mtl_gen_int_ppt_romberg - Romberg Rule of Integration...

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1/10/2010 http://numericalmethods.eng.usf.edu 1 Romberg Rule of Integration Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates
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Romberg Rule of Integration http://numericalmethods.eng.usf.edu
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http://numericalmethods.eng.usf.edu 3 Basis of Romberg Rule Integration = b a dx ) x ( f I The process of measuring the area under a curve. Where: f(x) is the integrand a= lower limit of integration b= upper limit of integration f(x) a b y x b a dx ) x ( f
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http://numericalmethods.eng.usf.edu 4 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.
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http://numericalmethods.eng.usf.edu 5 Error in Multiple Segment Trapezoidal Rule The true error in a multiple segment Trapezoidal Rule with n segments for an integral Is given by = b a dx ) x ( f I ( ) ( ) n f n a b E n i i t = ξ = 1 2 3 12 where for each i, is a point somewhere in the domain , . i ξ ( ) [ ] ih a , h i a + + 1
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http://numericalmethods.eng.usf.edu 6 Error in Multiple Segment Trapezoidal Rule The term can be viewed as an ( ) n f n i i = ξ 1 approximate average value of in . ( ) x f [ ] b , a This leads us to say that the true error, E t previously defined can be approximated as 2 1 n E t α
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7 Error in Multiple Segment Trapezoidal Rule Table 1 shows the results obtained for the integral using multiple segment Trapezoidal rule for n Value E t 1 11868 807 7.296 --- 2 11266 205 1.854 5.343 3 11153 91.4 0.8265 1.019 4 11113 51.5 0.4655 0.3594 5 11094 33.0 0.2981 0.1669 6 11084 22.9 0.2070 0.09082 7 11078 16.8 0.1521 0.05482 8 11074 12.9 0.1165 0.03560 = 30 8 8 9 2100 140000 140000 2000 dt t . t
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This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida - Tampa.

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mtl_gen_int_ppt_romberg - Romberg Rule of Integration...

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