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1 Chapter 5 Numerical Integration

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2 A Review of the Definite Integral

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

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

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

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6 Example 5.1

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7 Example 5.2

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8 h4h4

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9 Approximate Corrected Trapezoid Rule p. 176

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11 5.3 Simpson’s Rule and Degree of Precision

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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!!

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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

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25 5.4 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 f (1/4)f (3/4)

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29 h2h2

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31 5.5 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

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32 5.6 Gaussian Quadrature Gaussian quadrature is a very powerful tool for approximating integrals. The quadrature ( 求面積 ) rules are all base on special values of weights and Gauss points. The quadrature rule is written in the form weights Gauss points

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34 Example 5.8

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35 Question k

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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 Gauss points) equals the number of equations. Taking N=2n will yield a contradiction.

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37 Only to find an example

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38 Finding weights

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39 Finding Gauss Points

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40 Legendre polynomials

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41 Orthogonal polynomials

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42 Theorem 5.3

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43 Theorem 5.4 Any polynomial of degree n-1 can be interpolated by Lagrange interpolation with n data points

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45 Other Intervals, Other Rules

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46 Example 5.9 Table 5.5

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