Legendre modified moments for Euler's constant

Polynomial moments are often used for the computation of Gauss quadrature to stabilize the numerical calculation of the orthogonal polynomials, see [W. Gautschi, Computational aspects of orthogonal polynomials, in: P. Nevai (Ed.), Orthogonal Polynomials-Theory and Practice, NATO ASI Series, Series C: Mathematical and Physical Sciences, vol. 294. Kluwer, Dordrecht, 1990, pp. 181–216 [6]; W. Gautschi, On the sensitivity of orthogonal polynomials to perturbations in the moments, Numer. Math. 48(4) (1986) 369–382 [5]; W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3(3) (1982) 289–317 [4]] or numerical resolution of linear systems [C. Brezinski, Padé-type approximation and general orthogonal polynomials, ISNM, vol. 50, Basel, Boston, Stuttgart, Birkhäuser, 1980 [3]]. These modified moments can also be used to accelerate the convergence of sequences to a real or complex numbers if the error satisfies some properties as done in [C. Brezinski, Accélération de la convergence en analyse numérique, Lecture Notes in Mathematics, vol. 584. Springer, Berlin, New York, 1977; M. Prévost, Padé-type approximants with orthogonal generating polynomials, J. Comput. Appl. Math. 9(4) (1983) 333–346]. In this paper, we use Legendre modified moments to accelerate the convergence of the sequence H_n-log(n+1) to the Euler's constant γ. A formula for the error is given. It is proved that it is a totally monotonic sequence. At last, we give applications to the arithmetic property of γ.     

The moments of forward recurrence time

The moments of the forward recurrence time of an ordinary renewal process are derived in terms of the renewal function and the moments of the common lifetime distribution. The covariance between the forward recurrence time and the number of renewals is also obtained. Asymptotic formulae as the process is allowed to run on for a fixed long time are given.     

A recurrence relation for chebyshevianB-splines

The fundamental recurrence relation for polynomialB-splines is generalized to ChebyshevianB-splines.     

Asymptotics for orthogonal polynomials defined by a recurrence relation

Asymptotic expansions are given for orthogonal polynomials when the coefficients in the three-term recursion formula generating the orthogonal polynomials form sequences of bounded variation.     

An algorithm for Gaussian quadrature given modified moments

Summary An algebraic algorithm, the long quotient- modified difference (LQMD) algorithm, is described for the Gaussian quadrature of the one-dimensional product integral f(x)w(x)dx when the weight function (x) is known through modified momentsv _l =; theP _l (x) are any polynomials of degreel satisfying 3-term recurrence relations with known coefficients. The algorithm serves to establish the co-diagonal matrix, the eigenvalues of which are the Gaussian abscissas. Applied to ordinary moments it requires far fewer divisions than the quotient-difference algorithm; if theP _l (x) are themselves orthogonal with a kernelw ₀ 03F0;, there is no instability due to rounding errors. For smooth kernels (x) it is safe to use secondorder interpolation in determining the eigenvalues by Givens' method. The Christoffel weights can be expressed as ratios of two terms which are most easily calculated in a Sturm sequence beginning with the highest value ofl. A formula for the Christoffel weights applicable for rational versions of theQR algorithm is also derived. Convergence and the propagation of rounding errors are illustrated by several examples, and anAlgol procedure is given.Based in part on a project report presented by A. F. Donovan in partial fulfilment of the requirements for the degree of B. Sc. (Honours), University of Salford (1969).     

A recurrence relation for the square root

The behavior of the sequence x_{n + 1} = x_n(3N − x_n²)/2N is studied for N > 0 and varying real x₀. When 0 < x₀ < (3N)^1/2 the sequence converges quadratically to N^1/2. When x₀ > (5N)^1/2 the sequence oscillates infinitely. There is an increasing sequence β_r, with β₋₁ = (3N)^1/2 which converges to (5N)^1/2 and is such that when β_r < x₀ < β_{r + 1} the sequence {x_n} converges to (−1)^rN^1/2. For x₀ = 0, β₋₁, β₀,… the sequence converges to 0. For x₀ = (5N)^1/2 the sequence oscillates: x_n = (−1)ⁿ(5N)^1/2. The behavior for negative x₀ is obtained by symmetry.     

