DOCSLIB.ORG

Resurgence in Differential Equations, and Effective Summation Methods

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Resurgence in Differential Equations, and Effective Summation Methods Introductory Lectures: Resurgence in Differential Equations, and Effective Summation Methods Gerald Dunne University of Connecticut Isaac Newton Institute Spring School: Asymptotic Methods and Applications, March 22-26, 2021 Isaac Newton Institute Programme: Applicable Resurgent Asymptotics, 2021/2022 Basic Introduction to Resurgence: A Beginner’s Guide Lecture 3 Basic Introduction to Resurgence: A Beginner’s Guide 1. Lecture 1: Resurgence & Linear Differential Equations I Trans-series and Stokes Phenomenon in ODEs I Borel Summation basics I Recovering Non-perturbative Connection Formulas 2. Lecture 2: Resurgence & Nonlinear Differential Equations I Nonlinear Stokes Phenomenon I Painlevé Equation examples I Parametric Resurgence & Phase Transitions 3. Lecture 3: Effective Summation Methods I Probing the Borel Plane Numerically I The Physics of Padé Approximation I Optimal Summation and Extrapolation Effective Summation Methods: A Basic Introduction • computing perturbative coefficients is difficult • how do we make practical use of resurgence in a really difficult problem where we can only compute a small number of perturbative coefficients ? • consider a series which appears to be asymptotic, with generic leading large order behavior of the coefficients Γ(a n + b) c ∼ S ; n ! 1 n An • A ! location of the leading Borel singularity • b ! nature of the leading Borel singularity • a ! appropriate expansion variable • the Stokes constant S! normalization Effective Summation Methods: A Basic Introduction PN cn • given just N terms of such a series n xn+1 (i) how can we sum accurately ? (ii) how can we extrapolate accurately to x = 0 ? (iii) how can we extrapolate into the complex x plane to probe Stokes transitions ? (iv) how can we extract non-perturbative exponentially small effects (connection) ? • resurgence suggests that local analysis of perturbation theory encodes global information • How much global information can be decoded from a FINITE number of perturbative coefficients ? • resurgent functions have orderly structure in Borel plane ) develop extrapolation and summation methods that take advantage of this Effective Summation Methods: A Basic Introduction • conclusion: it can make a BIG difference how we sum • processing the same perturbative input data in different ways can lead to vastly different levels of precision • the basic toolkit: ratio tests, series acceleration methods (e.g. Richardson), Padé approximants, orthogonal polynomials, Szegö asymptotics, continued fractions, conformal maps, uniformization maps, ... • the good news: many of these are actually very easy to implement ! a simple set of exploratory procedures • the quality of the extrapolation of an asymptotic series is governed by the quality of the analytic continuation of the Borel transform Z 1 f(x) = dt e−x tB[f](t) 0 • lesson 1: it is better to work in the Borel plane Resurgence in Nonlinear Differential Equations • empirical observation: for “natural problems” the Borel plane often has structure Padé-Borel • simple & powerful summation/extrapolation method 2N Z 1 2N X cn X cn = dt e−xt tn xn+1 n! n=0 0 n=0 Z 1 −xt = dt e B2N [f](t) 0 • recall that the singularities of B2N [f](t) determine the non-perturbative physics • but B2N [f](t) is a polynomial ! • as N ! 1, B2N [f](t) develops singularities • Padé is an excellent “low resolution” detector of singularity structures: “Padé-Borel” method Basics of Padé Approximation • simple and efficient method to analytically continue a series beyond its radius of convergence P2N n • rational approximation to function F2N (t) = n cn t 2N RL(t) X P fF g (t) = = c tn + O t2N+1 [L;M] 2N S (t) n M n • completely algorithmic and algebraic (“built-in”) • near-diagonal Padé: polynomials RN (t) & SN (t) satisfy the same 3-term recursion relation • hence a deep connection to orthogonal polynomials, and their asymptotics (Szegö ...) Basics of Padé Approximation 1. at very high orders Padé can be numerically unstable: a ratio of polynomials with very large coefficients. It is often more stable to convert to an ‘equivalent’ partial fraction 2N PN n N X n n an t X rn cnt $ $ PN n t − t n n dn t n n 2. truncated series ! continued fraction, which often converges in all of C, minus a number of poles/cuts 2N 2N+1 1 2N+1 1+c1 t+···+c2N t +O t = +O t h1 t 1 + h2 t 1+ h t 1+ 3 .. 1 .