Acceleration of Born Series by Change of Variables Hector Lopez-Menchon, Juan M
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Ries Via the Generalized Weighted-Averages Method
Progress In Electromagnetics Research M, Vol. 14, 233{245, 2010 ACCELERATION OF SLOWLY CONVERGENT SE- RIES VIA THE GENERALIZED WEIGHTED-AVERAGES METHOD A. G. Polimeridis, R. M. Golubovi¶cNi¶ciforovi¶c and J. R. Mosig Laboratory of Electromagnetics and Acoustics Ecole Polytechnique F¶ed¶eralede Lausanne CH-1015 Lausanne, Switzerland Abstract|A generalized version of the weighted-averages method is presented for the acceleration of convergence of sequences and series over a wide range of test problems, including linearly and logarithmically convergent series as well as monotone and alternating series. This method was originally developed in a partition- extrapolation procedure for accelerating the convergence of semi- in¯nite range integrals with Bessel function kernels (Sommerfeld-type integrals), which arise in computational electromagnetics problems involving scattering/radiation in planar strati¯ed media. In this paper, the generalized weighted-averages method is obtained by incorporating the optimal remainder estimates already available in the literature. Numerical results certify its comparable and in many cases superior performance against not only the traditional weighted-averages method but also against the most proven extrapolation methods often used to speed up the computation of slowly convergent series. 1. INTRODUCTION Almost every practical numerical method can be viewed as providing an approximation to the limit of an in¯nite sequence. This sequence is frequently formed by the partial sums of a series, involving a ¯nite number of its elements. Unfortunately, it often happens that the resulting sequence either converges too slowly to be practically useful, or even appears as divergent, hence requesting the use of generalized Received 7 October 2010, Accepted 28 October 2010, Scheduled 8 November 2010 Corresponding author: Athanasios G. -
Convergence Acceleration of Series Through a Variational Approach
Convergence acceleration of series through a variational approach September 30, 2004 Paolo Amore∗, Facultad de Ciencias, Universidad de Colima, Bernal D´ıaz del Castillo 340, Colima, Colima, M´exico Abstract By means of a variational approach we find new series representa- tions both for well known mathematical constants, such as π and the Catalan constant, and for mathematical functions, such as the Rie- mann zeta function. The series that we have found are all exponen- tially convergent and provide quite useful analytical approximations. With limited effort our method can be applied to obtain similar expo- nentially convergent series for a large class of mathematical functions. 1 Introduction This paper deals with the problem of improving the convergence of a series in the case where the series itself is converging very slowly and a large number of terms needs to be evaluated to reach the desired accuracy. This is an interesting and challenging problem, which has been considered by many authors before (see, for example, [FV 1996, Br 1999, Co 2000]). In this paper we wish to attack this problem, although aiming from a different angle: instead of looking for series expansions in some small nat- ural parameter (i.e. a parameter present in the original formula), we in- troduce an artificial dependence in the formulas upon an (not completely) ∗[email protected] 1 arbitrary parameter and then devise an expansion which can be optimized to give faster rates of convergence. The details of how this work will be explained in depth in the next section. This procedure is well known in Physics and it has been exploited in the so-called \Linear Delta Expansion" (LDE) and similar approaches [AFC 1990, Jo 1995, Fe 2000]. -
Rate of Convergence for the Fourier Series of a Function with Gibbs Phenomenon
A Proof, Based on the Euler Sum Acceleration, of the Recovery of an Exponential (Geometric) Rate of Convergence for the Fourier Series of a Function with Gibbs Phenomenon John P. Boyd∗ Department of Atmospheric, Oceanic and Space Science and Laboratory for Scientific Computation, University of Michigan, 2455 Hayward Avenue, Ann Arbor MI 48109 [email protected]; http://www.engin.umich.edu:/∼ jpboyd/ March 26, 2010 Abstract When a function f(x)is singular at a point xs on the real axis, its Fourier series, when truncated at the N-th term, gives a pointwise error of only O(1/N) over the en- tire real axis. Such singularities spontaneously arise as “fronts” in meteorology and oceanography and “shocks” in other branches of fluid mechanics. It has been pre- viously shown that it is possible to recover an exponential rate of convegence at all | − σ |∼ − points away from the singularity in the sense that f(x) fN (x) O(exp( q(x)N)) σ where fN (x) is the result of applying a filter or summability method to the partial sum fN (x) and q(x) is a proportionality constant that is a function of d(x) ≡ |x − xs |, the distance from x to the singularity. Here we give an elementary proof of great generality using conformal mapping in a dummy variable z; this is equiva- lent to applying the Euler acceleration. We show that q(x) ≈ log(cos(d(x)/2)) for the Euler filter when the Fourier period is 2π. More sophisticated filters can in- crease q(x), but the Euler filter is simplest. -
