Numerical Integration

Total Page:16

File Type:pdf, Size:1020Kb

Numerical Integration Numerical Integration MAT 2310 Computational Mathematics Wm C Bauldry November, 2011 ICM 1 Contents 1. Numerical Integration 5. Gaussian Quadrature 2. Left- and Right Endpoint, 6. Gauss-Kronrod Quadrature and Midpoint Sums 7. Test Integrals 3. Trapezoid Sums 8. Exercises 4. Simpson’s Rule 9. Links and Others Today ICM 2 Numerical Integration What is Numerical Integration? Numerical integration or (numerical) 2.5 quadrature is the calculation of a definite 2 1.5 integral using numerical formulas, not the 1 fundamental theorem. The Greeks stud- 0.5 ied quadrature: given a figure, construct a -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -0.5 square that has the same area. The two -1 most famous are Hippocrates of Chios’ Quadrature of the Lune (c. 450 BC) and Archimedes’ Quadrature of the Parabola (c. 250 BC). Archimedes used the method of exhaustion — a precursor of calculus — invented by Eudoxus. Squaring the circle is one of the classical problems of constructing a square with the area of a given circle – it was shown impossible by Lindemann’s theorem (1882). ICM 3 Methods of Elementary Calculus Rectangle Methods Left endpoint sum Midpoint sum Right endpoint sum n n n X X X An ≈ f(xk−1) ∆xk An ≈ f(mk) ∆xk An ≈ f(xk) ∆xk k=1 k=1 k=1 1 mk = 2 (xk−1 + xk) (b−a)2 1 (b−a)3 1 (b−a)2 1 "n ≤ · M1 · "n ≤ · M2 · "n ≤ · M1 · 2 n 24 n2 2 n where M = max f (i)(x) R πf dx = 2 ≈ [1:984; 2:008; 1:984] i 0 n=10 ICM 4 Trapezoid Sums Trapezoid Sums Instead of the degree 0 rectangle approximations to the function, use a linear degree 1 approximation. The area of the trapezoid is given by 1 AT = 2 [f(xk−1) + f(xk)] ∆xk This gives an approximation for the integral Z b n X 1 f(x) dx ≈ 2 [f(xk−1) + f(xk)] ∆xk a k=1 [Midpoint: measure height at average x v. trapezoid: average the height measures] The formula is often written as " n−1 ! # X ∆xk T ≈ f(x ) + 2 f(x ) + f(x ) n 0 k n 2 k=1 Error for the trapezoid rule is (b − a)3 1 " ≤ · M · n 12 2 n2 ICM 5 Sample Trapezoid Example 1 1 1 Let f(x) = sin(x) + 2 sin(2x) − 4 sin(4x) + 16 sin(8x) over [0; π]. With an equipartition, ∆x = π=10 ≈ 0:314 Then " 10 ! # X ∆x T = f(0)+ 2 f( k π) +f(π) 10 9 2 k=1 which gives T10 = 1:984 with absolute error of 0:016. ICM 6 Simpson’s Rule Simpson’s Rule We now move to a degree 2 approximation. The easiest way to have 3 data pts is to take the panels in pairs: instead of rectangle base [xi; xi+1], use [xi; xi+1; xi+2]. So we require an even number of panels. The area under the parabola is 1 h i AS = 3 f(xi) + 4f(xi+1) + f(xi+2) ∆x This gives a 2n-panel approximation for the integral " n # Z b X ∆x f(x) dx ≈ f(x ) + 4f(x ) + f(x ) 2k−2 2k−1 2k 3 a k=1 often written as ∆x S = [f(x ) + 4f(x ) + 2f(x ) + 4f(x ) + ··· + 4(f(x ) + f(x )] 2n 0 1 2 3 2n−1 2n 3 The error is bounded by (b − a)5 1 " ≤ · M · n 180 4 n4 ICM 7 Sample Simpson Example 1 1 1 Let f(x) = sin(x) + 2 sin(2x) − 4 sin(4x) + 16 sin(8x) over [0; π]. With a 10 panel equipartition, ∆x = π=10 ≈ 0:314 Then, with yi = f(xi), 1 S10 = 3 [y0 +4y1 +2y2 +···+4y9 +y10]∆x which gives S10 = 2:000 with absolute error of 6:78 · 10−6. ICM 8 A Maple Comparison Approximating a Difficult Integral Z 2 10−4 Consider dx. The integrand has a sharp peak at π=2. 