Partial sum processes and continued fractions

The obvious way to compute the continued fraction of a real number α > 1 is to compute a very accurate numerical approximation of α, and then to iterate the well-known truncate-and-invert step which computes the next partial quotient a = bαc and the next complete quotient α′ = 1/(α − a). This method is called the basic method. In the course of this process precision is lost, and one has to take precautions to stop before the partial quotients become incorrect. Lehmer [6] gives a safe stopping-criterion, and a trick to reduce the amount of multi-length arithmetic. Schonhage [12] describes an algorithm for computing the greatest common divisor of u and v and the related continued fraction expansion of u/v in O(n log n log log n) steps if both u and v do not exceed 2. A disadvantage of this approach is that if we wish to extend the list of partial quotients computed from an initial approximation of α, we have to compute a more accurate initial approximation of α, compute the new complete quotient using this new approximation and the partial quotients already computed from the old approximation, and then extend the list of partial quotients using that new complete quotient (we notice that Shiu in [13, p.1312] incorrectly states that all the previous calculations have to be repeated, but the partial quotients computed so far don’t have to be recomputed). Bombieri and Van der Poorten [1], and Shiu [13] have recently proposed a remedy for this problem. They give a formula for computing a rational approximation of the next complete convergent from the first n partial quotients. From that complete convergent about n new partial quotients can be computed. So each step gives an approximate doubling of the number of partial quotients. To start the method, a few partial quotients have to be computed with the basic or indirect method. In [1] this method is proposed for algebraic numbers (which are zeros of polynomials) of degree ≥ 3, whereas Shiu also applies it to more general numbers, namely to transcendental numbers that can be defined as the zero of a function for which the logarithmic derivative at a rational point can be computed with arbitrary precision. This includes numbers like π, log π, and log 2. For each of thirteen different numbers, Shiu computes 10000 partial quotients. Their frequency distributions are compared with the one which almost all numbers should obey, according to the Khintchine– Levy theory [3, 7]. No significant deviations from this theory are reported. Shiu calls his method the direct method. Curiously, Shiu does not refer to what we would call the polynomial method for algebraic numbers [2, 5, 11] of degree ≥ 3, which computes the partial quotients of α using only the coefficients of its defining polynomial. Moreover, Shiu gives neither implementational

It is known that many structural schemes can be used to implement digital filters. Some structures of digital filters are characterized by low sensitivity of the characteristics to the accuracy of the representation of the coefficients. A separate group of methods for the structural synthesis of digital filters is based on the expansion of rational fractions, including transfer functions, into continued fractions. Because of this decomposition, ladder structures are obtained. The sensitivity of such ladder structures is analyzed in this paper. As a measure of the structure sensitivity, estimations of the partial derivatives of the transfer function coefficients with respect to the expansion coefficients in continued fraction were used. The analysis showed that the sensitivity of the transfer function coefficients to some expansion coefficients is very small. However, the presented examples show that the sensitivity to certain coefficients of expansion can exceed the sensitivity of the classical direct and the canonic forms.

Let a linear regression be given. For detecting change-points, it is usual to consider the sequence of partial sums of least squares residuals whence a partial sums process is defined. Given a sequence of exact experimental designs, we consider for each design the corresponding partial sums process. If the sequence of designs converges to a continuous design, we derive the explicit form of the limit process of the corresponding sequence of partial sums processes. This is a complicated function of the Brownian motion. These results are useful for the problem of testing for change of regression at known or unknown times.