Generating functions for vector partition functions and a basic recurrence relation

ABSTRACT

We define a generalized vector partition function and derive an identity for the generating series of such functions associated with solutions to basic recurrence relations of combinatorial analysis. As a consequence we obtain the generating function of the number of generalized lattice paths and a new version of the Chaundy-Bullard identity for the vector partition function.     

The evaluation and application of some modified moments

Recurrence formulas for the calculation of the modified moments $$\int\limits_{ - 1}^{ + 1} {(1 - x)^\alpha (1 + x)^\beta T_n (x)dx} $$ and $$\int\limits_{ - 1}^{ + 1} {(1 - x)^\alpha (1 + x)^\beta \ln \left( {\frac{{1 + x}}{2}} \right)T_n (x)dx} $$ are presented. Some applications of these modified moments are discussed, such as the numerical calculation of integrals of functions having branch points, the computation of Chebyshev series coefficients and the construction of Gaussian quadrature formulas for integrals with logarithmic singularity.     

A multidimensional continued fraction based on a high-order recurrence relation

The paper describes and studies an iterative algorithm for finding small values of a set of linear forms over vectors of integers. The algorithm uses a linear recurrence relation to generate a vector sequence, the basic idea being to choose the integral coefficients in the recurrence relation in such a way that the linear forms take small values, subject to the requirement that the integers should not become too large. The problem of choosing good coefficients for the recurrence relation is thus related to the problem of finding a good approximation of a given vector by a vector in a certain one-parameter family of lattices; the novel feature of our approach is that practical formulae for the coefficients are obtained by considering the limit as the parameter tends to zero. The paper discusses two rounding procedures to solve the underlying inhomogeneous Diophantine approximation problem: the first, which we call ``naive rounding' leads to a multidimensional continued fraction algorithm with suboptimal asymptotic convergence properties; in particular, when it is applied to the familiar problem of simultaneous rational approximation, the algorithm reduces to the classical Jacobi-Perron algorithm. The second rounding procedure is Babai's nearest-plane procedure. We compare the two rounding procedures numerically; our experiments suggest that the multidimensional continued fraction corresponding to nearest-plane rounding converges at an optimal asymptotic rate.

     

Abundant periodic and aperiodic solutions of a discontinuous three-term recurrence relation

In this note, we study a discontinuous three-term recurrence relation which arises from seeking the steady states of a cellular neural network with step control function. Several collections of periodic solutions are found. A necessary and sufficient condition for a solution to be periodic is stated and aperiodic solutions are found as consequences. We also show that any periodic solution can be derived from a primary periodic solution with least period not divisible by 5. Although the periodic or aperiodic solutions we found are not exhaustive, they are quite abundant and may reflect some of the rich physical phenomena in true biological systems. Our method in this note may also provide a general approach to analyze the periodicity of solutions of similar recurrence relations.     

The moments of the forward recurrence times of an alternating renewal process

Expressions for the moments of the forward recurrence times of an alternating renewal process are presented. The alternating phase-type renewal process is considered in detail.     

A recurrence relation for generalized divided differences with respect to ECT-systems

It is well known that ordinary divided differences can be computed recursively. This holds true also for generalized divided differences with respect to complete Chebyshev-systems. In this note for extended complete Chebyshev-systems and possibly repeated nodes for the recurrence relation a simple proof is given which also covers the case of complex valued functions. As an application, interpolation by linear combinations of certain complex exponential functions is considered. Moreover, it is shown that generalized divided differences are also continuous functions of their nodes. This revised version was published online in August 2006 with corrections to the Cover Date.     