1 2 −8 1 (x − 2 π) + 10 The exact value of the integral (FToC) is arctan(−10000 + 5000 π) + arctan(20000 − 5000 π) ≈ 3:14118447 Maple gives n = 50 n = 500 n = 5000 2 left 0:04972053 2 left 0:5393463 2 left 3:424973 6 right 0:04972537 6 right 0:5393457 6 right 3:424957 6 7 6 7 6 7 6 midpoint 3:09300007 6 midpoint 4:1670607 6 midpoint 2:880897 6 7 6 7 6 7 4trapezoid 0:04972285 4trapezoid 0:5393425 4trapezoid 3:424985 Simpson 2:0785900 Simpson 2:957850 Simpson 3:06226 To achieve relative error below 1% requires n ≥ 8000. ICM 9 Þe Charte Ob Anfeald Gang Quadrature “Height” Error Bound (b−a)2 1 Left end point f(xi) 2 · M1 · n (b−a)2 1 Right end point f(xi+1) 2 · M1 · n 3 f 1 (x + x ) (b−a) · M · 1 Midpoint 2 i i+1 24 2 n2 3 1 (f(x ) + f(x )) (b−a) · M · 1 Trapezoid Rule 2 i i+1 12 2 n2 1 (b−a)5 1 Simpson’s Rule 3 (f(xi) + 4f(xi+1) + f(xi+2)) 180 · M4 · n4 (i) where Mi ≥ max f (x) . ICM 10 Gaussian Quadrature Johann Carl Friedrich Gauss About 1815, while Gauss was finishing constructing astronomical observat- ory, he wrote a paper1 on approximating integrals. Gauss’s technique was studied and extended by Christoffel in 1858. There are several ways to develop his method. We’ll use the easiest . In Search of Improvements Write the rules we’ve seen as sums: 1 1 1 Left endpt: Ln= n f(x0) + n f(x1) + ··· + n f(xn−1) 1 1 1 Right endpt: Rn= n f(x1) + n f(x2) + ··· + n f(xn) 1 1 1 Midpoint:Mn= n f(xm1 ) + n f(xm2 ) + ··· + n f(xmn ) 1 1 1 1 Trapezoid: Tn= 2n f(x0) + n f(x1) + ··· + n f(xn−1) + 2n f(xn) 1 4 2 4 1 Simpson’s: Sn= 3n f(x0) + 3n f(x1) + 3n f(x2) + ··· + 3n f(xn−1) + 3n f(xn) 1 “Methodus nova integralium valores per approximationem inveniendi,” Werke, 3, K. Ges. Wissenschaft. Göttingen (1886) pp. 163–196. ICM 11 Patterns Observe 1. Each of the formulas has the same form An = w1 · f(x1) + w2 · f(x2) + ··· + wn · f(xn) with different sets of weights wi and different sets of nodes xi. 2. Any closed interval can be mapped to and from [−1; 1], so we R 1 can focus only on −1 f(x) dx. 3. Gauss posed the question: Is there a “best choice” of weights fwig and nodes fxig? 4. The answer depends on what “best” means. 5. Since we have 2n ‘unknowns’ wi and xi, let’s look for a set that integrates a 2n − 1 degree polynomial exactly. ICM 12 Sampling 3 Example (Third Degree) Set n = 3. Determine the choice of wi and of xi so that Z 1 3 p X p x dx = wk · (xk) −1 k=1 exactly for p = 0; 1;:::; 5 = 2 · 3 − 1. The range for p gives us six equations: 8 9 > w1 + w2 + w3 = 2 > > > > > > w x + w x + w x = 0 > > 1 1 2 2 3 3 > 8 p p > > 15 15 > 2 2 2 2 > > x1 = − 5 ; x2 = 0; x3 = 5 < w1x1 + w2x2 + w3x3 = = < 3 =) > w x3 + w x3 + w x3 = 0 > > > 1 1 2 2 3 3 > > w = 5 ; w = 8 ; w = 5 > > : 1 9 2 9 3 9 > 4 4 4 2 > > w1x1 + w2x2 + w3x3 = 5 > > > > 5 5 5 > : w1x1 + w2x2 + w3x3 = 0 ; p p 5 15 8 5 15 Our Gaussian quadrature is G3(f) = 9 f − 5 + 9 f(0) + 9 f 5 ICM 13 Testing Gauss Random Polynomials Generate and test a random 5th degree polynomial. p := unapply(sort(randpoly(x, degree = 5), x), x) x ! −7x5 + 22x4 − 55x3 − 94x2 + 87x − 56 G3 := 5/9*p(-sqrt(15)/5) + 8/9*p(0) + 5/9*p(sqrt(15)/5) 2488 − 15 Int(p(x), x = -1..1) = int(p(x), x = -1..1) Z 1 2488 p(x) dx = − 15 −1 Generate and test a random 7th degree polynomial. q := unapply(sort(randpoly(x, degree = 7), x), x) x ! 