Formulate the Matrix Continued Fractions and Some Applications
Keywords:continued fraction, A matrix continued fraction, Matrix polynomials, Approximation of irrational numbers, Fibonacci sequence.
A matrix continued fraction is a matrix representation of a continued fractions, It has the following formula:
The matrix can be used to convert a continued fraction to a rational number by using matrix multiplication to calculate the matrix product of the continuous fraction matrix and the vector [1, 0]. Additionally, it can be used to calculate the convergent of a continued fraction by using matrix multiplication to calculate the matrix product of the continuous fraction matrix and the vector [1, 1]. It can also be used to represent and calculate the solutions of some type of recursive equations. The use of matrix representation of continued fractions allows for efficient computation of continued fraction expansions using matrix multiplication, which can be easily parallelized in parallel computation algorithms. This can lead to significant speedup in the computation of continued fractions and can be useful in various fields such as computer graphics.
Handley, H.(2023), Continued Fractions: An Arithmetic and Analytic Study,
Cuyt, A.A., et al.(2008), Handbook of continued fractions for special functions: Springer Science & Business Media.
Frommer, A., K. Kahl, and M(2019). Tsolakis, Matrix functions via linear systems built from continued fractions. arXiv preprint arXiv:.03527, 2021.
Giscard, P.-L. and M. Foroozandeh, Exact solutions for the time-evolution of quantum spin systems under arbitrary waveforms using algebraic graph theory. Computer Physics Communications, 2023. 282: p. 108561.
Wall, H.S.(2018), Analytic theory of continued fractions. Courier Dover Publications.
Raissouli, M. and A. Kacha, Convergence of matrix continued fractions. Linear Algebra and its applications, 2000. 320(1-3): p. 115-129.
Ibran, Z.M., E.A. Aljatlawi, and A.M. Awin, On continued fractions and their applications. Journal of Applied Mathematics and Physics, 2022. 10(1): p. 142-159.
Arnoux, P. and S. Labbé, On some symmetric multidimensional continued fraction algorithms. Ergodic Theory and Dynamical Systems, 2018. 38(5): p. 1601-1626.
Jenkinson, O. and M. Pollicott, Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: a hundred decimal digits for the dimension of E2. Advances in Mathematics, 2018. 325: p. 87-115.
Rao, S.S.(2019), Vibration of continuous systems: John Wiley & Sons.
How to Cite
Copyright (c) 2023 Scientific Journal for Faculty of Science-Sirte University
This work is licensed under a Creative Commons Attribution 4.0 International License.