+h2N t 3. near-diagonal Padé are related to continued fractions Basics of Padé Approximation • a Padé approximant only has pole singularities. But for practical applications we are often interested in branch point singularities, e.g. Borel plane, critical exponents, ... • Padé represents a branch point as the accumulation point of an arc of poles: cf. potential theory (electrostatics) Im[t] Im[t] 10 10 5 5 Re[t] Re[t] -10 -5 5 10 -10 -5 5 10 -5 -5 -10 -10 1 1 1 1 Figure: Padé poles of 1 , 2πi 1 , and of 1 , 2πi 1 . (1−t) 5 (e 5 −t) 7 (1−t2) 5 (e 5 −t2) 7 Basics of Padé Approximation • symmetry of singularities is important for Padé Im[t] 2 1 Re[t] 5 10 15 -1 -2 Figure: Padé poles of 1=(1 − t + t2)1=5 • but the function has no singularities on the real t axis ! • Stahl (1997): in the limit N ! 1 Padé produces the “minimal capacitor” Im[t] 1.0 0.5 0.0 Re[t] 0.2 0.4 0.6 0.8 1.0 -0.5 -1.0 Figure: 1=t of Padé poles of 1=(1 − t + t2)1=5 Basics of Padé Approximation 1 • amazing sensitivity to exponent 2 1 1 inverse Padé poles of & 1 1 −10−6 (1 − t + t2) 2 (1 − t + t2) 2 Im[t] 1.0 0.5 0.0 Re[t] 0.2 0.4 0.6 0.8 1.0 -0.5 -1.0 • this sensitivity can be used to our advantage ... see later Basics of Padé Approximation Exercise 3.1: Explore the Padé pole structures for various functions with interesting singularities. Basics of Padé Approximation • Padé approximants are not very accurate near a branch point and branch cut, because of accumulating poles • Padé obscures repeated repeated singularities along the same line; but this is exactly what happens for resurgent Borel transforms in nonlinear problems • conformal mapping, before Padé, helps resolve these problems • make a conformal map into the unit disk, based on the leading Borel singularity/ies Conformal Mapping • conformal map of cut domains: t ! z Im[t] Im[t] Re[t] Re[t] Im[t] Im[z] Re[t] Re[z] Padé-Conformal-Borel Method • conformally map the Borel plane into the unit disk (based on the leading singularity), and then re-expand and do Padé inside the unit disk ) significant improvement for resurgent Borel transforms • precision can be quantified (Szegö asymptotics) • much better near singularities: higher precision determination of Stokes constants and resolution of the location and nature of the singularity • better extrapolation from x = +1 to x = 0 Basics of Padé Approximation 1 5 • cut of the Airy Borel function 2F1 6 ; 6 ; 1; −t • after the conformal map: Padé-Conformal-Borel: 10-term approximation to (1 + p)−1=3 Padé-Conformal-Borel: 10-term approximation to (1 + p)−1=3 • precision of the Borel transform along the cut Re[B(p)] 4 3 2 1 p -10 -8 -6 -4 -2 -1 -2 -3 blue: exact and Padé-Conformal-Borel approx. (10 terms) red: Padé-Borel approx. (10 terms) black-dashed: Taylor-Conformal-Borel approx. (10 terms) • Padé-Conformal-Borel is generically much more accurate near the singularity and along the cut Padé-Conformal-Borel: 10-term approx. in physical x plane • this translates into much better extrapolation of the Borel transform in the original physical x variable, from an expansion near x = +1 down to x ≈ 0 error 10 0.01 10-5 x 0.1 0.2 0.3 0.4 0.5 purple: 10 term x-Padé approx. (log of fractional error) red: 10 term Padé-Borel approx. blue: 10 term Padé-Conformal-Borel approx. black dotted line: 1% fractional error Padé-Conformal-Borel • the improved precision can be quantified −(Nx )1=2 cn ∼ Γ(n + α) ) frac. error ∼ e x-Padé 2 1=3 frac. error ∼ e−(N x) Padé-Borel 2 1=3 frac. error ∼ e−(N x) Taylor-Conformal-Borel 4 1=5 frac. error ∼ e−(N x) Padé-Conformal-Borel • for a chosen precision (e.g., 1% accuracy), with exactly the same input data we can extrapolate from an N-term expansion + at x = +1 down to xmin ! 0 , scaling with N as: extrapolation xmin scaling truncated series xmin ∼ N −1 x Padé xmin ∼ N −2 Padé-Borel xmin ∼ N −2 Taylor-Conformal-Borel xmin ∼ N −4 Padé-Conformal-Borel xmin ∼ N Padé-Conformal-Borel • another advantage of the conformal mapping is that it resolves repeated resurgent Borel singularities • recall that in a nonlinear problem a Borel singularity is expected to be repeated in (certain) integer multiples • these can be obscured by Padé’s attempt to represent a cut as an accumulation of poles Conformal Mapping of Borel Plane • map the doubly-cut Borel p plane to the unit disc p 2 z z = ! p = 1 + p1 + p2 1 − z2 Borel plane singularities: Padé-Borel transform Im[p] 10 5 Re[p] -0.4 -0.2 0.2 0.4 -5 -10 • before conformal map: resurgence is hidden After Conformal Map: Resurgent Poles in z Plane Im[z] 1.5 1.0 0.5 Re[z] -1.5 -1.0 -0.5 0.5 1.0 1.5 -0.5 -1.0 -1.5 • Conformal map reveals resurgent structure in Borel plane: Borel singularities separated into their resurgent patterns Optimal Method: Uniformizing Map • an even better procedure: replace the conformal map with a uniformizing map • Optimality Theorem (O.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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].
