbers to the formulas of classical hyperbolic matrix functions, we will dene hyperbolic Fibonacci matrix func- tions and we will deal with some of their properties.

Keywords

Matrix Function, hyperbolic functions, Fibonacci Numbers.

References

• Das, P., (1991), Optical Signal Processing, Springer, New York.
• Defez, E., Sastre, J., Ibanez, J.J. and Peinado, J., (2016), Solving engineering models using hyperbolic matrix functions, Appl. Math. Model., 40 (4), pp. 2837-2844.
• Pozar, D., (1991), Microwave Engineering, Addison-Wesley, New York.
• Shi, Y., Ding, F. and Chen, T., (2006), 2-Norm based recursive design of transmultiplexers with designable filter length, Circuits, Systems and Signal Processing, 25 (4), pp. 447-462.
• Shi, Y. and Fang, H., (2010), Kalman filter based identification for systems with randomly missing measurements in a network environment, International Journal of Control, 83 (3), pp. 538-551.
• Shi, Y. and Yu, B., (2009), Output feedback stabilization of networked control systems with random delays modeled by Markov chains, IEEE Transactions on Automatic Control, 54 (7), pp. 1668-1674.
• Zhang, J.B., Ding, F. and Shi, Y., (2009), Self-tuning control based on multi-innovation stochastic gradient parameter estimation, Systems & Control Letters, 58 (1), pp. 69-75.
• Defez, E., Sastre, J., Ibanez, J.J. and Ruiz, P.A., (2009), Computing matrix functions solving coupled differential models, Math. Comput. Model., 50 (5-6), pp. 831-839.
• Defez, E., Sastre, J., Ibanez, J.J. and Ruiz, P.A., (2013), Computing matrix functions arising in engineering models with orthogonal matrix polynomials, Math. Comput. Model., 57 (7-8), pp. 1738
• Higham, N.J., (2008), Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA.
• Jodar, L., Navarro, E., Posso, A. and Casaban, M., (2003), Constructive solution of strongly coupled continuous hyperbolic mixed problems, Appl. Numer. Math., 47 (3), pp. 477-492.
• Al-Mohy, A.H. and Higham, N.J., (2009), A new scaling and squaring algorithm for the matrix exponential, SIAM J. Matrix Anal. Appl., 31 (3), pp. 970-989.
• Defez, E. and J´odar, L., (1998), Some applications of Hermite matrix polynomials series expansions, J. Comput. Appl. Math., 99, pp. 105-117.
• Hargreaves, G.I. and Higham, N.J., (2005), Efficient algorithms for the matrix cosine and sine, Numer. Algorithms, 40, pp. 383-400.
• Moore, G., (2011), Orthogonal polynomial expansions for the matrix exponential, Linear Algebra Appl., 435 (2), pp. 537-559.
• Sastre, J., Ibanez, J.J., Defez, E. and Ruiz, P.A., (2011), Accurate matrix exponential computation to solve coupled differential models in engineering, Math. Comput. Model., 54, pp. 1835-1840.
• Sastre, J., Ibanez, J.J., Ruiz, P.A. and Defez, E., (2013), Efficient computation of the matrix cosine. Appl. Math. Comput., 219, pp. 7575-7585.
• Koshy, T., (2001), Fibonacci and Lucas Numbers with Applications, A Wiley-Interscience Publication.
• Vajda, S., (1989), Fibonacci & Lucas Numbers, and the Golden Section. Theory and Applications
• Ellis Horwood Limited. Bah¸si, M., (2016), Wilker-type inequalities for hyperbolic Fibonacci functions, Journal of Inequalities and Applications, 2016:146. Ko¸cer, E.G., Tu˘ glu, N. and Stakhov, A., (2008), Hyperbolic Functions with Second Order Recurrence
• Sequences, Ars Combinatoria, 88, pp. 65-81. Falc´on, S. and Plaza, ´A., (2008), The k−Fibonacci hyperbolic functions, Chaos, Solitons & Fractals, , pp. 409-420.
• Stakhov, A.P. and Tkachenko, I.S., (1993), Hyperbolic Fibonacci trigonometry, Rep Ukr. Acad. Sci., , pp. 9-14.
• Stakhov A. and Rozin B., (2005), On a new class of hyperbolic functions, Chaos, Solitons & Fractals, (2), pp. 379-389.
• Stakhov, A. and Rozin, B., (2005), The Golden Shofar, Chaos, Solitons & Fractals, 26 (3), pp. 677-84.
• Stakhov, A. and Rozin, B., (2007), The “golden” hyperbolic models of Universe, Chaos, Solitons & Fractals, 34 (2), pp. 159-171.