Romberg Rule of Integration

Transcription

1 Romberg Rule of ntegration Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker Transforming Numerical Methods Education for STEM Undergraduates 1/10/2010 1

2 Romberg Rule of ntegration

3 Basis of Romberg Rule ntegration The process of measuring the area under a curve. y b a f ( x )dx f(x) b = a f ( x )dx Where: f(x) is the integrand a= lower limit of integration b= upper limit of integration a b x

4 What is The Romberg Rule? Romberg ntegration is an extrapolation formula of the Trapezoidal Rule for integration. t provides a better approximation of the integral by reducing the True Error. 4

5 Error in Multiple Segment Trapezoidal Rule The true error in a multiple segment Trapezoidal Rule with n segments for an integral s given by E b = a t ξ i f ( x )dx n ( b a) i= 1 = 2 12n f n ( ξ ) where for each i, is a point somewhere in the domain, a i 1 h,a ih. [ ( ) ] i 5

6 The term Error in Multiple Segment Trapezoidal Rule n i= 1 f ( ξ ) i can be viewed as an n approximate average value of f x in a,b. ( ) [ ] This leads us to say that the true error, E t previously defined can be approximated as E t α 1 n 2 6

7 Error in Multiple Segment Trapezoidal Rule x = 0 8 Table 1 shows the results obtained for the integral using multiple segment Trapezoidal rule for ln 9. 8t dt t n Value E t % % t a Table 1: Multiple Segment Trapezoidal Rule Values 7

8 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. 8

9 Richardson s Extrapolation for Trapezoidal Rule The true error, E is estimated as t in the n-segment Trapezoidal rule E t where C is an approximate constant of proportionality. Since C 2 n E = TV t n Where TV = true value and n = approx. value 9

10 Richardson s Extrapolation for Trapezoidal Rule From the previous development, it can be shown that C ( 2n) 2 TV when the segment size is doubled and that 2n TV 2n 2n n which is Richardson s Extrapolation. 10

11 Example 1 The vertical distance covered by a rocket from 8 to 0 seconds is given by x = ln 9. 8t dt t 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, a for part (a). 11

12 Solution a) = m 1111m 4 = Using Richardson s extrapolation formula for Trapezoidal rule TV 2n 2n n and choosing n=2, TV = 1111 = 11062m 12

13 Solution (cont.) b) The exact value of the above integral is x = ln 9. 8t dt t = m Hence E t = True Value Approximate Value = = 1 m 1

14 Solution (cont.) c) The absolute relative true error t would then be t = = % Table 2 shows the Richardson s extrapolation results using 1, 2, 4, 8 segments. Results are compared with those of Trapezoidal rule. 14

15 Solution (cont.) Table 2: The values obtained using Richardson s extrapolation formula for Trapezoidal rule for x = ln 9. 8t dt t n Trapezoidal Rule t for Trapezoidal Rule Richardson s Extrapolation t for Richardson s Extrapolation Table 2: Richardson s Extrapolation Values 15

16 Romberg ntegration Romberg integration is same as Richardson s extrapolation formula as given previously. However, Romberg used a recursive algorithm for the extrapolation. Recall TV 2n 2n This can alternately be written as n ( ) 2n R = 2n 2n n 2 n = 2n n 1 16

17 Romberg ntegration ( 2n ) R 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 TV 4 ( 2 ) Ch n R Where Ch 4 is an approximation of the true error. 17

18 Romberg ntegration Determine another integral value with further halving the step size (doubling the number of segments), ( ) 4n R = 4n t follows from the two previous expressions that the true value TV can be written as TV ( ) 4n R 4n ( ) ( ) 4n R 15 2n R ( ) ( ) 4n R 2 n = 4n n R 18

19 Romberg ntegration A general expression for Romberg integration can be written as k, j k 1, j 1 k 1, j = k 1, j 1, k k 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. 19

20 Example 2 The vertical distance covered by a rocket from t = 8 to t = 0 seconds is given by x = ln 9. 8t dt t 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. 20

21 Solution From Table 1, the needed values from original Trapezoidal rule are , 1 = 1, 2 = , 4 = , = where the above four values correspond to using 1, 2, 4 and 8 segment Trapezoidal rule, respectively. 21

22 Solution (cont.) To get the first order extrapolation values, Similarly, 2,1 1,2 1,1 = 1, = = ,2 1, 1,2 = 1, = 1111 = , 1,4 1, = 1, = =

23 Solution (cont.) For the second order extrapolation values,,1 2,2 2,1 = 2, = = Similarly,,2 2, 2,2 = 2, = = 11061

24 Solution (cont.) For the third order extrapolation values,, 2, 1 4, 1 =, = = 11061m Table shows these increased correct values in a tree graph. 24

25 Solution (cont.) Table : mproved estimates of the integral value using Romberg ntegration First Order Second Order Third Order 1-segment 2-segment 4-segment 8-segment

26 Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATCA, MathCad and MAPLE, blogs, related physical problems, please visit /topics/romberg_ method.html

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