DOCSLIB.ORG

Quadrature Theory the Theory of Numerical Integration on a Compact Interval

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quadrature Theory the Theory of Numerical Integration on a Compact Interval Mathematical Surveys and Monographs Volume 178 Quadrature Theory The Theory of Numerical Integration on a Compact Interval Helmut Brass Knut Petras American Mathematical Society http://dx.doi.org/10.1090/surv/178 Quadrature Theory The Theory of Numerical Integration on a Compact Interval Mathematical Surveys and Monographs Volume 178 Quadrature Theory The Theory of Numerical Integration on a Compact Interval Helmut Brass Knut Petras American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Jordan S. Ellenberg Benjamin Sudakov MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 41A55; Secondary 65D30, 65D32. For additional information and updates on this book, visit www.ams.org/bookpages/surv-178 Library of Congress Cataloging-in-Publication Data Brass, Helmut. Quadrature theory : the theory of numerical integration on a compact interval / Helmut Brass, Knut Petras. p. cm. – (Mathematical surveys and monographs ; v. 178) Includes bibliographical references and index. ISBN 978-0-8218-5361-0 (alk. paper) 1. Gaussian quadrature formulas. 2. Numerical integration. I. Petras, Knut, 1962– II. Title. QA299.4.G3B73 2011 515.43–dc23 2011030201 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2011 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 161514131211 Contents Preface vii Chapter 1. Introduction 1 Chapter 2. The Abstract Framework 7 2.1. Standard estimation framework 7 2.2. Linear rules that are exact on a subspace 16 2.3. Strong optimality: inner product spaces 21 2.4. Varying the observation 26 Chapter 3. Norm and Kernel of the Remainder Functional 29 3.1. Norm of an estimation rule 29 3.2. The interpolation theorem 33 3.3. Quadrature formulas and one-sided approximation 36 3.4. Krein’s theory 40 Chapter 4. Co-observations 47 4.1. Survey 47 4.2. The Peano kernel theorem 50 4.3. Bounded derivatives as co-observation 59 4.4. Bounded variation as co-observation 63 4.5. Error bounds using the modulus of continuity 66 4.6. Derivatives of bounded variation 72 4.7. Sard’s co-observation 73 4.8. Co-observations of Davis type 76 4.9. Bounds in the complex plane as co-observations 82 4.10. Convex functions 93 Chapter 5. Quadrature Rules of Interpolatory Type 99 5.1. Recapitulation 99 5.2. The Newton–Cotes method 104 5.3. A theorem of Sloan and Smith 110 5.4. Error bounds for the Clenshaw–Curtis method 114 5.5. Relatives of the Clenshaw–Curtis method 121 5.6. The distribution of nodes 130 5.7. Bounds for the norms of Peano kernels of interpolatory rules 135 5.8. Asymptotic behaviour of a class of Peano kernels 144 Chapter 6. Gaussian Quadrature 149 6.1. Rules of high degree and orthogonal polynomials 149 6.2. Coefficients and nodes for general weights 156 v vi CONTENTS 6.3. Nodes and coefficients for w = 1 162 6.4. Peano kernels for general weights 166 6.5. Peano kernels for w = 1 173 6.6. Error bounds 180 6.7. Asymptotics of the error 189 6.8. Extremal properties of Gaussian rules 195 6.9. Why Gaussian quadrature? 197 6.10. The Kronrod method 201 6.11. Kronrod rules for w = 1 204 Chapter 7. Quadrature Rules with Equidistant Nodes 211 7.1. The trapezoidal method and the Euler–Maclaurin formula 211 7.2. More on the trapezoidal method 217 7.3. Simpson’s method 230 7.4. The Filon method 233 7.5. Gregory methods 236 7.6. Romberg methods 244 7.7. Equidistant nodes and the degree of polynomial exactness 254 7.8. The midpoint method 256 Chapter 8. Periodic Integrands 261 8.1. The special role of the trapezoidal rule for w = 1 261 8.2. Error bounds for the trapezoidal rule 264 8.3. Trigonometric