Continued Fraction Algorithm Research Articles

A two-dimensional continued fraction algorithm for best approximations with an application in cubic number fields.

Article A two-dimensional continued fraction algorithm for best approximations with an application in cubic number fields. was published on January 1, 1981 in the journal Journal für die reine und angewandte Mathematik (volume 1981, issue 326).

Minimal Solutions of Three-Term Recurrence Relations and Orthogonal Polynomials

We observe that the well-known recurrence relation ${p_{n + 1}}(z) = (z - {a_n}){p_n}(z) - {b_n}{p_{n - 1}}(z)$ for orthogonal polynomials admits a "minimal solution" if z is outside the spectrum of the mass distribution $ds(t)$ with respect to which the polynomials are orthogonal and if the moment problem for this distribution is determined. The minimal solution, indeed is ${f_n}(z) = \smallint {p_n}(t) ds(t)/(z - t)$, and can be computed accurately by means of the author’s continued fraction algorithm. An application is made to special Gauss-type quadrature formulas.

A continued fraction algorithm

Two existing algorithms for the evaluation of a finite sequence of convergents of a continued fraction are considered. Each method has a drawback concerning numerical stability or computational effort. A third algorithm is presented which requires less computations than the first method, and generally is more stable than the second one. The results are illustrated by numerical examples. The connection with Miklosko's algorithm is shown.

A canonical form for planar Farey sets

In a 1975 Acta Mathematica paper [3], Asmus Schmidt developed a new method for generating Gaussian rational approximations to complex irrational numbers. The approximations correspond to decreasing sequences of planar sets, which can be viewed as converging to a number (. These sets, which are called Farey sets, in turn correspond to complex unimodular transformations. Each Farey set has a canonical representation in terms of a composition of elements of a special set of seven unimodular maps. Schmidt's method can be viewed as analogous to the classical continued fraction algorithm for real numbers, with a sequence of the seven special maps corresponding to the usual sequence of partial quotients. Each complex irrational number has at most 2 sequences of maps associated with it. These sequences (called chains) are periodic if ( is quadratic and can be used to find the fundamental solution to the Pellian equation X2 Dy2 ? 1, to mention but two analogies to continued fractions. This paper contains a new proof [2] of the canonical form for Farey sets, on which Schmidt's method depends. The relevant definitions are introduced first, together with four fundamental lemmas. Let G be the set of unimodular transformations, i.e. complex bilinear maps of the form m: z -* (az + b)/(cz + d), where a, b, c, d E Z[i] and det m = ad bc E Qt1 = {-+I1,+i}. Let Gp and G, partition G into maps with det m = ?1 and det m = + i respectively. Let the closed upper half-plane be denoted by C, let * denote the set *= {z: O 0, N(z-2) > }

Minimal solutions of three-term recurrence relations and orthogonal polynomials

We observe that the well-known recurrence relation p n + 1 ( z ) = ( z − a n ) p n ( z ) − b n p n − 1 ( z ) {p_{n + 1}}(z) = (z - {a_n}){p_n}(z) - {b_n}{p_{n - 1}}(z) for orthogonal polynomials admits a "minimal solution" if z is outside the spectrum of the mass distribution d s ( t ) ds(t) with respect to which the polynomials are orthogonal and if the moment problem for this distribution is determined. The minimal solution, indeed is f n ( z ) = ∫ p n ( t ) d s ( t ) / ( z − t ) {f_n}(z) = \smallint {p_n}(t)\,ds(t)/(z - t) , and can be computed accurately by means of the author’s continued fraction algorithm. An application is made to special Gauss-type quadrature formulas.

Excitations in Heisenberg spin glasses

For the Eu x Sr1−x S-system in the spin glass regime (x=0.6 and 0.4) the shape functions of the inelastic neutron scattering cross sections at low temperatures are calculated from a numerical study of the imaginary partχ″(q,E) of the dynamic susceptibility at T=0 K. Within the calculation, the continued fraction algorithms of the preceding paper are applied. Additionally, for low temperatures, the spin-autocorrelation functionφ u (t) is calculated, averaged over all magnetic sitesl. From this quantity, it is also derived, how the (metastable) Edwards-Anderson order parameterq(T) decreases with increasing T, namely ∝(1-const.·T2). Finally, from a numerical study of the inverse participation ratio it is concluded that the spin excitations in the upper half of the energy range covered by the density of states should be localized.

