SE301_Topic 6Al-Amer20051 SE301:Numerical Methods Topic 6 Numerical Integration Dr. Samir Al-Amer Term 053

SE301_Topic 6Al-Amer20052 Lecture 17 Introduction to Numerical Integration  Definitions  Upper and Lower Sums  Trapezoid Method  Examples

SE301_Topic 6Al-Amer20053 Integration Indefinite Integrals Indefinite Integrals of a function are functions that differ from each other by a constant. Definite Integrals Definite Integrals are numbers.

SE301_Topic 6Al-Amer20054 Fundamental Theorem of Calculus

SE301_Topic 6Al-Amer20055 The Area Under the Curve One interpretation of the definite integral is Integral = area under the curve ab f(x)

SE301_Topic 6Al-Amer20056 Numerical Integration Methods Numerical integration Methods Covered in this course  Upper and Lower Sums  Newton-Cotes Methods:  Trapezoid Rule  Simpson Rules  Romberg Method  Gauss Quadrature

SE301_Topic 6Al-Amer20057 Upper and Lower Sums ab f(x) The interval is divided into subintervals

SE301_Topic 6Al-Amer20058 Upper and Lower Sums ab f(x)

SE301_Topic 6Al-Amer20059 Example

SE301_Topic 6Al-Amer Example

SE301_Topic 6Al-Amer Upper and Lower Sums Estimates based on Upper and Lower Sums are easy to obtain for monotonic functions (always increasing or always decreasing). For non-monotonic functions, finding maximum and minimum of the function can be difficult and other methods can be more attractive.

SE301_Topic 6Al-Amer Newton-Cotes Methods  In Newton-Cote Methods, the function is approximated by a polynomial of order n  Computing the integral of a polynomial is easy.

SE301_Topic 6Al-Amer Newton-Cotes Methods Trapezoid Method ( First Order Polynomial are used ) Simpson 1/3 Rule ( Second Order Polynomial are used ),

SE301_Topic 6Al-Amer Trapezoid Method f(x)

SE301_Topic 6Al-Amer Trapezoid Method Derivation-One interval

SE301_Topic 6Al-Amer Trapezoid Method f(x)

SE301_Topic 6Al-Amer Trapezoid Method Multiple Application Rule ab f(x) x

SE301_Topic 6Al-Amer Trapezoid Method General Formula and special case

SE301_Topic 6Al-Amer Example Given a tabulated values of the velocity of an object. Obtain an estimate of the distance traveled in the interval [0,3]. Time (s) Velocity (m/s) Distance = integral of the velocity

SE301_Topic 6Al-Amer Example 1 Time (s) Velocity (m/s)

SE301_Topic 6Al-Amer Estimating the Error For Trapezoid method

SE301_Topic 6Al-Amer Error in estimating the integral Theorem

SE301_Topic 6Al-Amer Example

SE301_Topic 6Al-Amer Example x f(x)

SE301_Topic 6Al-Amer Example x f(x)

SE301_Topic 6Al-Amer SE301: Numerical Method Lecture 18 Recursive Trapezoid Method Recursive formula is used

SE301_Topic 6Al-Amer Recursive Trapezoid Method f(x)

SE301_Topic 6Al-Amer Recursive Trapezoid Method f(x) Based on previous estimate Based on new point

SE301_Topic 6Al-Amer Recursive Trapezoid Method f(x) Based on previous estimate Based on new points

SE301_Topic 6Al-Amer Recursive Trapezoid Method Formulas

SE301_Topic 6Al-Amer Recursive Trapezoid Method

SE301_Topic 6Al-Amer Advantages of Recursive Trapezoid Recursive Trapezoid:  Gives the same answer as the standard Trapezoid method.  Make use of the available information to reduce computation time.  Useful if the number of iterations is not known in advance.

SE301_Topic 6Al-Amer SE301:Numerical Methods 19. Romberg Method Motivation Derivation of Romberg Method Romberg Method Example When to stop?

SE301_Topic 6Al-Amer Motivation for Romberg Method  Trapezoid formula with an interval h gives error of the order O(h 2 )  We can combine two Trapezoid estimates with intervals 2h and h to get a better estimate.

SE301_Topic 6Al-Amer Romberg Method First column is obtained using Trapezoid Method R(0,0) R(1,0)R(1,1) R(2,0)R(2,1)R(2,2) R(3,0)R(3,1)R(3,2)R(3,3) The other elements are obtained using the Romberg Method

SE301_Topic 6Al-Amer First Column Recursive Trapezoid Method

SE301_Topic 6Al-Amer Derivation of Romberg Method

SE301_Topic 6Al-Amer Romberg Method R(0,0) R(1,0)R(1,1) R(2,0)R(2,1)R(2,2) R(3,0)R(3,1)R(3,2)R(3,3)

SE301_Topic 6Al-Amer Property of Romberg Method R(0,0) R(1,0)R(1,1) R(2,0)R(2,1)R(2,2) R(3,0)R(3,1)R(3,2)R(3,3) Error Level

SE301_Topic 6Al-Amer Example /81/3

SE301_Topic 6Al-Amer Example 1 cont /81/3 11/321/3

SE301_Topic 6Al-Amer When do we stop?

SE301_Topic 6Al-Amer SE301:Numerical Methods 20. Gauss Quadrature Motivation General integration formula Read

SE301_Topic 6Al-Amer Motivation

SE301_Topic 6Al-Amer General Integration Formula

SE301_Topic 6Al-Amer Lagrange Interpolation

SE301_Topic 6Al-Amer Question What is the best way to choose the nodes and the weights?

SE301_Topic 6Al-Amer Theorem

SE301_Topic 6Al-Amer Weighted Gaussian Quadrature Theorem

SE301_Topic 6Al-Amer Determining The Weights and Nodes

SE301_Topic 6Al-Amer Determining The Weights and Nodes Solution

SE301_Topic 6Al-Amer Theorem

SE301_Topic 6Al-Amer Determining The Weights and Nodes Solution

SE301_Topic 6Al-Amer Determining The Weights and Nodes Solution

SE301_Topic 6Al-Amer Gaussian Quadrature See more in Table 22.1 (page 626) 626

SE301_Topic 6Al-Amer Error Analysis for Gauss Quadrature 627

SE301_Topic 6Al-Amer Example

SE301_Topic 6Al-Amer Example

SE301_Topic 6Al-Amer Improper Integrals 627

SE301_Topic 6Al-Amer Quiz x f(x)