Integration. Topic: Trapezoidal Rule. Major: General Engineering. Author: Autar Kaw, Charlie Barker.
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1 Integration Topic: Trapezoidal Rule Major: General Engineering Author: Autar Kaw, Charlie Barker 1
2 What is Integration Integration: The process of measuring the area under a function plotted on a graph. Where: f(x) is the integrand a= lower limit of integration b= upper limit of integration 2
3 Basis of Trapezoidal Rule Trapezoidal Rule is based on the Newton-Cotes Formula that states if one can approximate the integrand as an n th order polynomial where and 3
4 Basis of Trapezoidal Rule Then the integral of that function is approximated by the integral of that n th order polynomial. Trapezoidal Rule assumes n=1, that is, the area under the linear polynomial, 4
5 Derivation of the Trapezoidal Rule 5
6 Method Derived From Geometry The area under the curve is a trapezoid. The integral 6
7 Example 1 The vertical distance covered by a rocket from t=8 to t=30 seconds is given by: a) Use single segment Trapezoidal rule to find the distance covered. b) Find the true error, for part (a). c) Find the absolute relative true error, for part (a). 7
8 Solution a) 8
9 Solution (cont) a) b) The exact value of the above integral is 9
10 Solution (cont) b) c) The absolute relative true error,, would be 10
11 Multiple Segment Trapezoidal Rule In Example 1, the true error using single segment trapezoidal rule was large. We can divide the interval [8,30] into [8,19] and [19,30] intervals and apply Trapezoidal rule over each segment. 11
![We can divide the interval [8,30] into [8,19] and](/docs-images/46/21254732/images/page_11.jpg "[19,30] intervals and apply Trapezoidal rule over each")
12 With Multiple Segment Trapezoidal Rule Hence: 12
13 Multiple Segment Trapezoidal Rule The true error is: The true error now is reduced from -807 m to -205 m. Extending this procedure to divide the interval into equal segments to apply the Trapezoidal rule; the sum of the results obtained for each segment is the approximate value of the integral. 13
14 Multiple Segment Trapezoidal Rule Divide into equal segments as shown in Figure 4. Then the width of each segment is: The integral I is: Figure 4: Multiple (n=4) Segment Trapezoidal Rule 14
15 Multiple Segment Trapezoidal Rule The integral I can be broken into h integrals as: Applying Trapezoidal rule on each segment gives: 15
16 Example 2 The vertical distance covered by a rocket from to seconds is given by: a) Use two-segment Trapezoidal rule to find the distance covered. b) Find the true error, for part (a). c) Find the absolute relative true error, for part (a). 16
17 Solution a) The solution using 2-segment Trapezoidal rule is 17
18 Solution (cont) Then: 18
19 Solution (cont) b) The exact value of the above integral is so the true error is 19
20 Solution (cont) c) The absolute relative true error,, would be 20
21 Solution (cont) Table 1 gives the values obtained using multiple segment Trapezoidal rule for: n Value E t Table 1: Multiple Segment Trapezoidal Rule Values 21
22 Example 3 Use Multiple Segment Trapezoidal Rule to find the area under the curve. from to Using two segments, we get and 22
23 Solution Then: 23
24 Solution (cont) So what is the true value of this integral? Making the absolute relative true error: 24
25 Solution (cont) Table 2: Values obtained using Multiple Segment Trapezoidal Rule for: 25 n Approximate Value % % % % % % %
26 Error in Multiple Segment Trapezoidal Rule The true error for a single segment Trapezoidal rule is given by: where is some point in What is the error, then in the multiple segment Trapezoidal rule? It will be simply the sum of the errors from each segment, where the error in each segment is that of the single segment Trapezoidal rule. The error in each segment is 26
27 Error in Multiple Segment Trapezoidal Rule Similarly: It then follows that: 27
