DOCSLIB.ORG

Simplifying Integration for Logarithmic Singularities R.N.L. Smith

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Simplifying Integration for Logarithmic Singularities R.N.L. Smith Transactions on Modelling and Simulation vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-355X Simplifying integration for logarithmic singularities R.N.L. Smith Department of Applied Mathematics & OR, Cranfield University, RMCS, Shrivenham, Swindon, Wiltshire SN6 SLA, UK Introduction Any implementation of the boundary element method requires consideration of three types of integral - simple non-singular integrals, singular integrals of two kinds and those integrals which are 'nearly singular' (Brebbia and Dominguez 1989). If linear elements are used all integrals may be per- formed exactly (Jaswon and Symm 1977), but the additional complexity of quadratic or cubic element programs appears to give a significant gain in efficiency, and such elements are therefore often utilised; These elements may be curved and can represent complex shapes and changes in boundary variables more effectively. Unfortunately, more complex elements require more complex integration routines than the routines for straight elements. In dealing with BEM codes simplicity is clearly desirable, particularly when attempting to show others the value of boundary methods. Non-singular integrals are almost invariably dealt with by a simple Gauss- Legendre quadrature and problems with near-singular integrals may be overcome by using the same simple integration scheme given a sufficient number of integration points and some restriction on the minimum distance between a given element and the nearest off-element node. The strongly singular integrals which appear in the usual BEM direct method may be evaluated using the row sum technique which may not be the most efficient available but has the virtue of being easy to explain and implement. This Transactions on Modelling and Simulation vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-355X 234 Boundary Elements paper implements a recently developed one part Gauss integration scheme for singular logarithmic integrals which is both simpler to implement and more accurate than the usual two-part Gauss approach with separate Gauss- Legendre and logarithmically weighted Gauss schemes. We aim to present a description for the quadratic element which is self-contained and suitable as an introduction to this aspect of BEM. Quadratic elements Consider an isoparametric quadratic element defined on [0,1] as in figure 1. 92 93 # , # Figure 1: A straight quadratic element The variation of x, y, potential u arid potential derivative q — du/dn are all expressed in the same form in terms of their nodal values, for example: 4- 92^2 + %Ms (1) where the Lagrangian interpolation functions MI, A/2,A/3 are M, = (2f -!)(£-!) Mi = 4f(l - f) -1) In one dimension, a typical boundary element integral of q over an element Si on the boundary has the form (omitting a — l/(2ir) multiplier) / = / ln(r) q dS (2) J S, where dS is the element of arc length around the boundary S and r = i(x - xo)2 + (y - 3/0)2 is the distance between the source point (xo,yo) and the boundary point (,r,y) on the current element. Transactions on Modelling and Simulation vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 235 If g, r and dS are now all expressed in terms of the parameter £ using an isoparametric quadratic element then the integral (2) becomes (3) Jo where J(^) is the Jacobian transforming cK to If the point (xo,yo) coincides with the first node of the quadratic element, then r — 0 at this point and the integral including MI above is weakly singular. The other two shape functions MI and M^ are zero at f = 0 and those integrals are not therefore singular for this position of the source node - although they will be singular for the source node at node 2 and node 3 respectively. We discuss here only the case for a singularity on node 1, since integrals singular on the other two nodes are simple transforma- tions/combinations of the node 1 routine. Numerical integration Very simple elements allow the use of exact integration but for more com- plex elements, numerical integration is used. In general, an n-point Gauss- Legendre formula with abscissae x, and weights W{ (%)(& = ]T %,,/(%,) (4) i=i will give a result for the integral which is exact for all polynomials of degree 2n — 1 (neglecting any rounding errors). The high efficiency of such inte- gration schemes mean that they are almost invariably used for numerical integration in FEM and BEM. One weighted Gauss scheme which is particularly useful in the BEM is for integrals which include a log(£) term on the interval [0,1]. With abscissae Xi and weights W{ this scheme is written In order to incorporate this weighted Gauss scheme into the integral (3) we must isolate the £ part of ?