97x7 − 73x6 − 4x5 − 83x3 − 10x − 62 int(q(x),x= -1..1)= 5/9*q(-sqrt(15)/5)+8/9*q(0)+5/9*q(sqrt(15)/5) 722 2662 7 = 25 ICM 14 Gaussian Properties Theorem Z 1 Let f have 2n continuous derivatives. Then for "n = Gn − f(x)dx , 2 −1 " ≤ · M 2n n (2n + 1)! (2n) where M2n ≥ max f (x) . Values of Gaussian Weights and Nodes There are numerous sources online, e.g.,: 1. The classic Abramowitz and Stegun Handbook (see the entire book) 2. Holoborodko or Kamermans Or calculate the values directly: 1 dn 2 n Set Pn(x) = 2n n! · dxn x − 1 (the Legendre polynomials). Then n 2 fxigi=1 = fzeros of Png and wi = 2 0 2 (1 − xi )[Pn(xi)] ICM 15 Aleksandr Kronrod Kronrod’s Idea (1964) One difficulty in Gaussian quadrature is that increasing the number of nodes requires recomputing all the values of • nodes • weights • function evaluations Kronrod2 discovered he could interlace n + 1 new nodes with n original Gaussian nodes and have a rule of order 3n + 1. A 2n + 1 node Gaussian quadrature would have order 4n + 1, but with significant extra computation for an increase of only n in order over Kronrod’s method.
Recommended publications
  • 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]
  • The Legendre Transform Ross Bannister, May 2005 Orthogonality of the Legendre Polynomials the Legendre Polynomials Satisfy the Following Orthogonality Property [1], 1
    The Legendre Transform Ross Bannister, May 2005 Orthogonality of the Legendre polynomials The Legendre polynomials satisfy the following orthogonality property [1], 1 2 ¥ ¥§¦ ¥ ¤ ¤ d x Pn ¤ x Pm x mn 1 © 2n ¨ 1 x ¡£¢ 1 ¥ where Pn ¤ x is the nth order Legendre polynomial. In meteorology it is sometimes convenient to integrate over the latitude domain, , instead of over x. This is achieved with the following transform, d x ¦ ¦ ¦ ¥ ¤ x sin cos d x cos d 2 d In latitude (measured in radians), Eq. (1) becomes, /2 2 ¥ ¥ ¥§¦ ¤ ¤ d cos Pn ¤ Pm mn 3 ¨ © 2n 1 ¡£¢ /2 The Legendre transform and its inverse convert to and from the latitudinal and Legendre polynomial representations of functions (I call this these transforms a Legendre transform pair). In the following we shall develop these transforms in matrix form. The matrix form of the transforms is a compact and convenient means of performing summations over and over n (we shall assume that functions are represented discreetly in space and so integrals are represented by summations). In the notation to follow, functions are represented as vectors. A function in space shall be denoted by the vector x, and a function in n space shall be denoted by p. Let the matrix consisting of columns of Legendre polynomials be F. The orthogonality condition, Eq. (3), in matrix form is, T ¦ ¥ ¤ F PF 4 where P is the so-called diagonal inner product matrix whose diagonal elements are cos and is the ¨ ¥ diagonal matrix whose diagonal elements are the normalizations 2 / ¤ 2n 1 . Legendre transform pair ('unbalanced' version) The orthogonality condition, Eq.