Polylog and Hurwitz Zeta Algorithms Paper
AN EFFICIENT ALGORITHM FOR ACCELERATING THE CONVERGENCE OF OSCILLATORY SERIES, USEFUL FOR COMPUTING THE POLYLOGARITHM AND HURWITZ ZETA FUNCTIONS LINAS VEPŠTAS ABSTRACT. This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein’s “An efficient algorithm for computing the Riemann zeta function”[4, 5], to more general series. The algorithm provides a rapid means of evaluating Lis(z) for¯ general values¯ of complex s and a kidney-shaped region of complex z values given by ¯z2/(z − 1)¯ < 4. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex z-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler- Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evalua- tions of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler-Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor’s series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. -
A Probabilistic Perspective
Acceleration methods for series: A probabilistic perspective Jos´eA. Adell and Alberto Lekuona Abstract. We introduce a probabilistic perspective to the problem of accelerating the convergence of a wide class of series, paying special attention to the computation of the coefficients, preferably in a recur- sive way. This approach is mainly based on a differentiation formula for the negative binomial process which extends the classical Euler's trans- formation. We illustrate the method by providing fast computations of the logarithm and the alternating zeta functions, as well as various real constants expressed as sums of series, such as Catalan, Stieltjes, and Euler-Mascheroni constants. Mathematics Subject Classification (2010). Primary 11Y60, 60E05; Sec- ondary 33B15, 41A35. Keywords. Series acceleration, negative binomial process, alternating zeta function, Catalan constant, Stieltjes constants, Euler-Mascheroni constant. 1. Introduction Suppose that a certain function or real constant is given by a series of the form 1 X f(j)rj; 0 < r < 1: (1.1) j=0 From a computational point of view, it is not only important that the geo- metric rate r is a rational number as small as possible, but also that the coefficients f(j) are easy to compute, since the series in (1.1) can also be written as 1 X (f(2j) + rf(2j + 1)) r2j ; j=0 The authors are partially supported by Research Projects DGA (E-64), MTM2015-67006- P, and by FEDER funds. 2 Adell and Lekuona thus improving the geometric rate from r to r2. Obviously, this procedure can be successively implemented (see formula (2.8) in Section 2). -
Hermite Functions
Appendix A Hermite Functions Abstract Hermite functions play such a central role in equatorial dynamics that it is useful to collect information about them from a variety of sources. Hille-Watson- Boyd convergence and rate-of-convergence theorems, a table of explicit formulas for the Hermite coefficients of elementary functions, Hermite quadrature and inte- gral representations for the Hermite functions and so on are included. Another table lists numerical models employing Hermite functions for oceanographic and meteo- rological applications. Recurrence formulas are provided not only for the normalized Hermmite functions themselves, but also for computing derivatives and products of powers of y with Hermite functions. The Moore-Hutton series acceleration and Euler acceleration for slowly-converging Hermite series are also explained. Just because there’s an exact formula doesn’t mean it’s necessarily a good idea to use it. — Lloyd N. “Nick” Trefethen, FRS A.1 Normalized Hermite Functions: Definitions and Recursion The normalized Hermite functions ψn have the orthogonality property that ∞ ψn(y)ψm (y) dy = δmn (A.1) −∞ where δmn is the usual Kronecker δ, which is one when m = n and zero otherwise. The Hermite functions can be efficiently computed using a three-term recursion relation from two starting values: √ −1/4 2 −1/4 2 ψ0(y) ≡ π exp(−(1/2)y ); ψ1(y) ≡ π 2 y exp(−(1/2)y ) (A.2) 2 n ψ + (y) = y ψ (y) − ψ − (y) (A.3) n 1 n + 1 n n + 1 n 1 © Springer-Verlag GmbH Germany 2018 465 J.P. Boyd, Dynamics of the Equatorial Ocean, https://doi.org/10.1007/978-3-662-55476-0 -
One-Dimensional Semirelativistic Hamiltonian with Multiple Dirac Delta Potentials