••(£). One way of doing this (Smith and Mason Transactions on Modelling and Simulation vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-355X 236 Boundary Elements 1982) is as follows: When x is expressed as an isoparametric quadratic on [0,1] we can write z = A^ 4- Bf + zi; % = Cf 4- Df + 2/1 where C = 2%/i - 4?/2 + 2?/3, D = -3m + 4?/2 - 7/3 and if the element is singular on node 1 then r% = ^[(A" + C')(" + (2AB 4- The ln(r) integral singular on node 1 can thus be written as Jo This kind of separation by direct factorisation into the two components avoids division by small quantities or taking square roots and would thus seem a more robust approach than is sometimes used. The first integral on the right-hand side of (6) is evaluated using the weighted scheme (5) and the second non-singular integral using the simple quadrature (4). This two-part gauss quadrature with n points for each part is, of course, exact for polynomials of degree (2n — 1) and therefore for integrals on straight quadratic elements when n > 1. An alternative scheme for the singular logarithmic integral is based on sub- tracting out the singularity after expanding the integrand using Taylor se- ries (Aliabadi et al 1985) Whilst this approach is effective, it is substantially more difficult to explain or implement than a two-part Gauss quadrature. Crow (1993) developed a gauss quadrature scheme for a function of the form: which is exact where a and b are polynomials of degree at most N. Choosing a — 0 and 6=0 separately leads to the equations: TV 0 N i = / Jo 0 Transactions on Modelling and Simulation vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 237 The choice of basis polynomials for this problem is arbitrary since the un- known abscissae are not the roots of orthogonal polynomials but Legendre polynomials P*(x] on [0,1] produce lower condition numbers than powers of x in the following scheme: E o giving 2N+2 equations for the 2N+2 unknowns: Given an initial estimate for £, the first equation above is solved for w and the second equation for £ in an iterative fashion. The computation of quadrature points in this way seems to be particularly ill-conditioned and data is only available for N up to N = 7. The quadrature points for the case N — 6 are given below for convenience: The above quadrature scheme X .0030258021 .040978254 .170863300 w .011351339 .075241070 .18879004 X .41325571 .70909515 .93823959 w .28582072 .28448643 .15431040 Table 1: Gauss quadrature points covers the requirements for the BEM ln(r) singularity without the need to explicitly form the two components of the integrand in the singular case. Results The two-part log procedure and the single gauss quadrature are now com- pared for several elements with different profiles. Since exact solutions are not generally available, accurate comparison values were obtained from the adaptive numerical integration provided by the Maple symbolic algebra package. The other results were obtained using single precision arithmetic (c± 7dp) on a DEC MICRO VAX and include a straightforward 6-point Gauss-Legendre Transactions on Modelling and Simulation vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-355X 238 Boundary Elements rule which takes no account of the singularity. Both the singular rules also use 6 points, the two-part rule with 3 log -f 3 Legendre points. The integrals from all 3 nodes are treated in the same way although strictly only the first node integral is singular. nodes 1,2,3 x-coord 0.100 0.200 0.300 nodes 1,2,3 y-coord 0.100 0.200 0.300 6-pt Gauss 0.1887260 0.3954318 0.0516317 Two part log 0.1930966 0.3952629 0.0516753 Crow 6-pt 0.1930967 0.3952628 0.0516752 Maple 0.1930966 0.3952628 0.0516752 Table 2: log integrals on a straight element Table 2 gives integrals for a linear geometry quadratic element which can be evaluated exactly since the Jacobian is a constant. The singular nu- merical schemes should also be 'exact' to within numerical errors since the unaccounted for polynomial in the integrand is just the quadratic shape function of degree n = 2 and these schemes should therefore be exact with (2n — 1) > 2. This is borne out by table 2 which shows all 3 methods apart from the simple Gauss rule agreeing to machine precision. Observe that although the integrals from node 2 and node 3 are non-singular, the sin- gular rules appear to give significantly better estimates than simple Gauss- Legendre. Once the element deviates from a pure linear element, exact nodes 1,2,3 x-coord 0.100 0.200 0.300 nodes 1,2,3 y-coord 0.100 0.220 0.300 6-pt Gauss 0.2129289 0.3870630 0.0377493 Two part log 0.2182627 0.3868637 0.0377983 Crow 6-pt 0.2182668 0.3868548 0.0378030 Maple 0.2182667 0.3868547 0.0378031 Table 3: log integrals - small curvature integration is no longer generally possible and tables 3 and 4 show that the Crow rule performs significantly better than the other schemes; accu- rate to 5 or 6 places of decimals compared to 3 or 4 otherwise.
