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1.
Marc Prvost 《Journal of Computational and Applied Mathematics》2008,219(2):484-492
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 Hn-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. 相似文献
2.
Rodney Coleman 《European Journal of Operational Research》1982,9(2):181-183
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. 相似文献
3.
Tom Lyche 《Constructive Approximation》1985,1(1):155-173
The fundamental recurrence relation for polynomialB-splines is generalized to ChebyshevianB-splines. 相似文献
4.
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. 相似文献
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6.
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). 相似文献
7.
The behavior of the sequence xn + 1 = xn(3N − xn2)/2N is studied for N > 0 and varying real x0. When 0 < x0 < (3N)1/2 the sequence converges quadratically to N1/2. When x0 > (5N)1/2 the sequence oscillates infinitely. There is an increasing sequence βr, with β−1 = (3N)1/2 which converges to (5N)1/2 and is such that when βr < x0 < βr + 1 the sequence {xn} converges to (−1)rN1/2. For x0 = 0, β−1, β0,… the sequence converges to 0. For x0 = (5N)1/2 the sequence oscillates: xn = (−1)n(5N)1/2. The behavior for negative x0 is obtained by symmetry. 相似文献
8.
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. 相似文献
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10.
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. 相似文献
11.
Laurence A. Baxter 《European Journal of Operational Research》1983,12(2):205-207
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. 相似文献
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G. Mühlbach 《Numerical Algorithms》1999,22(3-4):317-326
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. 相似文献
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15.
J. W. Stairmand 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1985,36(5):658-668
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.
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.相似文献
16.
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,. . .,φr]⊤ be an r-dimensional o.n. scaling function vector with polynomial preservation of order (p.p.o.) m, and ψ=[ψ1,. . .,ψr]⊤ 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⊤,φr+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 φmD, then φ♯:=[φmD,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. 相似文献
17.
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. 相似文献
18.
19.
Summary.
Large, sparse nonsymmetric systems of linear equations with a
matrix whose eigenvalues lie in the right half plane may be solved by an
iterative method based on Chebyshev polynomials for an interval in the
complex plane. Knowledge of the convex hull of the spectrum of the
matrix is required in order to choose parameters upon which the
iteration depends. Adaptive Chebyshev algorithms, in which these
parameters are determined by using eigenvalue estimates computed by the
power method or modifications thereof, have been described by Manteuffel
[18]. This paper presents an adaptive Chebyshev iterative method, in
which eigenvalue estimates are computed from modified moments determined
during the iterations. The computation of eigenvalue estimates from
modified moments requires less computer storage than when eigenvalue
estimates are computed by a power method and yields faster convergence
for many problems.
Received May 13, 1992/Revised version received May 13,
1993 相似文献
20.
We describe a procedure for determining a few of the largest singular values of a large sparse matrix. The method by Golub and Kent which uses the method of modified moments for estimating the eigenvalues of operators used in iterative methods for the solution of linear systems of equations is appropriately modified in order to generate a sequence of bidiagonal matrices whose singular values approximate those of the original sparse matrix. A simple Lanczos recursion is proposed for determining the corresponding left and right singular vectors. The potential asynchronous computation of the bidiagonal matrices using modified moments with the iterations of an adapted Chebyshev semi-iterative (CSI) method is an attractive feature for parallel computers. Comparisons in efficiency and accuracy with an appropriate Lanczos algorithm (with selective re-orthogonalization) are presented on large sparse (rectangular) matrices arising from applications such as information retrieval and seismic reflection tomography. This procedure is essentially motivated by the theory of moments and Gauss quadrature.This author's work was supported by the National Science Foundation under grants NSF CCR-8717492 and CCR-910000N (NCSA), the U.S. Department of Energy under grant DOE DE-FG02-85ER25001, and the Air Force Office of Scientific Research under grant AFOSR-90-0044 while at the University of Illinois at Urbana-Champaign Center for Supercomputing Research and Development.This author's work was supported by the U.S. Army Research Office under grant DAAL03-90-G-0105, and the National Science Foundation under grant NSF DCR-8412314. 相似文献