PHYSICAL REVIEW D 95, 045004 (2017) One-dimensional semirelativistic Hamiltonian with multiple Dirac delta potentials Fatih Erman* Department of Mathematics, İzmir Institute of Technology, Urla 35430, İzmir, Turkey † Manuel Gadella Departamento de Física Teórica, Atómica y Óptica and IMUVA, Universidad de Valladolid, Campus Miguel Delibes, Paseo Belén 7, 47011 Valladolid, Spain ‡ Haydar Uncu Department of Physics, Adnan Menderes University, 09100 Aydın, Turkey (Received 18 August 2016; published 17 February 2017) In this paper, we consider the one-dimensional semirelativistic Schrödinger equation for a particle interacting with N Dirac delta potentials. Using the heat kernel techniques, we establish a resolvent formula in terms of an N × N matrix, called the principal matrix. This matrix essentially includes all the information about the spectrum of the problem. We study the bound state spectrum by working out the eigenvalues of the principal matrix. With the help of the Feynman-Hellmann theorem, we analyze how the bound state energies change with respect to the parameters in the model. We also prove that there are at most N bound states and explicitly derive the bound state wave function. The bound state problem for the two-center case is particularly investigated. We show that the ground state energy is bounded below, and there exists a self- adjoint Hamiltonian associated with the resolvent formula. Moreover, we prove that the ground state is nondegenerate. The scattering problem for N centers is analyzed by exactly solving the semirelativistic Lippmann-Schwinger equation. The reflection and the transmission coefficients are numerically and asymptotically computed for the two-center case. We observe the so-called threshold anomaly for two symmetrically located centers. -
Rapidly Convergent Summation Formulas Involving Stirling Series
Rapidly Convergent Summation Formulas involving Stirling Series Raphael Schumacher MSc ETH Mathematics Switzerland [email protected] Abstract This paper presents a family of rapidly convergent summation formulas for various finite sums of analytic functions. These summation formulas are obtained by applying a series acceleration transformation involving Stirling numbers of the first kind to the asymptotic, but divergent, expressions for the corresponding sums coming from the Euler-Maclaurin summation formula. While it is well-known that the expressions ob- tained from the Euler-Maclaurin summation formula diverge, our summation formulas are all very rapidly convergent and thus computationally efficient. 1 Introduction In this paper we will use asymptotic series representations for finite sums of various analytic functions, coming from the Euler-Maclaurin summation formula [13, 17], to obtain rapidly convergent series expansions for these finite sums involving Stirling series [1, 2]. The key tool for doing this will be the so called Weniger transformation [1] found by J. Weniger, which transforms an inverse power series into a Stirling series. n For example, two of our three summation formulas for the sum √k, where n N, look k=0 ∈ like P n ∞ k l (2l−3)!! (1) l l 2 3 1 1 3 k l=1( 1) 2 (l+1)! B +1Sk (l) √k = n 2 + √n ζ + √n ( 1) − 3 2 − 4π 2 − P(n + 1)(n + 2) (n + k) arXiv:1602.00336v1 [math.NT] 31 Jan 2016 Xk=0 Xk=1 ··· 2 3 1 1 3 √n √n 53√n = n 2 + √n ζ + + + 3 2 − 4π 2 24(n + 1) 24(n + 1)(n + 2) 640(n + 1)(n + 2)(n + 3) 79√n + + ..., 320(n + 1)(n + 2)(n + 3)(n + 4) 1 or n ∞ k l (2l−1)!! (1) l 2 3 1 1 3 1 k l=1( 1) 2l+1(l+2)! B +2Sk (l) √k = n 2 + √n ζ + + ( 1) − 3 2 − 4π 2 24√n − P√n(n + 1)(n + 2) (n + k) Xk=0 Xk=1 ··· 2 3 1 1 3 1 1 1 = n 2 + √n ζ + 3 2 − 4π 2 24√n − 1920√n(n + 1)(n + 2) − 640√n(n + 1)(n + 2)(n + 3) 259 115 ..., − 46080√n(n + 1)(n + 2)(n + 3)(n + 4) − 4608√n(n + 1)(n + 2)(n + 3)(n + 4)(n + 5) − (1) where the Bl’s are the Bernoulli numbers and Sk (l) denotes the Stirling numbers of the first kind. -
System Convergence in Transport Models: Algorithms Efficiency and Output Uncertainty
View metadata,Downloaded citation and from similar orbit.dtu.dk papers on:at core.ac.uk Dec 21, 2017 brought to you by CORE provided by Online Research Database In Technology System convergence in transport models: algorithms efficiency and output uncertainty Rich, Jeppe; Nielsen, Otto Anker Published in: European Journal of Transport and Infrastructure Research Publication date: 2015 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Rich, J., & Nielsen, O. A. (2015). System convergence in transport models: algorithms efficiency and output uncertainty. European Journal of Transport and Infrastructure Research, 15(3), 317-340. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Issue 15(3), 2015 pp. 317-340 ISSN: 1567-7141 EJTIR tlo.tbm.tudelft.nl/ejtir System convergence in transport models: algorithms efficiency and output uncertainty Jeppe Rich1 Department of Transport, Technical University of Denmark Otto Anker Nielsen2 Department of Transport, Technical University of Denmark Transport models most often involve separate models for traffic assignment and demand. -
Bound States in the Continuum in One-Dimensional Qubit Arrays