Numerical studies on the excitation spectra of heisenberg spin glasses

The excitations of Heisenberg spin glasses are studied by a continued fraction algorithm. Both the density of states g(E) and the structure function S( q, E) have been calculated (i) for Mattis spin glasses, and (ii) for the system Eu xSr 1−xS. S( q,E) has peaks at E ≠ 0, but g(E) remains finite for E→0. For a certain percentage of spins the thermal expectations value 〈 S l ( T) > does not vanish at sufficiently low temperatures.

Improving the speed of calculating the regulator of certain pure cubic fields

To calculate R, the regulator of a pure cubic field Q ( D ) 3 Q(\sqrt [3]{{D)}} , a complete period of Voronoi’s continued fraction algorithm over Q ( D ) 3 Q(\sqrt [3]{{D)}} is usually generated. In this paper it is shown how, in certain pure cubic fields, R can be determined by generating only about one third of this period. These results were used on a computer to find R and then the class number for all pure cubic fields Q ( p ) 3 Q(\sqrt [3]{{p)}} , where p is a prime, p ≡ − 1 ( mod 3 ) p \equiv - 1\;\pmod 3 , and p > 2 × 10 5 p > 2 \times {10^5} . Graphs illustrating the distribution of such cubic fields with class number one are presented.

Calculation of the Regulator of Q(√D) by use of the Nearest Integer Continued Fraction Algorithm

Calculation of the Regulator of Q(√D) by use of the Nearest Integer Continued Fraction Algorithm

Calculation of the regulator of 𝑄(√𝐷) by use of the nearest integer continued fraction algorithm

A computational method for determining the regulator of a real quadratic field Q ( D ) Q(\sqrt D ) is described. This method makes use of the properties of the nearest integer continued fraction of D \sqrt D and is about 25

Spin wave excitations and the magnetization curve of amorphous ferromagnets

Within the Heisenberg model it is studied, how the magnetic excitations and the magnetization curve of an amorphous ferromagnet depend on the exchange integralJ(r). For the “relaxed” structural model of L. von Heimendahl, consisting ofN=888 sites, the density of spin wave states is calculated by means of a continued fraction algorithm, using various model assumptions on the radial dependence ofJ(r). Although it is assumed thatJ(r) is different from zero only within the “nearest neighbour region” of the radial distributiong(r), the density of spin wave states and the critical temperature are strongly influenced by a positive or negative derivative ofJ(r) in this region. However, the magnetization curves, which are calculated in the Tyablikov decoupling approximation fors=1/2, show only a weak flattening, compared with the idealfcc case, when presented overT/Tc. The calculated low-temperature behaviour of the magnetization is compared with recent experiments on ferromagnetic metglasses, and it is found that a description with localized spins and a short-range Heisenberg interaction should be much better for these systems than for the corresponding crystalline metals.

A Generalization of the Simple Continued Fraction Algorithm

A Generalization of the Simple Continued Fraction Algorithm

A Continued Fraction Algorithm for Approximating All Real Polynomial Roots

It is almost presumptuous to write another paper on approximating the real roots of a polynomial with real coefficients because so much is known, theoretically and practically, about the subject. However, the advent of computers makes it worthwhile to look again at some long forgotten theorems that can now be efficiently implemented on a computer to find all the real roots (and their multiplicities) of a real polynomial. Since a real number can be accommodated explicitly on the computer only by rational approximations, the generality of our method is restricted by the computer to polynomials with rational coefficients. Indeed, all the examples we tested had integral coefficients because a polynomial with rational coefficients is easily made equivalent to a polynomial with integral coefficients. For all practical purposes, therefore, our polynomials have integral coefficients. The theorem that lies at the heart of our algorithm was proved in 1836 by Vincent ([6], [7]) and uses a continued fraction algorithm to separate the roots. Once a root is separated out, we continue with a continued fraction algorithm that approximates the root to any desired accuracy. An interesting feature of the program is that there is no problem with round-off error because not only are the coefficients integers, but integer arithmetic is used and all computations are exact. We programmed in APL because it handles polynomials so efficiently. Our algorithm for finding all real roots first tests for rational roots, using the rational root theorem, and then factors out the linear factors, leaving a polynomial whose real roots are all irrational. The multiplicity of each rational root is also noted. The algorithm then tests the new polynomial for multiple irrational roots using an elegant procedure described in Uspensky [6, p. 65]. Let X,.denote the product of all linear factors corresponding to simple roots; let X2 denote the product of all linear factors corresponding to double roots, etc. If there are no roots of multiplicity k, we set Xk = 1. If, for example, P(x) = (x 1)(x 2)(x 3)2(x 4)2(x 5)3, then P(x) can be written in the form P(x) = X2X2X3, where X, = (x 1)(x 2), X2 = (x 3)(x 4), and XA = (x 5). In general a polynomial P(x) with r multiple roots can be written in the form P(x) = a,,X,X2 ... X'where a,, is a constant and X, = (x b,) ... (x bj) is a polynomial whose simple roots are all the roots of multiplicity i of P. The X, are obtained as follows: Let D, = gcd{P, P'} (P, P'), where P' is the derivative of P. Generate a sequence of polynomials D2= (D, D ), D3 =(D2, D2), etc. An easy calculation shows that