28 Error in Multiple Segment Trapezoidal Rule Hence the total error in multiple segment Trapezoidal rule is. The term is an approximate average value of the Hence: 28
29 Error in Multiple Segment Trapezoidal Rule Below is the table for the integral as a function of the number of segments. You can visualize that as the number of segments are doubled, the true error gets approximately quartered. n Value
30 Integration Topic: Simpson s 1/3 rd Rule Major: General Engineering 30
31 Basis of Simpson s 1/3 rd Rule Trapezoidal rule was based on approximating the integrand by a first order polynomial, and then integrating the polynomial in the interval of integration. Simpson s 1/3rd rule is an extension of Trapezoidal rule where the integrand is approximated by a second order polynomial. Hence Where is a second order polynomial. 31
32 Basis of Simpson s 1/3 rd Rule Choose and as the three points of the function to evaluate a 0, a 1 and a 2. 32
33 Basis of Simpson s 1/3 rd Rule Solving the previous equations for a 0, a 1 and a 2 give 33
34 Basis of Simpson s 1/3 rd Rule Then 34
35 Basis of Simpson s 1/3 rd Rule Substituting values of a 0, a 1, a 2 give Since for Simpson s 1/3rd Rule, the interval [a, b] is broken into 2 segments, the segment width 35
36 Basis of Simpson s 1/3 rd Rule Hence Because the above form has 1/3 in its formula, it is called Simpson s 1/3rd Rule. 36
37 Example 1 The distance covered by a rocket from t=8 to t=30 is given by a) Use Simpson s 1/3rd Rule to find the approximate value of x b) Find the true error, c) Find the absolute relative true error, 37
38 Solution a) 38
39 Solution (cont) b) The exact value of the above integral is True Error 39
40 Solution (cont) a)c) Absolute relative true error, 40
41 Multiple Segment Simpson s 1/3rd Rule 41
42 Multiple Segment Simpson s 1/3 rd Rule Just like in multiple segment Trapezoidal Rule, one can subdivide the interval [a, b] into n segments and apply Simpson s 1/3rd Rule repeatedly over every two segments. Note that n needs to be even. Divide interval [a, b] into equal segments, hence the segment width where 42
43 Multiple Segment Simpson s 1/3 rd Rule f(x)..... x x 0 x 2 x n-2 x n Apply Simpson s 1/3rd Rule over each interval, 43
44 Multiple Segment Simpson s 1/3 rd Rule Since 44
45 Multiple Segment Simpson s 1/3 rd Rule Then 45
46 Multiple Segment Simpson s 1/3 rd Rule 46
47 Example 2 Use 4-segment Simpson s 1/3rd Rule to approximate the distance covered by a rocket from t= 8 to t=30 as given by a) Use four segment Simpson s 1/3rd Rule to find the approximate value of x. b) Find the true error, for part (a). c) Find the absolute relative true error, for part (a). 47
48 Solution a) Using n segment Simpson s 1/3rd Rule, So 48
49 Solution (cont.) 49
50 Solution (cont.) cont. 50
51 Solution (cont.) b) In this case, the true error is c) The absolute relative true error 51
52 Solution (cont.) Table 1: Values of Simpson s 1/3rd Rule for Example 2 with multiple segments n Approximate Value E t Є t % % % % % 52
53 Error in the Multiple Segment Simpson s 1/3 rd Rule The true error in a single application of Simpson s 1/3rd Rule is given as In Multiple Segment Simpson s 1/3rd Rule, the error is the sum of the errors in each application of Simpson s 1/3rd Rule. The error in n segment Simpson s 1/3rd Rule is given by 53
54 Error in the Multiple Segment Simpson s 1/3 rd Rule... 54
55 Error in the Multiple Segment Simpson s 1/3 rd Rule Hence, the total error in Multiple Segment Simpson s 1/3rd Rule is 55
56 Error in the Multiple Segment Simpson s 1/3 rd Rule The term is an approximate average value of Hence where 56
57 Integration Topic: Romberg Rule Major: General Engineering 57
58 What is The Romberg Rule? Romberg Integration is an extrapolation formula of the Trapezoidal Rule for integration. It provides a better approximation of the integral by reducing the True Error. 58
59 Error in Multiple Segment Trapezoidal Rule The true error in a multiple segment Trapezoidal Rule with n segments for an integral Is given by 59 where for each i, is a point somewhere in the domain,.