interpolation 271 8.4. Universality 273 8.5. Standard rules for Fourier coefficients 276 8.6. Other rules for Fourier coefficients 283 Chapter 9. Variance and Chebyshev-type Rules 291 9.1. Fundamentals 291 9.2. Chebyshev methods 296 9.3. The special case of w = 1 299 9.4. Variance 303 Chapter 10. Problems 307 Appendix A. Orthogonal Polynomials 315 Appendix B. Bernoulli Polynomials 325 Appendix C. Validation of Co-observations 329 C.1. Automatic generation of Taylor coefficients 329 C.2. Real interval arithmetic 331 C.3. Complex interval arithmetic 333 Bibliography 335 Books on quadrature 335 References 336 Symbols 357 Index 361 Preface Methods for the approximate calculation of definite integrals are covered in every book on numerical analysis. Our intention here is to provide a complementary treatment of this topic by presenting a coherent theory that encompasses many deep and elegant results as well as a large number of interesting (solved and open) problems. The inclusion of the word “theory” in the title highlights our emphasis on concepts rather than numerical recipes. Thus, no computer programs and only a few numerical examples are given in the book. The focus on theory does not, however, mean that we pass over concrete practical problems, merely that we choose to restrict our attention to problems for which a guaranteed result can be obtained in a systematic manner. Systematic analyses of this kind rely on certain properties of the integrand, over and beyond the knowledge of finitely many function values. Such additional information about the integrand (called “co-observations”) forms the central organizing principle for our theory, and distinguishes our book from other texts on quadrature. A wide variety of co-observations are examined in this monograph, as we believe such information will be very useful for solving problems in practical contexts. While quadrature theory is often viewed as a branch of numerical analysis, its influence extends much further: it has been the starting point of many far-reaching generalizations in various directions, as well as a testing ground for new ideas and concepts; in fact, in many instances the extensions seem more “natural” than the original motivating problem. We shall discuss such generalizations, although the classical problem will remain our guiding star throughout the book. Working on quadrature has given us great pleasure over the years, and we hope we can convey our enthusiasm for the subject to the readers of this book. The mathematical prerequisites for engaging with this text are knowledge (at the level taught in most undergraduate courses) of linear algebra, advanced calculus and real analysis. We thank our wives for their patience and assistance in many ways. We are grateful to Alice Yew for her help in editing our manuscript. Helmut Brass and Knut Petras Technische Universit¨at Braunschweig vii Bibliography Books on Quadrature Brass, H. (1977) Quadraturverfahren. Vandenhoeck & Ruprecht, G¨ottingen. Davis, P.J.; Rabinowitz, P. (1984) Methods of Numerical Integration, second edition. Academic Press, Orlando. Engels, H. (1980) Numerical Quadrature and Cubature. Academic Press, London. Evans, G. (1993) Practical Numerical Integration. Wiley, Chichester. Ghizzetti, A.; Ossicini, A. (1970) Quadrature Formulae. Birkh¨auser, Basel. Kowalewski, G. (1932) Interpolation und gen¨aherte Quadratur. Teubner, Leipzig. Krommer, A.R.; Ueberhuber, C.W. (1994) Numerical Integration on Advanced Computer Systems. Springer, Berlin. Krommer, A.R.; Ueberhuber, C.W. (1998) Computational Integration. SIAM, Philadelphia. Kronrod, A.S. (1964) Nodes and Weights of Quadrature Formulas (in Russian). Izdat Nauka, Moscow. English translation: Consultants Bureau, New York (1965). Krylov, V.J. (1962) Approximate Calculation of Integrals. (Translated from the Russian by A.H. Stroud.) Macmillan, New York. Kythe, P.K.; Schaferkotter,¨ M.R. (2005) Handbook of Computational Meth- ods for Integration. Chapman & Hall/CRC Press, Boca Raton, FL. Levin, M.; Girshovich, J. (1979) Optimal Quadrature Formulas. Teubner, Leipzig. Nikol’skii, S.M. (1964) Quadrature Formulae. Hindustan Publishing Corpora- tion, Delhi. Piessens, R.; de Doncker-Kapenga, E.; Uberhuber,¨ C.W.; Kahaner, D.K. (1983) QUADPACK: A Subroutine Package for Automatic Integration. Springer, Berlin. Smith, H.V. (1993) Numerical Methods of Integration. Chartwell-Bratt, Bromley. Stroud, A.H.; Secrest, D. (1966) Gaussian Quadrature Formulas. Prentice- Hall, Englewood Cliffs, NJ. Zielas, W. (1977) Verallgemeinerte Interpolation und Quadraturen. Akademie- Verlag, Berlin. 335 336 BIBLIOGRAPHY References Abramowitz, M.; Stegun, I.A. (1964) Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC. Achieser, N.J. (1956) Theory of Approximation. Frederick Ungar Publishing Co., New York. Achieser, N.J.; Krein, M.G. (1938) Some remarks on the coefficients in quad- rature formulae of the type of Gauss (in Ukrainian). TrudiOdes.Derz.Univ. Mat. 2, 29–38. Akrivis, G. (1984) Fehlerabsch¨atzungen f¨ur Gauss-Quadraturformeln. Numer. Math. 44, 261–278. Akrivis, G. (1985) The error norm of certain Gaussian quadrature formulae. Math. Comp. 45, 513–519. Akrivis, G.; Forster,¨ K.-J. (1984) On the definiteness of quadrature formulae of Clenshaw-Curtis type. Computing 33, 363–366. Albrecht, J. (1972) Intervallschachtelungen beim Romberg-Verfahren. Z.
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]
  • 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]
  • 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]
  • 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]
  • Convolution Quadrature and Discretized Operational Calculus. II*
    Numerische Numer. Math. 52, 413-425 (1988) Mathematik Springer-Verlag 1988 Convolution Quadrature and Discretized Operational Calculus. II* C. Lubich Institut ffir Mathematik und Geometrie,Universit~it Innsbruck, Technikerstrasse 13, A-6020 Innsbruck, Austria Summary. Operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations: singular inte- grals (power and logarithmic singularities, finite part integrals), multiple time- scale convolution, Volterra integral equations, Wiener-Hopf integral equa- tions. Frequency domain conditions, which determine the stability of such equations, can be carried over to the discretization. Subject Classifications: AMS(MOS): 65R20, 65D30; CR: G1.9. 6. Introduction In this paper we give applications of the operational quadrature rules of Part I to numerical integration and the numerical solution of integral equations. It is the aim to indicate problem classes where such methods can be applied success- fully, and to point out features which distinguish them from other approaches. Section 8 deals with the approximation of integrals with singular kernel. This comes as a direct application of the results of Part I. In Sect. 9 we study the numerical approximation of convolution integrals whose kernel has components at largely differing time-scales. In Sect. 10 we give a case study for a problem in chemical absorption kinetics. There a system of an ordinary differential equation coupled with a diffusion equation is reduced to a single Volterra integral equation. This reduction is achieved in a standard way via Laplace transform techniques and, as is typical in such situations, it is the Laplace transform of the kernel (rather than the kernel itself) which is known a priori, and which is of a simple form.
    [Show 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.