Load more

Recommended publications
  • Differential Calculus and by Era Integral Calculus, Which Are Related by in Early Cultures in Classical Antiquity the Fundamental Theorem of Calculus
    History of calculus - Wikipedia, the free encyclopedia 1/1/10 5:02 PM History of calculus From Wikipedia, the free encyclopedia History of science This is a sub-article to Calculus and History of mathematics. History of Calculus is part of the history of mathematics focused on limits, functions, derivatives, integrals, and infinite series. The subject, known Background historically as infinitesimal calculus, Theories/sociology constitutes a major part of modern Historiography mathematics education. It has two major Pseudoscience branches, differential calculus and By era integral calculus, which are related by In early cultures in Classical Antiquity the fundamental theorem of calculus. In the Middle Ages Calculus is the study of change, in the In the Renaissance same way that geometry is the study of Scientific Revolution shape and algebra is the study of By topic operations and their application to Natural sciences solving equations. A course in calculus Astronomy is a gateway to other, more advanced Biology courses in mathematics devoted to the Botany study of functions and limits, broadly Chemistry Ecology called mathematical analysis. Calculus Geography has widespread applications in science, Geology economics, and engineering and can Paleontology solve many problems for which algebra Physics alone is insufficient. Mathematics Algebra Calculus Combinatorics Contents Geometry Logic Statistics 1 Development of calculus Trigonometry 1.1 Integral calculus Social sciences 1.2 Differential calculus Anthropology 1.3 Mathematical analysis
    [Show full text]
  • Squaring the Circle a Case Study in the History of Mathematics the Problem
    Squaring the Circle A Case Study in the History of Mathematics The Problem Using only a compass and straightedge, construct for any given circle, a square with the same area as the circle. The general problem of constructing a square with the same area as a given figure is known as the Quadrature of that figure. So, we seek a quadrature of the circle. The Answer It has been known since 1822 that the quadrature of a circle with straightedge and compass is impossible. Notes: First of all we are not saying that a square of equal area does not exist. If the circle has area A, then a square with side √A clearly has the same area. Secondly, we are not saying that a quadrature of a circle is impossible, since it is possible, but not under the restriction of using only a straightedge and compass. Precursors It has been written, in many places, that the quadrature problem appears in one of the earliest extant mathematical sources, the Rhind Papyrus (~ 1650 B.C.). This is not really an accurate statement. If one means by the “quadrature of the circle” simply a quadrature by any means, then one is just asking for the determination of the area of a circle. This problem does appear in the Rhind Papyrus, but I consider it as just a precursor to the construction problem we are examining. The Rhind Papyrus The papyrus was found in Thebes (Luxor) in the ruins of a small building near the Ramesseum.1 It was purchased in 1858 in Egypt by the Scottish Egyptologist A.
    [Show full text]
  • Numerical Solution of Ordinary Differential Equations
    NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS Kendall Atkinson, Weimin Han, David Stewart University of Iowa Iowa City, Iowa A JOHN WILEY & SONS, INC., PUBLICATION Copyright c 2009 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created ore extended by sales representatives or written sales materials. The advice and strategies contained herin may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
    [Show full text]
  • Numerical Integration
    Chapter 12 Numerical Integration Numerical differentiation methods compute approximations to the derivative of a function from known values of the function. Numerical integration uses the same information to compute numerical approximations to the integral of the function. An important use of both types of methods is estimation of derivatives and integrals for functions that are only known at isolated points, as is the case with for example measurement data. An important difference between differen- tiation and integration is that for most functions it is not possible to determine the integral via symbolic methods, but we can still compute numerical approx- imations to virtually any definite integral. Numerical integration methods are therefore more useful than numerical differentiation methods, and are essential in many practical situations. We use the same general strategy for deriving numerical integration meth- ods as we did for numerical differentiation methods: We find the polynomial that interpolates the function at some suitable points, and use the integral of the polynomial as an approximation to the function. This means that the truncation error can be analysed in basically the same way as for numerical differentiation. However, when it comes to round-off error, integration behaves differently from differentiation: Numerical integration is very insensitive to round-off errors, so we will ignore round-off in our analysis. The mathematical definition of the integral is basically via a numerical in- tegration method, and we therefore start by reviewing this definition. We then derive the simplest numerical integration method, and see how its error can be analysed. We then derive two other methods that are more accurate, but for these we just indicate how the error analysis can be done.