We consider the process of partial sums of moving averages of finite order with a regular varying memory function, constructed from a stationary sequence, variance of the sum of which is a regularly varying function. We study the Gaussian approximation of this process of partial sums with the aid of a certain class of Gaussian processes, and obtain sufficient conditions for the $C$-convergence in the invariance principle in the Donsker form

The paper deals with the problem of representation of Horn’s hypergeometric functions via continued fractions and branched continued fractions. We construct the formal continued fraction expansions for three ratios of Horn’s hypergeometric functions H7. The method employed is a two-dimensional generalization of the classical method of constructing a Gaussian continued fraction. It is proved that the continued fraction, which is an expansion of each ratio, uniformly converges to a holomorphic function of two variables on every compact subset of some domain of C2, and that this function is an analytic continuation of such a ratio in this domain. To illustrate this, we provide some numerical experiments at the end.

The development of bounds on the complex effective conductivity tensor σ* (that relates the average current to the average electric field in a multicomponent composite) has been hindered by lack of a suitable continued-fraction representation for σ*. Here a new field equation recursion method is developed which gives an expression for σ* as a continued fraction of a novel form incorporating as coefficients the component conductivities and a set of fundamental geometric parameters reflecting the composite geometry. A hierarchy of field equations is set up such that the solutions of the (j+1)th-order equation generate the solutions of thejth-order equation. Consequently the effective tensor Ω(j) associated with thejth-order field equation is expressible as a fractional linear matrix transformation of Ω(j+1). These transformations combine to form the continued fraction expansion for σ*=Ω(0) which is exploited in the following paper, Part II, to obtain bounds: crude bounds on Ω(j), forj≧1, give narrow bounds on σ*. The continued fraction is a generalization to multivariate functions of the continued fraction expansion of single variable Stieltjes functions that proved important in the development of the theory of Páde approximants, asymptotic analysis, and the theory of orthogonal polynomials in the last century. The results extend to other transport problems, including conduction in polycrystalline media, the viscoelasticity of composites, and the response of multicomponent, multiterminal linear electrical networks.

A digital computer program for the design of an all-pass function, with n natural modes, to correct the phase of a prescribed transfer function is discussed. The program is based on the expansion of both the desired phase and that of the all-pass function as a Tchebycheff polynomial series, together with the selection of the all-pass natural modes, so that the first n coefficients in the two series agree. If the over-all delay is unimportant, it may be chosen either to increase the number of matched coefficients by one, or to seek a minimum peak error over the approximating bandwidth. The design process is based on a continued fraction expansion, and a detailed technique for adjusting the over-all delay is discussed. Recurrence relationships are established for expanding the phase of any transfer function as a Tchebycheff polynomial series. The dependence of peak error on approximating bandwidth is discussed, and a routine for the automatic adjustment of the bandwith to meet a specified peak error is described. The design accuracy depends primarily on the root extraction technique, and on the loss of significant figures during continued fraction expansion. Adequate results have been obtained with 8 or 10 natural modes. The speed of the program depends on the number of basic iterations of the design process, which varies from one to ten. On a Ferranti Mercury Computer with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n = 8</tex> , the basic iteration time is 20 seconds.

Hausdorff dimension of certain sets arising in continued fraction expansions

In this paper, the design of half sample delay IIR filter is presented. First, the continued fraction expansion (CFE) and its recursive computation are reviewed briefly. Then, the CFE of square root function is applied to design half sample delay IIR filters with various orders. Next, the comparison with conventional maximally flat IIR allpass filter is made and implementation issue is also addressed. Finally, several design examples are illustrated to demonstrate the effectiveness of the proposed method.

A random walk with counterbalanced steps is a process of partial sums \check S(n)=\check X_{1}+ \cdots + \check X_{n} whose steps \check X_{n} are given recursively as follows. For each n\geq 2 , with a fixed probability p , \check X_{n} is a new independent sample from some fixed law \mu , and with complementary probability 1-p , \check X_{n}= -\check X_{v(n)} counterbalances a previous step, with v(n) a uniform random pick from \{1, \ldots, n-\nobreak 1\} . We determine the asymptotic behavior of \check S(n) in terms of p and the first two moments of \mu . Our approach relies on a coupling with a reinforcement algorithm due to H. A. Simon, and on properties of random recursive trees and Eulerian numbers, which may be of independent interest. The method can be adapted to the situation where the step distribution \mu belongs to the domain of attraction of a stable law.

