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# che 250 numeric methods - PowerPoint PPT Presentation

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

• 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

• 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

• 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

• 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

• 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

• 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:

• 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?

• 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

• Approximating the curve with a second order Lagrange polynomial

• So, this will break n points into n/2 even base parabolic intervals (n must be even)

• Example 21.5

• Questions?

• 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

• 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

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

• Example 21.7, 21.8

• 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

• 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

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

• Example 22.10

• Questions?

• 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