Multidimensional Continued Fraction Algorithms are generalizations of the Euclid algorithm and find iteratively the gcd of two or more numbers. They are defined as linear applications on some subcone of R. We consider multidimensional continued fraction algorithms that acts symmetrically on the positive cone R+ for d = 3. We include well-known and old ones (Poincaré, Brun, Selmer, Fully Subtrac...

Many kinds of algorithms of continued fraction expansions of dimension s(≥ 2) have been studied starting with K.G.J.Jacobi(1804-1851), for example, see [14]. For s = 1, we know Lagrange’s theorem related to periodic continued fractions and real quadratic irrationals. But, even for real cubic irrationalities, there appeared no suitable algorithms (of dimension 2). In this section, we roughly exp...

We compute explicitly the density of the invariant measure for the Reverse algorithm which is absolutely continuous with respect to Lebesgue measure, using a method proposed by Arnoux and Nogueira. We also apply the same method on the unsorted version of Brun algorithm and Cassaigne algorithm. We illustrate some experimentations on the domain of the natural extension of those algorithms. For so...

Continued fractions (CFs) play a fundamental role in many investigations due to its applications to diverse fields like number theory, special functions, approximations, moment problems, digital networks, statistics, and signal processing. Their importance has grown further with the advent of fast computing facilities. The problem of converting a continued fraction into a power series is import...