    [Show full text]
  • On the Numerical Solution of Equations Involving Differential Operators with Constant Coefficients 1
    ON THE NUMERICAL SOLUTION OF EQUATIONS 219 The author acknowledges with thanks the aid of Dolores Ufford, who assisted in the calculations. Yudell L. Luke Midwest Research Institute Kansas City 2, Missouri 1 W. E. Milne, "The remainder in linear methods of approximation," NBS, Jn. of Research, v. 43, 1949, p. 501-511. 2W. E. Milne, Numerical Calculus, p. 108-116. 3 M. Bates, On the Development of Some New Formulas for Numerical Integration. Stanford University, June, 1929. 4 M. E. Youngberg, Formulas for Mechanical Quadrature of Irrational Functions. Oregon State College, June, 1937. (The author is indebted to the referee for references 3 and 4.) 6 E. L. Kaplan, "Numerical integration near a singularity," Jn. Math. Phys., v. 26, April, 1952, p. 1-28. On the Numerical Solution of Equations Involving Differential Operators with Constant Coefficients 1. The General Linear Differential Operator. Consider the differential equation of order n (1) Ly + Fiy, x) = 0, where the operator L is defined by j» dky **£**»%- and the functions Pk(x) and Fiy, x) are such that a solution y and its first m derivatives exist in 0 < x < X. In the special case when (1) is linear the solution can be completely determined by the well known method of varia- tion of parameters when n independent solutions of the associated homo- geneous equations are known. Thus for the case when Fiy, x) is independent of y, the solution of the non-homogeneous equation can be obtained by mere quadratures, rather than by laborious stepwise integrations. It does not seem to have been observed, however, that even when Fiy, x) involves the dependent variable y, the numerical integrations can be so arranged that the contributions to the integral from the upper limit at each step of the integration, at the time when y is still unknown at the upper limit, drop out.
    [Show full text]
  • The Original Euler's Calculus-Of-Variations Method: Key
    Submitted to EJP 1 Jozef Hanc, [email protected] The original Euler’s calculus-of-variations method: Key to Lagrangian mechanics for beginners Jozef Hanca) Technical University, Vysokoskolska 4, 042 00 Kosice, Slovakia Leonhard Euler's original version of the calculus of variations (1744) used elementary mathematics and was intuitive, geometric, and easily visualized. In 1755 Euler (1707-1783) abandoned his version and adopted instead the more rigorous and formal algebraic method of Lagrange. Lagrange’s elegant technique of variations not only bypassed the need for Euler’s intuitive use of a limit-taking process leading to the Euler-Lagrange equation but also eliminated Euler’s geometrical insight. More recently Euler's method has been resurrected, shown to be rigorous, and applied as one of the direct variational methods important in analysis and in computer solutions of physical processes. In our classrooms, however, the study of advanced mechanics is still dominated by Lagrange's analytic method, which students often apply uncritically using "variational recipes" because they have difficulty understanding it intuitively. The present paper describes an adaptation of Euler's method that restores intuition and geometric visualization. This adaptation can be used as an introductory variational treatment in almost all of undergraduate physics and is especially powerful in modern physics. Finally, we present Euler's method as a natural introduction to computer-executed numerical analysis of boundary value problems and the finite element method. I. INTRODUCTION In his pioneering 1744 work The method of finding plane curves that show some property of maximum and minimum,1 Leonhard Euler introduced a general mathematical procedure or method for the systematic investigation of variational problems.
    [Show full text]
  • Improving Numerical Integration and Event Generation with Normalizing Flows — HET Brown Bag Seminar, University of Michigan —