DIPARTIMENTO INTERATENEO DI FISICA "M. MERLIN" Dottorato di ricerca in Fisica - Ciclo XXXII Settore Scientifico Disciplinare FIS/02 Bound states in the continuum in one-dimensional qubit arrays Dottorando: Domenico Pomarico Supervisori: Ch.mo Prof. Saverio Pascazio Dott. Francesco V. Pepe Coordinatore: Ch.mo Prof. Giuseppe Iaselli ESAME FINALE 2020 Contents Introduction1 I Friedrichs-Lee model simulators3 1 The Friedrichs-Lee model5 1.1 Unstable vacua............................6 1.2 Minimal coupling........................... 10 1.2.1 Canonical quantization................... 12 1.3 Dipole Hamiltonian and Rotating waves.............. 15 1.4 Waveguide QED in a nutshell.................... 19 1.4.1 Quasi-1D free field and interaction Hamiltonian..... 20 2 The resolvent formalism 23 2.1 Resolvent operator.......................... 23 2.2 Schrödinger equation......................... 25 2.3 Survival amplitude.......................... 26 2.4 Fermi golden rule........................... 28 2.5 Diagrammatics and Self-energy................... 30 2.6 Projection of the resolvent in a subspace.............. 33 3 Single emitter decay 35 3.1 Selection rules............................. 35 3.1.1 Approximations summary................. 37 3.2 Spontaneous photon emission.................... 39 3.2.1 Analytical continuation................... 42 3.2.2 Weisskopf-Wigner approximation............. 44 3.2.3 Plasmonic eigenstates.................... 46 iii iv Contents II Emitters pair: bound states and their dissociation 49 4 Entangled bound states generation 51 4.1 Resonant bound states in the continuum.............. 52 4.1.1 Entanglement by relaxation................. 55 4.1.2 Energy density........................ 57 4.1.3 Atomic population...................... 57 4.2 Time evolution and bound state stability.............. 59 4.3 Off-resonant bound states...................... 64 4.4 Extension to generic dispersion relations.............. 69 5 Correlated photon emission by two excited atoms in a waveguide 71 5.1 The two-excitation sector...................... -
Spectral Theory - Wikipedia, the Free Encyclopedia Spectral Theory from Wikipedia, the Free Encyclopedia
6/2/12 Spectral theory - Wikipedia, the free encyclopedia Spectral theory From Wikipedia, the free encyclopedia In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces.[1] It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations.[2] The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter.[3] Contents 1 Mathematical background 2 Physical background 3 A definition of spectrum 4 What is spectral theory, roughly speaking? 5 Resolution of the identity 6 Resolvent operator 7 Operator equations 8 Notes 9 General references 10 See also 11 External links Mathematical background The name spectral theory was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid, in an infinite-dimensional setting. The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was therefore fortuitous. There have been three main ways to formulate spectral theory, all of which retain their usefulness. After Hilbert's initial formulation, the later development of abstract Hilbert space and the spectral theory of a single normal operator on it did very much go in parallel with the requirements of physics; particularly at the hands of von Neumann.[4] The further theory built on this to include Banach algebras, which can be given abstractly. -
Euler's Constant: Euler's Work and Modern Developments
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 50, Number 4, October 2013, Pages 527–628 S 0273-0979(2013)01423-X Article electronically published on July 19, 2013 EULER’S CONSTANT: EULER’S WORK AND MODERN DEVELOPMENTS JEFFREY C. LAGARIAS Abstract. This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ··· bearing his name, together with some of his related work on the gamma function, values of the zeta function, and diver- gent series. The second part describes various mathematical developments involving Euler’s constant, as well as another constant, the Euler–Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations, and random matrix products. It also includes recent results on Diophantine approximation and transcendence related to Euler’s constant. Contents 1. Introduction 528 2. Euler’s work 530 2.1. Background 531 2.2. Harmonic series and Euler’s constant 531 2.3. The gamma function 537 2.4. Zeta values 540 2.5. Summing divergent series: the Euler–Gompertz constant 550 2.6. Euler–Mascheroni constant; Euler’s approach to research 555 3. Mathematical developments 557 3.1. Euler’s constant and the gamma function 557 3.2. Euler’s constant and the zeta function 561 3.3. Euler’s constant and prime numbers 564 3.4. Euler’s constant and arithmetic functions 565 3.5. Euler’s constant and sieve methods: the Dickman function 568 3.6. Euler’s constant and sieve methods: the Buchstab function 571 3.7.