The paper is related to the classical problem of the rational approximation of analytic functions of one or several variables, particulary the issues that arise in the construction and studying of continued fraction expansions and their multidimensional generalizations—branched continued fraction expansions. We used combinations of three- and four-term recurrence relations of the generalized hypergeometric function 3F2 to construct the branched continued fraction expansions of the ratios of this function. We also used the concept of correspondence and the research method to extend convergence, already known for a small region, to a larger region. As a result, we have established some convergence criteria for the expansions mentioned above. It is proved that the branched continued fraction expansions converges to the functions that are an analytic continuation of the ratios mentioned above in some region. The constructed expansions can approximate the solutions of certain differential equations and analytic functions, which are represented by generalized hypergeometric function 3F2. To illustrate this, we have given a few numerical experiments at the end.

In order to implement the continued fraction algorithm on a highly parallel computer, like the Massively Parallel Processor, it is necessary to be able to compute certain numbers which occur at widely-spaced intervals within the continued fraction expansion of f~, where N is the number to be factored.In this paper several properties of the continued fraction expansion of a quadratic irrational are developed.These results are then applied to the development of a very simple algorithm for finding the widely-spaced numbers referred to above.

The paper considers the problem of approximating Lauricella--Saran's hypergeometric func\-tions $F_K$ in specialcases by bran\-ched continued fractions as a special family offunctions. Under the certain conditions on the elements ofbran\-ched continued fraction expansions of some ratios of thesefunctions, it is proven that every expansion converges to thefunction that is analytic in the domain of analytic continuation\[\mathfrak{D}_\eta=\{\mathbf{z}\in\mathbb{R}^3\colon z_1\le\eta,\;z_2\le\eta,\;z_3\le0\},\quad0&lt;\eta&lt;1,\]at least as fast as a geometric series with a ratio less thenunity. For this purpose, the method based on the formula for thedifference of two approximants of a branched continued fractionand the PF method (based on the so-called property of fork for abranched continued fraction with positive elements) was used.

In order to implement the continued fraction algorithm on a highly parallel computer, like the Massively Parallel Processor, it is necessary to be able to compute certain numbers which occur at widely-spaced intervals within the continued fraction expansion of N \sqrt N . where N is the number to be factored. In this paper several properties of the continued fraction expansion of a quadratic irrational are developed. These results are then applied to the development of a very simple algorithm for finding the widely-spaced numbers referred to above.

Let q, fj, t2, be a sequence of elements of a field F. We give a 2 3 continued fraction algorithm for t^x + txx + t2x + .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 s ',+/*/.0</<*-l, /=0 for every positive integer s. 1.Let t0, tx, t2, be a sequence of elements of a field F. Set T-\ttft./=0 Let d be a nonnegative integer.We say that T* is an approximation of T of degree d if there exist polynomials V and W such that T* = V/W, deg V < d, deg W < d, x\w, and x2d\WT -V.We shall give a continued fraction expansion for xT.This yields polynomials V-, W, and integers d-, 0 = dx < d2 < d3 < , such that (V, W) = 1 and V,/W-is an approximation of T of degree d.Suppose T* is any approximation of T of some degree d.Then T* = V/W for that value of i such that d < d<di+x.If the sequence of the t satisfies a linear recurrence of degree d, but not one of smaller degree, then there is an m such that dm = d and the linear recurrence is given by the polynomial Wm.In this case, WmT = Vm, the continued fraction expansion, terminates at i = m, and we can determine Wm from the first 2d of the t:, i.e., from those t such that 0 </ < 2d.Our algorithm is closely related to Zierler's version of Berlekamp's algorithm [1].2. We consider continued fraction expansions of the form a = A, +-l A, + 2 A3+-where A,, A2, A3, are elements from some field E. We can write