    Improving Numerical Integration and Event Generation with Normalizing Flows | HET Brown Bag Seminar, University of Michigan | Claudius Krause Fermi National Accelerator Laboratory September 25, 2019 In collaboration with: Christina Gao, Stefan H¨oche,Joshua Isaacson arXiv: 191x.abcde Claudius Krause (Fermilab) Machine Learning Phase Space September 25, 2019 1 / 27 Monte Carlo Simulations are increasingly important. https://twiki.cern.ch/twiki/bin/view/AtlasPublic/ComputingandSoftwarePublicResults MC event generation is needed for signal and background predictions. ) The required CPU time will increase in the next years. ) Claudius Krause (Fermilab) Machine Learning Phase Space September 25, 2019 2 / 27 Monte Carlo Simulations are increasingly important. 106 3 10− parton level W+0j 105 particle level W+1j 10 4 W+2j particle level − W+3j 104 WTA (> 6j) W+4j 5 10− W+5j 3 W+6j 10 Sherpa MC @ NERSC Mevt W+7j / 6 Sherpa / Pythia + DIY @ NERSC 10− W+8j 2 10 Frequency W+9j CPUh 7 10− 101 8 10− + 100 W +jets, LHC@14TeV pT,j > 20GeV, ηj < 6 9 | | 10− 1 10− 0 50000 100000 150000 200000 250000 300000 0 1 2 3 4 5 6 7 8 9 Ntrials Njet Stefan H¨oche,Stefan Prestel, Holger Schulz [1905.05120;PRD] The bottlenecks for evaluating large final state multiplicities are a slow evaluation of the matrix element a low unweighting efficiency Claudius Krause (Fermilab) Machine Learning Phase Space September 25, 2019 3 / 27 Monte Carlo Simulations are increasingly important. 106 3 10− parton level W+0j 105 particle level W+1j 10 4 W+2j particle level − W+3j 104 WTA (>
    [Show full text]
  • J. Wallis the Arithmetic of Infinitesimals Series: Sources and Studies in the History of Mathematics and Physical Sciences
    J. Wallis The Arithmetic of Infinitesimals Series: Sources and Studies in the History of Mathematics and Physical Sciences John Wallis was appointed Savilian Professor of Geometry at Oxford University in 1649. He was then a relative newcomer to mathematics, and largely self-taught, but in his first few years at Oxford he produced his two most significant works: De sectionibus conicis and Arithmetica infinitorum. In both books, Wallis drew on ideas originally developed in France, Italy, and the Netherlands: analytic geometry and the method of indivisibles. He handled them in his own way, and the resulting method of quadrature, based on the summation of indivisible or infinitesimal quantities, was a crucial step towards the development of a fully fledged integral calculus some ten years later. To the modern reader, the Arithmetica Infinitorum reveals much that is of historical and mathematical interest, not least the mid seventeenth-century tension between classical geometry on the one hand, and arithmetic and algebra on the other. Newton was to take up Wallis’s work and transform it into mathematics that has become part of the mainstream, but in Wallis’s text we see what we think of as modern mathematics still 2004, XXXIV, 192 p. struggling to emerge. It is this sense of watching new and significant ideas force their way slowly and sometimes painfully into existence that makes the Arithmetica Infinitorum such a relevant text even now for students and historians of mathematics alike. Printed book Dr J.A. Stedall is a Junior Research Fellow at Queen's University. She has written a number Hardcover of papers exploring the history of algebra, particularly the algebra of the sixteenth and 159,99 € | £139.99 | $199.99 ▶ seventeenth centuries.
    [Show full text]
  • A Brief Introduction to Numerical Methods for Differential Equations
    A Brief Introduction to Numerical Methods for Differential Equations January 10, 2011 This tutorial introduces some basic numerical computation techniques that are useful for the simulation and analysis of complex systems modelled by differential equations. Such differential models, especially those partial differential ones, have been extensively used in various areas from astronomy to biology, from meteorology to finance. However, if we ignore the differences caused by applications and focus on the mathematical equations only, a fundamental question will arise: Can we predict the future state of a system from a known initial state and the rules describing how it changes? If we can, how to make the prediction? This problem, known as Initial Value Problem(IVP), is one of those problems that we are most concerned about in numerical analysis for differential equations. In this tutorial, Euler method is used to solve this problem and a concrete example of differential equations, the heat diffusion equation, is given to demonstrate the techniques talked about. But before introducing Euler method, numerical differentiation is discussed as a prelude to make you more comfortable with numerical methods. 1 Numerical Differentiation 1.1 Basic: Forward Difference Derivatives of some simple functions can be easily computed. However, if the function is too compli- cated, or we only know the values of the function at several discrete points, numerical differentiation is a tool we can rely on. Numerical differentiation follows an intuitive procedure. Recall what we know about the defini- tion of differentiation: df f(x + h) − f(x) = f 0(x) = lim dx h!0 h which means that the derivative of function f(x) at point x is the difference between f(x + h) and f(x) divided by an infinitesimal h.
