1 Chapter 5 Numerical Integration

2 A Review of the Definite Integral

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

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

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

6 Example 5.1

7 Example 5.2

8 h4h4

9 Approximate Corrected Trapezoid Rule

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Simpson’s Rule and Degree of Precision

12 let

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14 Example 5.3

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16 The Composite Rule Assume Example 5.4

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18 Example 5.5 h4h4 It’s OK!!

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”?

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24 Example 5.6

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

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28 Example 5.7

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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

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

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34 Example 5.8

35 Question k ×

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 Only to find an example

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39 Finding Gauss Points

40 Theorem 5.3

41 Theorem 5.4

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43 Other Intervals, Other Rules

44 Example 5.9 Table 5.5

45 Error estimation!! See exercise!!

46 Example 5.10

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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 Approximation

50 Estimating p

51 Example 5.11

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53 Example 5.12

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

56 Example 5.13

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