60 Error in Multiple Segment Trapezoidal Rule The term can be viewed as an approximate average value of in. This leads us to say that the true error, E t previously defined can be approximated as 60
61 Error in Multiple Segment Trapezoidal Rule n Value E t Table 1 shows the results obtained for the integral using multiple segment Trapezoidal rule for Table 1: Multiple Segment Trapezoidal Rule Values 61
62 Error in Multiple Segment Trapezoidal Rule The true error gets approximately quartered as the number of segments is doubled. This information is used to get a better approximation of the integral, and is the basis of Richardson s extrapolation. 62
63 Richardson s Extrapolation for Trapezoidal Rule The true error, is estimated as in the n-segment Trapezoidal rule where C is an approximate constant of proportionality. Since Where TV = true value and = approx. value 63
64 Richardson s Extrapolation for Trapezoidal Rule From the previous development, it can be shown that when the segment size is doubled and that which is Richardson s Extrapolation. 64
65 Example 1 The vertical distance covered by a rocket from 8 to 30 seconds is given by a) Use Richardson s rule to find the distance covered. Use the 2-segment and 4-segment Trapezoidal rule results given in Table 1. b) Find the true error, E t for part (a). c) Find the absolute relative true error, for part (a). 65
66 Solution a) Using Richardson s extrapolation formula for Trapezoidal rule and choosing n=2, 66
67 Solution (cont.) b) The exact value of the above integral is Hence 67
68 Solution (cont.) c) The absolute relative true error would then be. Table 2 shows the Richardson s extrapolation results using 1, 2, 4, 8 segments. Results are compared with those of Trapezoidal rule. 68
69 Solution (cont.) Table 2: The values obtained using Richardson s extrapolation formula for Trapezoidal rule for. n Trapezoidal Rule for Trapezoidal Rule Richardson s Extrapolation for Richardson s Extrapolation Table 2: Richardson s Extrapolation Values 69
70 Romberg Integration Romberg integration is same as Richardson s extrapolation formula as given previously. However, Romberg used a recursive algorithm for the extrapolation. Recall This can alternately be written as 70
71 Romberg Integration Note that the variable TV is replaced by as the value obtained using Richardson s extrapolation formula. Note also that the sign is replaced by = sign. Hence the estimate of the true value now is Where Ch 4 is an approximation of the true error. 71
72 Romberg Integration Determine another integral value with further halving the step size (doubling the number of segments), It follows from the two previous expressions that the true value TV can be written as 72
73 Romberg Integration A general expression for Romberg integration can be written as The index k represents the order of extrapolation. k=1 represents the values obtained from the regular Trapezoidal rule, k=2 represents values obtained using the true estimate as O(h 2 ). The index j represents the more and less accurate estimate of the integral. 73
74 Example 2 The vertical distance covered by a rocket from to seconds is given by Use Romberg s rule to find the distance covered. Use the 1, 2, 4, and 8-segment Trapezoidal rule results as given in the Table 1. 74
75 Solution From Table 1, the needed values from original Trapezoidal rule are where the above four values correspond to using 1, 2, 4 and 8 segment Trapezoidal rule, respectively. 75
76 Solution (cont.) To get the first order extrapolation values, Similarly, 76
77 Solution (cont.) For the second order extrapolation values, Similarly, 77
78 Solution (cont.) For the third order extrapolation values, Table 3 shows these increased correct values in a tree graph. 78
79 Solution (cont.) Table 3: Improved estimates of the integral value using Romberg Integration First Order Second Order Third Order 1-segment 2-segment 4-segment 8-segment
