An algorithm for shifted continued fraction expansions in parallel linear time

Polynomials with Odd Orthogonal Multiplicity

This paper presents an efficient algorithm for encoding and decoding Shortened Reed-Solomon (SRS) codes based on evaluation. The encoding and syndrome computations are performed using polynomial evaluation and the decoding is performed using a recursive extension of the original algorithm. The proposed decoding algorithm initially calculates the error locator polynomial by using the Berlekamp Massey (BM) algorithm and with the recursive computation of discrepancies, it decodes directly the message polynomial. Compared to the traditional decoding algorithm using BM algorithm, Chien search and Forney formula, the proposed algorithm results in area savings for hardware implementations due to hardware re-use and the reduced number of stages. Also, the savings in memory for the software implementation makes it suitable for memory constrained applications such as Wireless Sensor Networks (WSN) applications. For the hardware implementations, the number of clock cycles required are lower for the proposed scheme, which results in time savings. Also, software implementation results of RS (32, 24) for resource constrained WSN application on ATMEL Atmega 128 microcontroller show that the proposed encoder and decoder is only about 52% and 72% of the memory footprint of a traditional encoder and decoder respectively.

The Berlekamp—Massey algorithm (further, the BMA) is interpreted as an algorithm for constructing Pade approximations to the Laurent series over an arbitrary field with singularity at infinity. It is shown that the BMA is an iterative procedure for constructing the sequence of polynomials orthogonal to the corresponding space of polynomials with respect to the inner product determined by the given series. The BMA is used to expand the exponential in continued fractions and calculate its Pade approximations.

We study the distribution of s-dimensional points of digital explicit inversive pseudorandom numbers with arbitrary lags. We prove a discrepancy bound and derive results on the pseudorandomness of the binary threshold sequence derived from digital explicit inversive pseudorandom numbers in terms of bounds on the correlation measure of order k and the linear complexity profile. The proofs are based on bounds on exponential sums and earlier relations of Mauduit, Niederreiter and Sárközy between discrepancy and correlation measure of order k and of Brandstätter and the third author between correlation measure of order k and linear complexity profile, respectively.

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

We essentially improve lower bounds on the linear complexity of a sequence representing the residues of the discrete logarithm in a finite field modulo a divisor of the order of the multiplicative group. More generally, we present the result as a bound on the linear complexity profile. The proof is based on character sum bounds.

Linear complexity profile of m-ary pseudorandom sequences with small correlation measure

Following the idea of Xing et al., we investigate a general method for constructing families of pseudorandom sequences with low correlation and large linear complexity from elliptic curves over finite fields in this correspondence. With the help of the tool of exponential sums on elliptic curves, we study their periods, linear complexities, linear complexity profiles, distributions of r-patterns, periodic correlation, partial period distributions, and aperiodic correlation in detail. The results show that they have nice randomness.

The Berlekamp-Massey algorithm [L. S. De Jong, Numerical aspects of realization algorithms in linearsystems theory, Ph.D. thesis, T. H. Eindhoven, the Netherlands, 1975] and [J. L. Massey, Shift-register synthesis and BCH decoding, IEEE Trans. Information Theory, IT-15 (1969), pp. 122–127] for minimal realization problems is a special case of the Padé approximation problem. As a matter of fact, it computes among other polynomials the denominators of the elements of the Padé table that are on the descending diagonal $\{ [0/1],[1/2], \cdots ,[k/k + 1], \cdots \} $ as far as they exist and this algorithm works for nonnormal Padé tables too. This algorithm does not seem to be well-known in Padé approximation literature. It is not very difficult to generalize this algorithm so as to compute the other Padé approximants of a nonnormal table. Some variants will lead to a generalization of the algorithm of Brezinski [Computation of Padé approximants continued fractions, J. Comput. Appl. Math., 2 (1976), pp. 113–123], which computes the descending diagonals of a normal Padé table, and of the algorithm of Watson [D. Bussonnais, “Tous” les algorithmes de calcul par recurrence des approximants de Padé d’une serie, Construction de fractions continues correspondantes, Séminaire d’Analyse Numérique, No. 293, Grenoble,1978], [G. Claessens, new look at the Padé table and the different methods for computing its elements, J. Comput. Appl. Math., 1 (1975), pp. 141–152], [P. J. S. Watson, Algorithms for differentiation and integration, Padé Approximants and their Applications, P. R. Groves-Morris, ed., Academic Press London, 1973, pp. 93–98] that computes the descending staircases of normal tables. This work is the dual of Cordellier’s [algorithmes de calcul recursif des elements d’une table de Padé non normale, Conference on Padé approximation, Lille, France, 1978], and of McEliece and Shearer [A property of Euclid’s algorithm and an application to Padé approximation, SIAM J. Appl. Math., 34 (1978), pp. 611–616] deriving similar results for ascending diagonals and staircases. A more continued fraction-like approach would generalize the Thatcher algorithm and may be found in [J. A. Murphy and H. R. O’Donohoe, A class of algorithms for obtaining rational approximants to functions which are defined by power series, Z. Angew. Math. Phys., 28 (1977), pp. 1121–1131] and [A. Bultheel, Division algorithms for continued fractions and the Padé table, Applied Mathematics and Programming Division, Rep. TW41, Katholicke Universitiet Leuven, Heverlee, Belgium, August1978].

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.

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.

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.

Another viewpoint is presented on the derivation of the Berlekamp-Massey algorithm. Our approach differs from previous ones in the following manner. The properties of the shortest linear feedback shift register that generates a given sequence are first derived Without reference to the Berlekamp-Massey algorithm. The Berlekamp-Massey algorithm is then derived using these properties. Our approach has the advantage of being easier to understand.