A Continued Fraction Algorithm for Approximating All Real Polynomial Roots

A Continued Fraction Algorithm for Approximating All Real Polynomial Roots

A generalization of the simple continued fraction algorithm

In this paper we present a generalization of the continued fraction algorithm, based on a geometric and matrix-theoretic approach. We first give a geometric representation in the plane R 2 {R^2} , of the simple continued fraction algorithm, described in terms of geometric and arithmetic properties of 2 × 2 2 \times 2 matrices with nonnegative integer entries and determinant 1. The algorithm of this paper is then derived as a natural generalization of the situation in R 2 {R^2} . We describe a computational procedure for our algorithm, and give several examples.

Excitations and critical phenomena of amorphous magnetic systems

Abstract The properties of low-temperature excitations and of the critical phenomena, especially the critical dynamics, are studied within the framework of phenomenological models, namely the quantum mechnaical Heisenberg model, and classical Langevin equations based on a Ginzburg-Landau-Wilson type Hamiltonian. The structure function S(q, ω), representing essentially the differential cross section for inelastic scattering of neurrons as a function of the energy loss hω and momentum tum transfer hq, is calculated together with related quantities: For low temperatures, a continued fraction algorithm is applied to an 888-atoms “relaxed” hard sphere structural model of an amorphous ferromagnet, whereas for temperatures in the vicinity o fthe magnetic phase transition renomalization group methods are used. In both cases, S(q, ω) is strongly influenced by the disorder.

Simple proof of the continued fraction algorithm for decoding Reed-Solomon codes

It was shown recently that BCH and RS codes can be implemented by Berlekamp's algorithm using continued fraction approximations. A simple transparent proof of Berlekamp's algorithm that uses such a development is given in this paper.

Lambert's Problem Revisited

Two classical foms of the time of flight equation for the two body, two point boundary value known as Lambert's problem, are combined to produce an elegant formulation which may serve as the nucleus of an extremely efficient cornputat.- ion algorithm. (i.e., includes elliptic, parabolic and hyperbolic orbits), is a well-behaved function of a single convenient independent variable, and requires the evaluation of a single hypergeometric function. New recursive identities f or hypergeometric functions are developed and efFectively exploited to enhance computation speed. top-down continued fraction algorithm, used for efficient evaluation of the hypergeometric function, which avoids the necessity of repeated scaling. This formulation represents a considerable improve- ment over other methods known to the author, The time equation is universal

Continued fractions and linear recurrences

Let t 0 , t 1 , t 2 , ⋯ {t_0},{t_1},{t_2}, \cdots be a sequence of elements of a field F. We give a continued fraction algorithm for t 0 x + t 1 x 2 + t 2 x 3 + ⋯ {t_0}x + {t_1}{x^2} + {t_2}{x^3} + \cdots . If our sequence satisfies a linear recurrence, then the continued fraction algorithm is finite and produces this recurrence. More generally the algorithm produces a nontrivial solution of the system \[ ∑ j = 0 s t i + j λ j , 0 ⩽ i ⩽ s − 1 , \sum \limits _{j = 0}^s {{t_{i + j}}{\lambda _j},\quad 0 \leqslant i \leqslant s - 1,} \] for every positive integer s.

Matrix interpretations and applications of the continued fraction algorithm

Matrix interpretations and applications of the continued fraction algorithm