scholarly journals Structured Matrices, Continued Fractions, and Root Localization of Polynomials

SIAM Review ◽  
2012 ◽  
Vol 54 (3) ◽  
pp. 421-509 ◽  
Author(s):  
Olga Holtz ◽  
Mikhail Tyaglov
Keyword(s):  
Continued Fractions ◽  
Structured Matrices ◽  
Root Localization

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

scholarly journals Extreme Value Theory for Hurwitz Complex Continued Fractions

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 840
Author(s):  
Maxim Sølund Kirsebom
Keyword(s):  
Invariant Measure ◽  
Extreme Value Theory ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Value Theory ◽  
Extreme Value ◽  
Poisson Law ◽  
Complex Continued Fractions

The Hurwitz complex continued fraction is a generalization of the nearest integer continued fraction. In this paper, we prove various results concerning extremes of the modulus of Hurwitz complex continued fraction digits. This includes a Poisson law and an extreme value law. The results are based on cusp estimates of the invariant measure about which information is still limited. In the process, we obtained several results concerning the extremes of nearest integer continued fractions as well.

scholarly journals Global Properties of Eigenvalues of Parametric Rank One Perturbations for Unstructured and Structured Matrices

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
André C. M. Ran ◽  
Michał Wojtylak
Keyword(s):  
Unit Circle ◽  
Structured Matrices ◽  
Additional Structure ◽  
General Properties ◽  
Global Properties ◽  
Classes Of Matrices ◽  
Rank One

AbstractGeneral properties of eigenvalues of $$A+\tau uv^*$$ A + τ u v ∗ as functions of $$\tau \in {\mathbb {C} }$$ τ ∈ C or $$\tau \in {\mathbb {R} }$$ τ ∈ R or $$\tau ={{\,\mathrm{{e}}\,}}^{{{\,\mathrm{{i}}\,}}\theta }$$ τ = e i θ on the unit circle are considered. In particular, the problem of existence of global analytic formulas for eigenvalues is addressed. Furthermore, the limits of eigenvalues with $$\tau \rightarrow \infty $$ τ → ∞ are discussed in detail. The following classes of matrices are considered: complex (without additional structure), real (without additional structure), complex H-selfadjoint and real J-Hamiltonian.

scholarly journals Calculating astronomical refraction by means of continued fractions

1979 ◽  
Vol 89 ◽  
pp. 95-101
Author(s):  
S. Mikkola
Keyword(s):  
Asymptotic Expansion ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Astronomical Refraction

A continued fraction was derived for the summation of the asymptotic expansion of astronomical refraction. Using simple approximations for the last denominator of the fraction, accurate formulae, useful down to the horizon, were obtained. The method is not restricted to any model of the atmosphere and can thus be used in calculations based on actual aerological measurements.

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