CISE301_Topic71 SE301: Numerical Methods Topic 7 Numerical Integration Lecture KFUPM (Term 101) Section 04 Read Chapter 21, Section 1 Read Chapter 22, Sections 2-3

CISE301_Topic72 L ecture 24 Introduction to Numerical Integration  Definitions  Upper and Lower Sums  Trapezoid Method (Newton-Cotes Methods)  Romberg Method  Gauss Quadrature  Examples

CISE301_Topic73 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_Topic74 Fundamental Theorem of Calculus

CISE301_Topic75 The Area Under the Curve One interpretation of the definite integral is: Integral = area under the curve ab f(x)

CISE301_Topic76 Upper and Lower Sums ab f(x) The interval is divided into subintervals.

CISE301_Topic77 Upper and Lower Sums ab f(x)

CISE301_Topic78 Example

9 Example

10 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_Topic711 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_Topic712 Newton-Cotes Methods Trapezoid Method ( First Order Polynomials are used ) Simpson 1/3 Rule ( Second Order Polynomials are used )

CISE301_Topic713 L ecture 25 Trapezoid Method  Derivation-One Interval  Multiple Application Rule  Estimating the Error  Recursive Trapezoid Method Read 21.1

CISE301_Topic714 Trapezoid Method f(x)

CISE301_Topic715 Trapezoid Method Derivation-One Interval

CISE301_Topic716 Trapezoid Method f(x)

CISE301_Topic717 Trapezoid Method Multiple Application Rule ab f(x) x

CISE301_Topic718 Trapezoid Method General Formula and Special Case

CISE301_Topic719 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_Topic720 Example 1 Time (s) Velocity (m/s)

CISE301_Topic721 Error in estimating the integral Theorem

CISE301_Topic722 Estimating the Error For Trapezoid Method

CISE301_Topic723 Example

CISE301_Topic724 Example x f(x)

CISE301_Topic725 Example x f(x)

CISE301_Topic726 Recursive Trapezoid Method f(x)

CISE301_Topic727 Recursive Trapezoid Method f(x) Based on previous estimate Based on new point

CISE301_Topic728 Recursive Trapezoid Method f(x) Based on previous estimate Based on new points

CISE301_Topic729 Recursive Trapezoid Method Formulas

CISE301_Topic730 Recursive Trapezoid Method

Example on Recursive Trapezoid CISE301_Topic731 nhR(n,0) 0 (b-a)=/2(/4)[sin(0) + sin(/2)]= (b-a)/2=/4R(0,0)/2 + (/4) sin(/4) = (b-a)/4=/8R(1,0)/2 + (/8)[sin(/8)+sin(3/8)] = (b-a)/8=/16R(2,0)/2 + (/16)[sin(/16)+sin(3/16)+sin(5/16)+ sin(7/16)] = Estimated Error = |R(3,0) – R(2,0)| =

CISE301_Topic732 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_Topic733 L ecture 26 Romberg Method  Motivation  Derivation of Romberg Method  Romberg Method  Example  When to stop? Read 22.2

CISE301_Topic734 Motivation for Romberg Method  Trapezoid formula with a sub-interval h gives an error of the order O(h 2 ).  We can combine two Trapezoid estimates with intervals h and h/2 to get a better estimate.

CISE301_Topic735 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_Topic736 First Column Recursive Trapezoid Method

CISE301_Topic737 Derivation of Romberg Method

CISE301_Topic738 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_Topic739 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_Topic740 Example 0.5 3/81/3

CISE301_Topic741 Example (cont.) 0.5 3/81/3 11/321/3

CISE301_Topic742 When do we stop?

CISE301_Topic743 L ecture 27 Gauss Quadrature  Motivation  General integration formula Read 22.3

CISE301_Topic744 Motivation

CISE301_Topic745 General Integration Formula

CISE301_Topic746 Lagrange Interpolation

Example  Determine the Gauss Quadrature Formula of If the nodes are given as (-1, 0, 1)  Solution: First we need to find l 0 (x), l 1 (x), l 2 (x)  Then compute: CISE301_Topic747

Solution CISE301_Topic748

Using the Gauss Quadrature Formula CISE301_Topic749

Using the Gauss Quadrature Formula CISE301_Topic750

CISE301_Topic751 Improper Integrals