    [Show full text]
  • Calculation of Highly Oscillatory Integrals by Quadrature Methods
    CALCULATION OF HIGHLY OSCILLATORY INTEGRALS BY QUADRATURE METHODS A Senior Scholars Thesis by KRISHNA THAPA Submitted to Honors and Undergraduate Research Texas A&M University in partial fulfillment of the requirements for the designation as UNDERGRADUATE RESEARCH SCHOLAR May 2012 Major: Physics CALCULATION OF HIGHLY OSCILLATORY INTEGRALS BY QUADRATURE METHODS A Senior Scholars Thesis by KRISHNA THAPA Submitted to Honors and Undergraduate Research Texas A&M University in partial fulfillment of the requirements for the designation as UNDERGRADUATE RESEARCH SCHOLAR Approved by: Research Advisor: Stephen Fulling Associate Director, Honors and Undergraduate Research: Duncan MacKenzie May 2012 Major: Physics iii ABSTRACT Calculation of Highly Oscillatory Integrals by Quadrature Methods. (May 2012) Krishna Thapa Department of Physics Texas A&M University Research Advisor: Dr. Stephen Fulling Department of Mathematics R 1 i!g(x) Highly oscillatory integrals of the form I(f) = 0 dxf(x)e arise in various problems in dynamics, image analysis, optics, and other fields of physics and math- ematics. Conventional approximation methods for such highly oscillatory integrals tend to give huge errors as frequency (!) ! 1. Over years, various attempts have been made to get over this flaw by considering alternative quadrature methods for integration. One such method was developed by Filon in 1928, which Iserles et al. have recently extended. Using this method, Iserles et al. show that as ! ! 1, the error decreases further as the error is inversely proportional to !. We use methods developed by Iserles' group, along with others like Newton-Cotes, Clenshaw-Curtis and Levin's methods with the aid of Mathematica. Our aim is to find a systematic way of calculating highly oscillatory integrals.
    [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]
  • Numerical Construction of Gaussian Quadrature Formulas For
    mathematics of computation, VOLUME 27, NUMBER 124, OCTOBER 1973 Numerical Construction of Gaussian Quadrature Formulas for / (-Log *)• xa-f(x)-dx and / Em(x)-j(x)-dx Jo Jo By Bernard Danloy Abstract. Most nonclassical Gaussian quadrature rules are difficult to construct because of the loss of significant digits during the generation of the associated orthogonal poly- nomials. But, in some particular cases, it is possible to develop stable algorithms. This is true for at least two well-known integrals, namely ¡l-(Loêx)-x°f(x)dx and ¡Ô Em(x)f(x)-dx. A new approach is presented, which makes use of known classical Gaussian quadratures and is remarkably well-conditioned since the generation of the orthogonal polynomials requires only the computation of discrete sums of positive quantities. Finally, some numerical results are given. 1. Introduction. Let w(x) be a nonnegative weight function on (a, b) such that all its moments (1.1) M*= / w(x)-x"-dx, k = 0, 1, 2, ••■ , exist. The «-point Gaussian quadrature rule associated with w(x) and (a, b) is that uniquely defined linear functional (1.2) G„-/= ¿X, •/(*,) ¿-i which satisfies (1.3) G„/ = [ w(x)-j(x)-dx Ja whenever / is a polynomial of degree ^ 2n — 1. It is a well-known result [8] that the Gaussian abscissas *¿ are the roots of the polynomials orthogonal on (a, b) with respect to w(x), and that the associated coeffi- cients X,, called Christoffel constants, can also be expressed in terms of these poly- nomials. A direct exploitation of these results is still the most widely recommended procedure, even though alternative approaches have been suggested by Rutishauser [7], Golub and Welsch [6].
    [Show full text]