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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

Lecture #20

Part 6 Introduction

Chapra, Chapter 21: Newton-Cotes

Chapter 22: Integration of Equations

20070326


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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


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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


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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


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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


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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


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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:


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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?


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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


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Simpson’s 1/3 Rule

  • Approximating the curve with a second order Lagrange polynomial


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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?


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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


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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


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Newton-Cotes Formulas

  • The formulas are summarized on page 604 along with the truncation error estimates

  • Example 21.7, 21.8


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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


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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


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Integration of Equations

  • This process is repeated until the approximation error between successive iterations falls below a stopping criterion

  • Example 22.10

  • Questions?


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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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