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1 Chapter 5 Numerical Integration. 2 A Review of the Definite Integral.

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Presentation on theme: "1 Chapter 5 Numerical Integration. 2 A Review of the Definite Integral."— Presentation transcript:

1 1 Chapter 5 Numerical Integration

2 2 A Review of the Definite Integral

3 3 Riemann Sum  A summation of the form is called a Riemann sum.

4 4 5.2 Improving the Trapezoid Rule  The trapezoid rule for computing integrals:  The error:

5 5 5.2 Improving the Trapezoid Rule  So that  Therefore,  Error estimation:  Improvement of the approximation: the corrected trapezoid rule

6 6 Example 5.1

7 7 Example 5.2

8 8 h4h4

9 9 Approximate Corrected Trapezoid Rule

10 10

11 Simpson’s Rule and Degree of Precision

12 12 let

13 13

14 14 Example 5.3

15 15

16 16 The Composite Rule Assume Example 5.4

17 17

18 18 Example 5.5 h4h4 It’s OK!!

19 19 Discussion  From our experiments:  From the definition of Simpson’s rule:  Why? Why Simpson’s rule is “more accurate than it ought to be”?

20 20

21 21

22 22

23 23

24 24 Example 5.6

25 The Midpoint Rule  Consider the integral:  And the Taylor approximation:  The midpoint rule:  Its composite rule: because

26 26

27 27

28 28 Example 5.7

29 29

30 30

31 Application: Stirling’s Formula  Stirling’s formula is an interesting and useful way to approximate the factorial function, n!, for large values of n. Use Stirling ’ s formula to show that for all x. Example

32 Gaussian Quadrature  Gaussian quadrature is a very powerful tool for approximating integrals.  The quadrature rules are all base on special values of weights and abscissas (called Gauss points)  The quadrature rule is written in the form weights Gauss points

33 33

34 34 Example 5.8

35 35 Question k ×

36 36 Discussion  The high accuracy of Gaussian quadrature then comes from the fact that it integrates very-high-degree polynomials exactly.  We should choose N=2n-1, because a polynomial of degree 2n-1 has 2n coefficients, and thus the number of unknowns ( n weights plus n abscissas) equals the number of equations.  Taking N=2n will yield a contradiction.

37 37 Only to find an example

38 38

39 39 Finding Gauss Points

40 40 Theorem 5.3

41 41 Theorem 5.4

42 42

43 43 Other Intervals, Other Rules

44 44 Example 5.9 Table 5.5

45 45 Error estimation!! See exercise!!

46 46 Example 5.10

47 47

48 Extrapolation Methods  One of the most important ideas in computational mathematics is— We can take the information from a few approximations and Use that to both estimate the error in the approximation and generate a significantly improved approximation  In this section we will embark on a more detailed study of some of these ideas.

49 49 Approximation

50 50 Estimating p

51 51 Example 5.11

52 52

53 53 Example 5.12

54 54

55 55 Error Estimation and an Improved Approximation I2n-InI2n-In

56 56 Example 5.13

57 57 Example 5.14 O(h 2 ): Error of trapezoid rule O(h 4 ): Error of Richardson extrapolation


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