Lecture on Numerical Analysis

1. Lecture on Numerical Analysis Dr.-Ing. Michael Dumbser 24 / 09 / 2008

2. Numerical Integration (Quadrature) of Functions - Motivation Task: compute approximately f Solution strategy: • Divide interval [a;b] into n smaller subintervals • Approximate f(x) by interpolation polynomials on the subintervals, e.g. using Lagrange interpolation • Integrate these polynomials exactly on each subinterval and sum up • So-called Newton-Cotes formulae x h a b

3. Transformation of the Integration Interval The computation of numerical quadrature formulae for each sub-interval can be technically considerably simplified using the following variable substitution: Therefore, it is sufficient, without the loss of generality, to consider from now on the case of numerical integration in the reference interval [0;1].

4. The Newton-Cotes Formulae • The integral of f(x) is approximated in the following steps: • First, the function f(x) is interpolated by a polynomial of degree k insideeach sub-interval. The interpolation points are distributed in an equidistantmanner in each sub-interval. • Second, the interpolation polynomial is integrated analytically. • Steps 1 and 2 produce an approximation of the integral of f(x) in termsof the function values fi at the interpolation points and the step size h.

5. The Trapezium or Trapezoid Rule Use linear interpolation polynomials, i.e. polynomials of degree one in the subintervals [xi;xi+1]  [0;1]: f b a x h

6. The Simpson Rule • (Originally Discovered by Johannes Kepler in 1615) Use quadratic interpolation polynomials, i.e. polynomials of degree two: f b a x h

7. The 3/8 Rule Use cubic interpolation polynomials, i.e. polynomials of degree three:

8. Isaac Newton • Sir Isaac Newton - (* 4. January 1643 in Woolsthorpe; † 31. March 1727 in Kensington) • Physicist, Mathematician, Astronomer and Philosopher • Together with Leibniz, Newton is one of the inventors of infinitesimal calculus (differentiation and integration) • 1667: Fellow of Trinity College, Cambridge • 1687: Philosophiae Naturalis Principia Mathematica.Newton discovered the universal law of gravitation and the laws of motion of classical mechanics • 1704: Opticks. A corpuscular theory of light. • From 1703 president of the Royal Society • Buried in Westminster abbey in London

9. The Gaussian Quadrature Formulae The previously derived formulae were all of the form and were very easy to obtain since an equidistant spacing of the nodes was imposed and only the weights wj had to be computed. The aim of the Gaussian quadrature formulae is now to obtain an optimal quadrature formula with a given number of points by making also the nodes an unknown in the derivation procedure of the quadrature formula and to come up with an optimal set of nodes xj and weights wj. An explicit construction strategy for Gaussian integration formulae: (1) Using M quadrature points, we have M unknowns for the positions and also M unknowns for the weights, i.e. a total of 2M unknowns. (2) We need 2M equations to determine uniquely the 2M unknowns. (3) The equations are obtained by requiring that the integration formula isexact for polynomials from degree 0 up to degree 2M-1 !

10. The Gaussian Quadrature Formulae This means we have 2M equations of the form to solve for wj and xj. For Pi(x), any polynomial of degree i can be used, in particular also the monomial xi. Example 1: One integration point, i.e. M = 1, leading to the two equations:

11. The Gaussian Quadrature Formulae Example 2: Two integration points, i.e. M = 2, leading to the 4 equations:

12. The Gaussian Quadrature Formulae A more efficient and more general way of obtaining the Gaussian quadrature formulaemakes use of so-called orthogonal polynomials Li(x), which are the so-called Legendrepolynomials. First, we define the scalar-product of two functions f and g as follows: With this scalar product available, we can define the L2norm of a function f as The set of polynomials Li(x) is called orthogonal, if it satisfies the relation

13. The Gaussian Quadrature Formulae The polynomials Li(x) can be constructed via Gram-Schmidt orthogonalization from the monomials M0 = 1, M1 = x2, M3 = x3, … Mn = x nas follows: We first use the analogy of the scalar product of two functions with thescalar product already known for vectors: The Gram-Schmidt orthogonalization then proceeds as follows:

14. The Gaussian Quadrature Formulae Instead of performing orthogonalization of vectors, we now perform the orthogonalization of functions as follows:

15. The Gaussian Quadrature Formulae Using the orthogonality property of the Legendre polynomials, we find that (1) The Gaussian quadrature formulae are written as (2) If we now apply formula (2) to the integrals given in (1), we obtain the following linear equation system for the weights wj, if we suppose the positions xj to be known: (3) (3‘)

16. The Gaussian Quadrature Formulae Theorem: If the n positions xj are given be the nroots of the polynomialLn ( Ln( xj ) = 0 ), and the weights are given by the solution of system (3), then the Gaussian quadrature rules are exact for polynomials up to degree 2n-1, i.e. Proof: Suppose p(x) is an arbitrary polynomial of the space of polynomials of degree 2n-1, i.e. Then we can write the polynomial p(x) as

17. The Gaussian Quadrature Formulae Proof (continued): We have We also have This finishes the proof. QED

18. Integration of Improper Integrals If the integration interval goes to infinity, it can be very useful to change the integration variables use the following substitution: Example: If the integrand is singular at a known position c, than it is usually useful to split the integral as: Note: Gaussian quadrature formulae never use the interval endpoints,which makes them very useful for the computation of improper integrals!