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Fortran Optimizing Compiler 6
    Numerical Libraries for General Interest Seminars 2015 Scientific Computing April 1, 2015 Ge Baolai SHARCNET Faculty of Science Western University Topics . Numerical libraries for scientific computing . Numerical libraries at SHARCNET . Examples – Linear algebra – FFTW – ODE solvers – Optimization – GSL – Multiprecision with MPFUN – PETSc (PDE solvers) . Q&A Overview Numerical Libraries SEMINARS 2015 Numerical Computing . Linear algebra . Nonlinear equations . Optimization . Interpolation/Approximation . Integration and differentiation . Solving ODEs . Solving PDEs . FFT . Random numbers and stochastic simulations . Special functions Copyright © 2001-2015 Western University Seminar Series on Scientific and High Performance Computing, London, Ontario, 2015 Numerical Libraries SEMINARS 2015 More fundamental problems: . Linear algebra . Nonlinear equations . Numerical integration . ODE . FFT . Random numbers . Special functions Copyright © 2001-2015 Western University Seminar Series on Scientific and High Performance Computing, London, Ontario, 2015 Numerical Libraries SEMINARS 2015 Top Ten Algorithms for Science (Jack Dongarra 2000) 1. Metropolis Algorithm for Monte Carlo 2. Simplex Method for Linear Programming 3. Krylov Subspace Iteration Methods 4. The Decompositional Approach to Matrix Computations 5. The Fortran Optimizing Compiler 6. QR Algorithm for Computing Eigenvalues 7. Quicksort Algorithm for Sorting 8. Fast Fourier Transform 9. Integer Relation Detection 10. Fast Multipole Method Copyright © 2001-2015 Western University Seminar
    [Show full text]
  • Accelerating Convergence of Series
    ACCELERATING CONVERGENCE OF SERIES KEITH CONRAD 1. Introduction An infinite series is the limit of its partial sums. However, it may take a large number of terms to get even a few correct digits for the series from its partial sums. For example, 1 X 1 (1.1) n2 n=1 2 converges but the partial sums sN = 1 + 1=4 + 1=9 + ··· + 1=N take a long time to settle down, as the table below illustrates, where sN is truncated to 8 digits after the decimal point. The 1000th partial sum s1000 winds up matching the full series (1.1) only in 1.64. N 10 20 25 50 100 1000 sN 1.54976773 1.59616324 1.60572340 1.62513273 1.63498390 1.64393456 That the partial sums sN converge slowly is related to the error bound from the integral 1 X 1 test: = s + r where n2 N N n=1 1 X 1 Z 1 dx 1 (1.2) r = < = : N n2 x2 N n=N+1 N To approximate (1.1) by sN correctly to 3 digits after the decimal point means rN < :0001 = 1=104, so the bound in (1.2) suggests we make 1=N ≤ 1=104, so N ≥ 10000.1 In the era before electronic computers, computing the 1000th partial sum of (1.1) was not feasible. P Our theme is speeding up convergence of a series S = n≥1 an. This means rewriting P 0 0 P 0 S in a new way, say S = n≥1 an, so that the new tail rN = n>N an goes to 0 faster P than the old tail rN = n>N an.
    [Show full text]
  • Arxiv:Math/0702243V4
    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.
    [Show full text]