    [Show full text]
  • Handout on Gaussian Quadrature, Parts I and II
    Theoretical foundations of Gaussian quadrature 1 Inner product vector space Definition 1. A vector space (or linear space) is a set V = {u, v, w,...} in which the following two operations are defined: (A) Addition of vectors: u + v ∈ V , which satisfies the properties (A1) associativity: u + (v + w) = (u + v) + w ∀ u, v, w in V ; (A2) existence of a zero vector: ∃ 0 ∈ V such that u + 0 = u ∀u ∈ V ; (A3) existence of an opposite element: ∀u ∈ V ∃(−u) ∈ V such that u + (−u) = 0; (A4) commutativity: u + v = v + u ∀ u, v in V ; (B) Multiplication of a number and a vector: αu ∈ V for α ∈ R, which satisfies the properties (B1) α(u + v) = αu + αv ∀ α ∈ R, ∀u, v ∈ V ; (B2) (α + β)u = αu + βu ∀ α, β ∈ R, ∀u ∈ V ; (B3) (αβ)u = α(βu) ∀ α, β ∈ R, ∀u ∈ V ; (B4) 1u = u ∀u ∈ V . Definition 2. An inner product linear space is a linear space V with an operation (·, ·) satisfying the properties (a) (u, v) = (v, u) ∀ u, v in V ; (b) (u + v, w) = (u, w) + (v, w) ∀ u, v, w in V ; (c) (αu, v) = α(u, v) ∀α ∈ R, ∀ u, v, w in V ; (d) (u, u) ≥ 0 ∀u ∈ V ; moreover, (u, u) = 0 if and only if u = 0. d Example. The “standard” inner product of the vectors u = (u1, u2, . , ud) ∈ R and d v = (v1, v2, . , vd) ∈ R is given by d X (u, v) = uivi . i=1 1 Example. Let G be a symmetric positive-definite matrix, for example 5 4 1 G = (gij) = 4 7 0 .
    [Show full text]
  • [Draft] 4.4.3. Examples of Gaussian Quadrature. Gauss–Legendre
    CAAM 453 · NUMERICAL ANALYSIS I Lecture 26: More on Gaussian Quadrature [draft] 4.4.3. Examples of Gaussian Quadrature. Gauss{Legendre quadrature. The best known Gaussian quadrature rule integrates functions over the interval [−1; 1] with the trivial weight function w(x) = 1. As we saw in Lecture 19, the orthogonal polynomials for this interval and weight are called Legendre polynomials. To construct a Gaussian quadrature rule with n + 1 points, we must determine the roots of the degree-(n + 1) Legendre polynomial, then find the associated weights. First, consider the case of n = 1. The quadratic Legendre polynomial is 2 φ2(x) = x − 1=3; and from this polynomialp one can derive the 2-point quadrature rule that is exact for cubic poly- nomials, with roots ±1= 3. This agrees with the special 2-point rule derived in Section 4.4.1. The values for the weights follow simply, w0 = w1 = 1, giving the 2-point Gauss{Legendre rule p p I(f) = f(−1= 3) + f(1= 3): For Gauss{Legendre quadrature rules based on larger numbers of points, there are various ways to compute the nodes and weights. Traditionally, one would consult a book of mathematical tables, such as the venerable Handbook of Mathematical Functions, edited by Milton Abramowitz and Irene Stegun for the National Bureau of Standards in the early 1960s. Now, one can look up these quadrature nodes and weights on web sites, or determine them to high precision using a symbolic mathematics package such as Mathematica. Most effective of all, one can compute these nodes and weights by solving a symmetric tridiagonal eigenvalue problem.
    [Show full text]
  • Chapter 6 Numerical Integration
    Chapter 6 Numerical Integration In many computational economic applications, one must compute the de¯nite integral of a real-valued function f de¯ned on some interval I of n. Some applications call for ¯nding the area under the function, as in comp<uting the change in consumer welfare caused by a price change. In these instances, the integral takes the form f(x) dx: ZI Other applications call for computing the expectation of a function of a random variable with a continuous probability density p. In these instances the integral takes the form f(x)p(x) dx: ZI We will discuss only the three numerical integration techniques most com- monly encountered in practice. Newton-Cotes quadrature techniques employ a strategy that is a straightforward generalization of Riemann integration principles. Newton-Cotes methods are easy to implement. They are not, however, the most computationally e±cient way of computing the expec- tation of a smooth function. In such applications, the moment-matching strategy embodied by Gaussian quadrature is often substantially superior. Finally, Monte Carlo integration methods are discussed. Such methods are simple to implement and are particularly useful for complex high-dimensional integration, but should only be used as a last resort. 1 6.1 Newton-Cotes Quadrature Consider the problem of computing the de¯nite integral f(x) dx Z[a;b] of a real-valued function f over a bounded interval [a; b] on the real line. Newton-Cotes quadrature rules approximate the integrand f with a piece- wise polynomial function f^ and take the integral of f^ as an approximation of the integral of f.
    [Show full text]