    [Show full text]
  • Arxiv:1809.06300V1 [Hep-Ph] 17 Sep 2018
    CERN-TH-2018-205, TTP18-034 Double-real contribution to the quark beam function at N3LO QCD Kirill Melnikov,1, ∗ Robbert Rietkerk,1, y Lorenzo Tancredi,2, z and Christopher Wever1, 3, x 1Institute for Theoretical Particle Physics, KIT, Karlsruhe, Germany 2Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland 3Institut f¨urKernphysik, KIT, 76344 Eggenstein-Leopoldshafen, Germany Abstract We compute the master integrals required for the calculation of the double-real emission contri- butions to the matching coefficients of jettiness beam functions at next-to-next-to-next-to-leading order in perturbative QCD. As an application, we combine these integrals and derive the double- real emission contribution to the matching coefficient Iqq(t; z) of the quark beam function. arXiv:1809.06300v1 [hep-ph] 17 Sep 2018 ∗ Electronic address: [email protected] y Electronic address: [email protected] z Electronic address: [email protected] x Electronic address: [email protected] 1 I. INTRODUCTION The absence of any evidence for physics beyond the Standard Model at the LHC implies a growing importance of indirect searches for new particles and interactions. An integral part of this complex endeavour are first-principles predictions for hard scattering processes in proton collisions with controllable perturbative accuracy. In recent years, we have seen a remarkable progress in an effort to provide such predictions. Indeed, robust methods for one-loop computations developed during the past decade, that allowed the theoretical description of a large number of processes with multi-particle final states through NLO QCD [1{6], were followed by the development of practical NNLO QCD subtraction and slicing schemes [7{17] and advances in computations of two-loop scattering amplitudes [18{29].
    [Show full text]
  • Further Results on the Dirac Delta Approximation and the Moment Generating Function Techniques for Error Probability Analysis in Fading Channels
    International Journal of Computer Networks & Communications (IJCNC) Vol.5, No.1, January 2013 FURTHER RESULTS ON THE DIRAC DELTA APPROXIMATION AND THE MOMENT GENERATING FUNCTION TECHNIQUES FOR ERROR PROBABILITY ANALYSIS IN FADING CHANNELS Annamalai Annamalai 1, Eyidayo Adebola 2 and Oluwatobi Olabiyi 3 Center of Excellence for Communication Systems Technology Research Department of Electrical & Computer Engineering, Prairie View A&M University, Texas [email protected], [email protected], [email protected] ABSTRACT In this article, we employ two distinct methods to derive simple closed-form approximations for the statistical expectations of the positive integer powers of Gaussian probability integral E[ Q p (βΩ γ )] with γ respect to its fading signal-to-noise ratio (SNR) γ random variable. In the first approach, we utilize the shifting property of Dirac delta function on three tight bounds/approximations for Q (.) to circumvent the need for integration. In the second method, tight exponential-type approximations for Q (.) are exploited to simplify the resulting integral in terms of only the weighted sum of moment generating function (MGF) of γ. These results are of significant interest in the development of analytically tractable and simple closed- form approximations for the average bit/symbol/block error rate performance metrics of digital communications over fading channels. Numerical results reveal that the approximations based on the MGF method are much more versatile and can achieve better accuracy compared to the approximations derived via the asymptotic Dirac delta technique for a wide range of digital modulations schemes and wireless fading environments. KEYWORDS Moment generating function method, Dirac delta approximation, Gaussian quadrature approximation.
    [Show full text]
  • Numerical Integration of Functions with Logarithmic End Point
    FACTA UNIVERSITATIS (NIS)ˇ Ser. Math. Inform. 17 (2002), 57–74 NUMERICAL INTEGRATION OF FUNCTIONS WITH ∗ LOGARITHMIC END POINT SINGULARITY Gradimir V. Milovanovi´cand Aleksandar S. Cvetkovi´c Abstract. In this paper we study some integration problems of functions involv- ing logarithmic end point singularity. The basic idea of calculating integrals over algebras different from the standard algebra {1,x,x2,...} is given and is applied to evaluation of integrals. Also, some convergence properties of quadrature rules over different algebras are investigated. 1. Introduction The basic motive for our work is a slow convergence of the Gauss-Legendre quadrature rule, transformed to (0, 1), 1 n (1.1) I(f)= f(x) dx Qn(f)= Aνf(xν), 0 ν=1 in the case when f(x)=xx. It is obvious that this function is continuous (even uniformly continuous) and positive over the interval of integration, so that we can expect a convergence of (1.1) in this case. In Table 1.1 we give relative errors in Gauss-Legendre approximations (1.1), rel. err(f)= |(Qn(f) − I(f))/I(f)|,forn = 30, 100, 200, 300 and 400 nodes. All calculations are performed in D- and Q-arithmetic, with machine precision ≈ 2.22 × 10−16 and ≈ 1.93 × 10−34, respectively. (Numbers in parentheses denote decimal exponents.) Received September 12, 2001. The paper was presented at the International Confer- ence FILOMAT 2001 (Niˇs, August 26–30, 2001). 2000 Mathematics Subject Classification. Primary 65D30, 65D32. ∗The authors were supported in part by the Serbian Ministry of Science, Technology and Development (Project #2002: Applied Orthogonal Systems, Constructive Approximation and Numerical Methods).
    [Show full text]