EE3561_Unit 7Al-Dhaifallah EE 3561 : Computational Methods Unit 7 Numerical Integration Dr. Mujahed AlDhaifallah ( Term 342)
EE3561_Unit 7Al-Dhaifallah14352 Lecture 19 Introduction to Numerical Integration Definitions Upper and Lower Sums Trapezoid Method Examples
EE3561_Unit 7Al-Dhaifallah14353 Integration Indefinite Integrals Indefinite Integrals of a function are functions that differ from each other by a constant. Definite Integrals Definite Integrals are numbers.
EE3561_Unit 7Al-Dhaifallah14354 Fundamental Theorem of Calculus
EE3561_Unit 7Al-Dhaifallah14355 The Area Under the Curve One interpretation of the definite integral is Integral = area under the curve ab f(x)
EE3561_Unit 7Al-Dhaifallah14356 Riemann Sums
EE3561_Unit 7Al-Dhaifallah14357 Numerical Integration Methods Numerical integration Methods Covered in this course Upper and Lower Sums Newton-Cotes Methods: Trapezoid Rule Romberg Method
EE3561_Unit 7Al-Dhaifallah14358 Upper and Lower Sums ab f(x) The interval is divided into subintervals
EE3561_Unit 7Al-Dhaifallah14359 Upper and Lower Sums ab f(x)
EE3561_Unit 7Al-Dhaifallah Example
EE3561_Unit 7Al-Dhaifallah Example
EE3561_Unit 7Al-Dhaifallah 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.
EE3561_Unit 7Al-Dhaifallah 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.
EE3561_Unit 7Al-Dhaifallah Newton-Cotes Methods Trapezoid Method ( First Order Polynomial are used ) Simpson 1/3 Rule ( Second Order Polynomial are used ),
EE3561_Unit 7Al-Dhaifallah Trapezoid Method f(x)
EE3561_Unit 7Al-Dhaifallah Trapezoid Method Derivation-One interval
EE3561_Unit 7Al-Dhaifallah Trapezoid Method f(x)
EE3561_Unit 7Al-Dhaifallah Trapezoid Method Multiple Application Rule ab f(x) x
EE3561_Unit 7Al-Dhaifallah Trapezoid Method General Formula and special case
EE3561_Unit 7Al-Dhaifallah 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
EE3561_Unit 7Al-Dhaifallah Example Time (s) Velocity (m/s)
EE3561_Unit 7Al-Dhaifallah Estimating the Error For Trapezoid method
EE3561_Unit 7Al-Dhaifallah Error in estimating the integral Theorem
EE3561_Unit 7Al-Dhaifallah Example
EE3561_Unit 7Al-Dhaifallah Example x f(x)
EE3561_Unit 7Al-Dhaifallah Example x f(x)
EE3561_Unit 7Al-Dhaifallah Lecture 21 Romberg Method Motivation Derivation of Romberg Method Romberg Method Example When to stop?
EE3561_Unit 7Al-Dhaifallah 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 h and h/2 to get a better estimate.
EE3561_Unit 7Al-Dhaifallah 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
EE3561_Unit 7Al-Dhaifallah First Column Recursive Trapezoid Method f(x)
EE3561_Unit 7Al-Dhaifallah Recursive Trapezoid Method f(x) Based on previous estimate Based on new point
EE3561_Unit 7Al-Dhaifallah Recursive Trapezoid Method f(x) Based on previous estimate Based on new points
EE3561_Unit 7Al-Dhaifallah Recursive Trapezoid Method Formulas
EE3561_Unit 7Al-Dhaifallah Recursive Trapezoid Method
EE3561_Unit 7Al-Dhaifallah Derivation of Romberg Method
EE3561_Unit 7Al-Dhaifallah 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)
EE3561_Unit 7Al-Dhaifallah 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
EE3561_Unit 7Al-Dhaifallah Example /81/3
EE3561_Unit 7Al-Dhaifallah Example 1 cont /81/3 11/321/3
EE3561_Unit 7Al-Dhaifallah When do we stop?