CISE301_Topic7KFUPM1 SE301: Numerical Methods Topic 7 Numerical Integration Lecture KFUPM Read Chapter 21, Section 1 Read Chapter 22, Sections 2-3

CISE301_Topic7KFUPM2 L ecture 24 Introduction to Numerical Integration  Definitions  Upper and Lower Sums  Trapezoid Method (Newton-Cotes Methods)  Romberg Method  Gauss Quadrature  Examples

CISE301_Topic7KFUPM3 Integration Indefinite Integrals Indefinite Integrals of a function are functions that differ from each other by a constant. Definite Integrals Definite Integrals are numbers.

CISE301_Topic7KFUPM4 Fundamental Theorem of Calculus

CISE301_Topic7KFUPM5 The Area Under the Curve One interpretation of the definite integral is: Integral = area under the curve ab f(x)

CISE301_Topic7KFUPM6 Upper and Lower Sums ab f(x) The interval is divided into subintervals.

CISE301_Topic7KFUPM7 Upper and Lower Sums ab f(x)

CISE301_Topic7KFUPM8 Example

CISE301_Topic7KFUPM9 Example

CISE301_Topic7KFUPM10 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.

CISE301_Topic7KFUPM11 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.

CISE301_Topic7KFUPM12 Newton-Cotes Methods Trapezoid Method ( First Order Polynomials are used ) Simpson 1/3 Rule ( Second Order Polynomials are used )

CISE301_Topic7KFUPM13 L ecture 25 Trapezoid Method  Derivation-One Interval  Multiple Application Rule  Estimating the Error  Recursive Trapezoid Method Read 21.1

CISE301_Topic7KFUPM14 Trapezoid Method f(x)

CISE301_Topic7KFUPM15 Trapezoid Method Derivation-One Interval

CISE301_Topic7KFUPM16 Trapezoid Method f(x)

CISE301_Topic7KFUPM17 Trapezoid Method Multiple Application Rule ab f(x) x

CISE301_Topic7KFUPM18 Trapezoid Method General Formula and Special Case

CISE301_Topic7KFUPM19 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

CISE301_Topic7KFUPM20 Example 1 Time (s) Velocity (m/s)

CISE301_Topic7KFUPM21 Estimating the Error For Trapezoid Method

CISE301_Topic7KFUPM22 Error in estimating the integral Theorem

CISE301_Topic7KFUPM23 Example

CISE301_Topic7KFUPM24 Example x f(x)

CISE301_Topic7KFUPM25 Example x f(x)

CISE301_Topic7KFUPM26 Recursive Trapezoid Method f(x)

CISE301_Topic7KFUPM27 Recursive Trapezoid Method f(x) Based on previous estimate Based on new point

CISE301_Topic7KFUPM28 Recursive Trapezoid Method f(x) Based on previous estimate Based on new points

CISE301_Topic7KFUPM29 Recursive Trapezoid Method Formulas

CISE301_Topic7KFUPM30 Recursive Trapezoid Method

CISE301_Topic7KFUPM31 Advantages of Recursive Trapezoid Recursive Trapezoid:  Gives the same answer as the standard Trapezoid method.  Makes use of the available information to reduce the computation time.  Useful if the number of iterations is not known in advance.

CISE301_Topic7KFUPM32 L ecture 26 Romberg Method  Motivation  Derivation of Romberg Method  Romberg Method  Example  When to stop? Read 22.2

CISE301_Topic7KFUPM33 Motivation for Romberg Method  Trapezoid formula with an interval h gives an error of the order O(h 2 ).  We can combine two Trapezoid estimates with intervals 2h and h to get a better estimate.

CISE301_Topic7KFUPM34 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

CISE301_Topic7KFUPM35 First Column Recursive Trapezoid Method

CISE301_Topic7KFUPM36 Derivation of Romberg Method

CISE301_Topic7KFUPM37 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)

CISE301_Topic7KFUPM38 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

CISE301_Topic7KFUPM39 Example /81/3

CISE301_Topic7KFUPM40 Example 1 (cont.) 0.5 3/81/3 11/321/3

CISE301_Topic7KFUPM41 When do we stop?

CISE301_Topic7KFUPM42 L ecture 27 Gauss Quadrature  Motivation  General integration formula Read 22.3

CISE301_Topic7KFUPM43 Motivation

CISE301_Topic7KFUPM44 General Integration Formula

CISE301_Topic7KFUPM45 Lagrange Interpolation

CISE301_Topic7KFUPM46 Question What is the best way to choose the nodes and the weights?

CISE301_Topic7KFUPM47 Theorem

CISE301_Topic7KFUPM48 Weighted Gaussian Quadrature Theorem

CISE301_Topic7KFUPM49 Determining The Weights and Nodes

CISE301_Topic7KFUPM50 Determining The Weights and Nodes Solution

CISE301_Topic7KFUPM51 Determining The Weights and Nodes Solution

CISE301_Topic7KFUPM52 Determining The Weights and Nodes Solution

CISE301_Topic7KFUPM53 Gaussian Quadrature See more in Table 22.1 (page 626)

CISE301_Topic7KFUPM54 Error Analysis for Gauss Quadrature

CISE301_Topic7KFUPM55 Example

CISE301_Topic7KFUPM56 Example

CISE301_Topic7KFUPM57 Improper Integrals