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mws_ele_int_ppt_romberg

# mws_ele_int_ppt_romberg - Authors:AutarKaw,CharlieBarker...

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11/12/10 http://numericalmethods.eng.usf.edu 1 Romberg Rule of Integration Electrical 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
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.
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 ( 29 ( 29 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 ( 29 [ ] 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  ( 29 n f n i i = ξ 1 approximate average value of           in          . ( 29 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 2245
http://numericalmethods.eng.usf.edu 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 .

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mws_ele_int_ppt_romberg - Authors:AutarKaw,CharlieBarker...

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