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ChE 250 Numeric Methods. Lecture #20 Part 6 Introduction Chapra, Chapter 21: Newton-Cotes Chapter 22: Integration of Equations 20070326. Class Schedule. Two weeks of lecture left This week, Chapter 21-23 Next week, Chapter 24-25 We need to schedule your final exam

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che 250 numeric methods

ChE 250 Numeric Methods

Lecture #20

Part 6 Introduction

Chapra, Chapter 21: Newton-Cotes

Chapter 22: Integration of Equations

20070326

class schedule
Class Schedule
  • Two weeks of lecture left
    • This week, Chapter 21-23
    • Next week, Chapter 24-25
  • We need to schedule your final exam
    • Cumulative chapters 1 through 22
    • Choices: 11th, 13th,…other?
  • Project 2
    • Due April 20th
    • Should be posted on course webpage next weekend
today s lecture
Today’s Lecture
  • Introduction
    • What integration and differentiation is used for
  • Chapter 21: Newton-Cotes Integration
    • Trapezoid Rule
    • Simpson’s 1/3 Rule
    • Simpson’s 3/8 Rule
    • Composite Methods
  • Chapter 22: Integration of Equations
    • Romberg Integration
numeric integration differentiation
Numeric Integration & Differentiation
  • Many times, it is not efficient to analytically derive the solution to an integral problem so numeric methods are used
  • If analyzing a data set, two approaches are available:
    • you can use interval methods like the trapezoidal rule or Simpson’s
    • you can use techniques discussed in part 5 to fit the data to a curve, then use calculus to calculate the definite integral
numeric integration differentiation5
Numeric Integration & Differentiation
  • If the derivative is needed there are finite difference methods that can be used
  • Finite difference works for data or functions
    • it is preferred to use a curve instead of raw so that error does not propagate
newton cotes integration formulas
Newton-Cotes Integration Formulas
  • For data or a function, the easy way to integrate is to approximate the values in the range of interest with a polynomial, then integrate the polynomial
    • The trapezoidal method uses first-order polynomial (straight lines) to connect adjacent points
trapezoidal rule
Trapezoidal Rule
  • For a function that can be evaluated throughout the range from x0 to xn, where n is the number of equal base trapezoids, we can calculate the area:
trapezoidal rule8
Trapezoidal Rule
  • The error of approximating a curve with a straight line can be large
  • To reduce error, we can increase the number of intervals, however, round off error will eventually dominate, so a reasonable number for n is 100-200
  • Example 21.2
  • Questions?
simpson s rules
Simpson’s Rules
  • For greater accuracy, each interval could be estimated using a higher order polynomial instead of a straight line

Simpson’s 3/8

Simpson’s 1/3

simpson s 1 3 rule
Simpson’s 1/3 Rule
  • Approximating the curve with a second order Lagrange polynomial
simpson s 1 3 rule11
Simpson’s 1/3 Rule
  • So, this will break n points into n/2 even base parabolic intervals (n must be even)
  • Example 21.5
  • Questions?
simpson s 3 8 rule
Simpson’s 3/8 Rule
  • If a third order Lagrange interpolating polynomial is used, 4 data points are used for each interval
  • Because the accuracy is not much higher than the 1/3 rule and more points calculations are required, typically 3/8 is not used for the entire interval
multiple rules
Multiple ‘Rules’
  • For a set of data with odd intervals, one set of four points is used with the 3/8 Rule and the remaining even number of intervals use 1/3 Rule
  • This method becomes more important with unevenly spaced intervals
newton cotes formulas
Newton-Cotes Formulas
  • The formulas are summarized on page 604 along with the truncation error estimates
  • Example 21.7, 21.8
integration of equations
Integration of Equations
  • Romberg Integration is based on the trapezoidal rule, but it uses a recursive approach to refine the integral estimate quickly based on the values calculated by successive n=2j trapezoidal rule estimates
integration of equations16
Integration of Equations
  • This method uses a call to a trapaziod subroutine to generate estimates
  • The estimates are then combined to form improved estimates with lower error n-1 times
integration of equations17
Integration of Equations
  • This process is repeated until the approximation error between successive iterations falls below a stopping criterion
  • Example 22.10
  • Questions?
preparation for 28mar
Preparation for 28Mar
  • Reading
    • Chapter 22: Integration of Equations
    • Chapter 23: Numeric Differentiation
  • Homework: Due Friday April 6
    • Chapter 21
      • 9, 13, 20
    • Chapter 22
      • 3, 7, 14
    • Chapter 23
      • 1, 9, 12, 13, 14, 27
    • Chapter 24
      • 4, 5, 12
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