80 Integration Topic: Gauss Quadrature Rule of Integration Major: General Engineering 80
81 Two-Point Gaussian Quadrature Rule 81
82 Basis of the Gaussian Quadrature Rule Previously, the Trapezoidal Rule was developed by the method of undetermined coefficients. The result of that development is summarized below. 82
83 Basis of the Gaussian Quadrature Rule The two-point Gauss Quadrature Rule is an extension of the Trapezoidal Rule approximation where the arguments of the function are not predetermined as a and b but as unknowns x 1 and x 2. In the two-point Gauss Quadrature Rule, the integral is approximated as 83
84 Basis of the Gaussian Quadrature Rule The four unknowns x 1, x 2, c 1 and c 2 are found by assuming that the formula gives exact results for integrating a general third order polynomial, Hence 84
85 Basis of the Gaussian Quadrature Rule It follows that Equating Equations the two previous two expressions yield 85
86 Basis of the Gaussian Quadrature Rule Since the constants a 0, a 1, a 2, a 3 are arbitrary 86
87 Basis of Gauss Quadrature The previous four simultaneous nonlinear Equations have only one acceptable solution, 87
88 Basis of Gauss Quadrature Hence Two-Point Gaussian Quadrature Rule 88
89 Higher Point Gaussian Quadrature Formulas 89
90 Higher Point Gaussian Quadrature Formulas is called the three-point Gauss Quadrature Rule. The coefficients c 1, c 2, and c 3, and the functional arguments x 1, x 2, and x 3 are calculated by assuming the formula gives exact expressions for integrating a fifth order polynomial General n-point rules would approximate the integral 90
91 Arguments and Weighing Factors for n-point Gauss Quadrature Formulas In handbooks, coefficients and arguments given for n-point Gauss Quadrature Rule are given for integrals Table 1: Weighting factors c and function arguments x used in Gauss Quadrature Formulas. Points Weighting Factors 2 c 1 = c 2 = Function Arguments x 1 = x 2 = as shown in Table 1. 3 c 1 = c 2 = c 3 = c 1 = c 2 = c 3 = c 4 = x 1 = x 2 = x 3 = x 1 = x 2 = x 3 = x 4 =
92 Arguments and Weighing Factors for n-point Gauss Quadrature Formulas Table 1 (cont.) : Weighting factors c and function arguments x used in Gauss Quadrature Formulas. 92 Points Weighting Factors 5 c 1 = c 2 = c 3 = c 4 = c 5 = c 1 = c 2 = c 3 = c 4 = c 5 = c 6 = Function Arguments x 1 = x 2 = x 3 = x 4 = x 5 = x 1 = x 2 = x 3 = x 4 = x 5 = x 6 =
93 Arguments and Weighing Factors for n-point Gauss Quadrature Formulas So if the table is given for integrals, how does one solve? The answer lies in that any integral with limits of can be converted into an integral with limits Let If If then then Such that: 93
94 Arguments and Weighing Factors for n-point Gauss Quadrature Formulas Then Hence Substituting our values of x, and dx into the integral gives us 94
95 Example 1 For an integral derive the one-point Gaussian Quadrature Rule. Solution The one-point Gaussian Quadrature Rule is 95
96 Solution Assuming the formula gives exact values for integrals and Since the other equation becomes 96
97 Solution (cont.) Therefore, one-point Gauss Quadrature Rule can be expressed as 97
98 Example 2 a) Use two-point Gauss Quadrature Rule to approximate the distance covered by a rocket from t=8 to t=30 as given by b) Find the true error, for part (a). c) Also, find the absolute relative true error, 98 for part (a).
99 Solution First, change the limits of integration from [8,30] to [-1,1] by previous relations as follows 99
100 Solution (cont) Next, get weighting factors and function argument values from Table 1. for the two point rule,. 100
101 Solution (cont.) Now we can use the Gauss Quadrature formula 101
102 Solution (cont) since 102
103 Solution (cont) b) The true error,, is c) The absolute relative true error,, is (Exact value = m) 103
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