1. M. Dumbser 1 / 18 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser Lecture on Numerical Analysis Dr.-Ing. Michael Dumbser 24 / 09 / 2008

2. M. Dumbser 2 / 18 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser Numerical Integration (Quadrature) of Functions - Motivation ab x f h Task: compute approximately 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

3. M. Dumbser 3 / 18 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser 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. M. Dumbser 4 / 18 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser The Newton-Cotes Formulae The integral of f(x) is approximated in the following steps: 1.First, the function f(x) is interpolated by a polynomial of degree k inside each sub-interval. The interpolation points are distributed in an equidistant manner in each sub-interval. 2.Second, the interpolation polynomial is integrated analytically. 3.Steps 1 and 2 produce an approximation of the integral of f(x) in terms of the function values f i at the interpolation points and the step size h.

5. M. Dumbser 5 / 18 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser The Trapezium or Trapezoid Rule a b f h Use linear interpolation polynomials, i.e. polynomials of degree one in the subintervals [x i ;x i+1 ] [0;1]: x

6. M. Dumbser 6 / 18 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser The Simpson Rule (Originally Discovered by Johannes Kepler in 1615) a b f h Use quadratic interpolation polynomials, i.e. polynomials of degree two: x

7. M. Dumbser 7 / 18 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser The 3/8 Rule Use cubic interpolation polynomials, i.e. polynomials of degree three:

8. M. Dumbser 8 / 18 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser 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. M. Dumbser 9 / 18 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser 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 j 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 x j and weights w j. 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 is exact for polynomials from degree 0 up to degree 2M-1 !

10. M. Dumbser 10 / 18 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser The Gaussian Quadrature Formulae This means we have 2M equations of the form to solve for j and j. For P i ( ), any polynomial of degree i can be used, in particular also the monomial i. Example 1: One integration point, i.e. M = 1, leading to the two equations:

11. M. Dumbser 11 / 18 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser The Gaussian Quadrature Formulae Example 2: Two integration points, i.e. M = 2, leading to the 4 equations:

12. M. Dumbser 12 / 18 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser The Gaussian Quadrature Formulae A more efficient and more general way of obtaining the Gaussian quadrature formulae makes use of so-called orthogonal polynomials L i ( ), which are the so-called Legendre polynomials. The set of polynomials L i ( ) is called orthogonal, if it satisfies the relation First, we define the scalar-product of two functions f and g as follows: With this scalar product available, we can define the L 2 norm of a function f as

13. M. Dumbser 13 / 18 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser The Gaussian Quadrature Formulae The polynomials L i ( ) can be constructed via Gram-Schmidt orthogonalization from the monomials M 0 = 1, M 1 = 2, M 3 = 3, … M n = n as follows: We first use the analogy of the scalar product of two functions with the scalar product already known for vectors: The Gram-Schmidt orthogonalization then proceeds as follows:

14. M. Dumbser 14 / 18 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser The Gaussian Quadrature Formulae Instead of performing orthogonalization of vectors, we now perform the orthogonalization of functions as follows:

15. M. Dumbser 15 / 18 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser The Gaussian Quadrature Formulae Using the orthogonality property of the Legendre polynomials, we find that The Gaussian quadrature formulae are written as If we now apply formula (2) to the integrals given in (1), we obtain the following linear equation system for the weights j, if we suppose the positions j to be known: (1) (2) (3)

16. M. Dumbser 16 / 18 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser The Gaussian Quadrature Formulae Theorem: If the n positions j are given be the n roots of the polynomial L n ( L n ( j ) = 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( ) is an arbitrary polynomial of the space of polynomials of degree 2n-1, i.e. Then we can write the polynomial p( ) as

17. M. Dumbser 17 / 18 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser The Gaussian Quadrature Formulae Proof (continued): We have We also have This finishes the proof. QED

18. M. Dumbser 18 / 18 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser 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!