# Formulate the Matrix Continued Fractions and Some Applications

## DOI:

https://doi.org/10.37375/sjfssu.v3i1.1100## Keywords:

continued fraction, A matrix continued fraction, Matrix polynomials, Approximation of irrational numbers, Fibonacci sequence.## Abstract

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.

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*Scientific Journal for Faculty of Science-Sirte University*,

*3*(1), 35–46. https://doi.org/10.37375/sjfssu.v3i1.1100

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