Construction of orthonormal multi-wavelets with additional vanishing moments

An iterative scheme for constructing compactly supported orthonormal (o.n.) multi-wavelets with vanishing moments of arbitrarily high order is established. Precisely, let φ=[φ₁,. . .,φ_r]^⊤ be an r-dimensional o.n. scaling function vector with polynomial preservation of order (p.p.o.) m, and ψ=[ψ₁,. . .,ψ_r]^⊤ an o.n. multi-wavelet corresponding to φ, with two-scale symbols P and Q, respectively. Then a new (r+1)-dimensional o.n. scaling function vector φ^♯:=[φ^⊤,φ_r+1]^⊤ and some corresponding o.n. multi-wavelet ψ^♯ are constructed in such a way that φ^♯ has p.p.o.=n>m and their two-scale symbols P^♯ and Q^♯ are lower and upper triangular block matrices, respectively, without increasing the size of the supports. For instance, for r=1, if we consider the mth order Daubechies o.n. scaling function φ_m^D, then φ^♯:=[φ_m^D,φ₂]^⊤ is a scaling function vector with p.p.o. >m. As another example, for r=2, if we use the symmetric o.n. scaling function vector φ in our earlier work, then we obtain a new pair of scaling function vector φ^♯=[φ^⊤,φ₃]^⊤ and multi-wavelet ψ^♯ that not only increase the order of vanishing moments but also preserve symmetry. Dedicated to Charles A. Micchelli in Honor of His Sixtieth Birthday Mathematics subject classifications (2000) 42C15, 42C40. Charles K. Chui: Supported in part by NSF grants CCR-9988289 and CCR-0098331 and Army Research Office under grant DAAD 19-00-1-0512. Jian-ao Lian: Supported in part by Army Research Office under grant DAAD 19-01-1-0739.     

A modified moments analysis of dispersion in steady and oscillatory flows

A modification of Aris's method of moments solution to the heat/mass dispersion equation has been developed. In addition to taking axial moments of concentration, the present approach also defines mean cross-sectionally weighted values of the concentration. This leads to a set of ordinary differential equations rather than the partial differential equations generated by Aris's method. Approximate solutions have been developed for the cases of steady and oscillatory flows in a flat channel and values of the dispersion coefficient have been in good agreement with exact analytical predictions calculated by Dewey and Sullivan and by Watson.

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Zusammenfassung Eine Modifaktion der Momenten-Methode von Aris für die Lösung der Dispersionsgleichung für Wärme/Masse ist entwickelt worden. Die neue Methode benützt nicht nur axiale Momente der Konzentration, sondern definiert zusätzlich im Querschnitt gewogene Durchschnittswerte der Konzentration. Dieses führt zu einem Satz gewöhnlicher Differentialgleichungen, an Stelle der partiellen Differentialgleichung nach der Methode von Aris. Angenäherte Lösungen sind entwickelt worden für die Fälle von stationären und oszillatorischen Strömungen in einem ebenen Kanal. Ergebnisse für Dispersions-Koeffizienten stimmen gut überein mit den exakten analytischen Werten, die von Dewey und Sullivan sowohl wie von Watson berechnet worden sind.

     

Approximating dominant singular triplets of large sparse matrices via modified moments

A procedure for determining a few of the largest singular values and corresponding singular vectors of large sparse matrices is presented. Equivalent eigensystems are solved using a technique originally proposed by Golub and Kent based on the computation of modified moments. The asynchronicity in the computations of moments and eigenvalues makes this method attractive for parallel implementations on a network of workstations. Although no obvious relationship between modified moments and the corresponding eigenvectors is known to exist, a scheme to approximate both eigenvalues and eigenvectors (and subsequently singular values and singular vectors) has been produced. This scheme exploits both modified moments in conjunction with the Chebyshev semi-iterative method and deflation techniques to produce approximate eigenpairs of the equivalent sparse eigensystems. The performance of an ANSI-C implementation of this scheme on a network of UNIX workstations and a 256-processor Cray T3D is presented.This research was supported in part by the National Science Foundation under grant numbers NSF-ASC-92-03004 and NSF-ASC-94-11394.