Computational Methods EML 3041
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This 4 page Class Notes was uploaded by Kristina Mayer on Wednesday September 23, 2015. The Class Notes belongs to EML 3041 at University of South Florida taught by Staff in Fall. Since its upload, it has received 30 views. For similar materials see /class/212674/eml-3041-university-of-south-florida in Engineering Mechanical at University of South Florida.
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Date Created: 09/23/15
Richardson s Extrapolation Formula for Trapezoidal Rule The true error in a multiple segment Trapezoidal Rule with 71 segments for an integral 1 fxdx 1 is given by 3 i Mi bl 1 2 n n where for each 139 51 is a point somewhere in the domain a 139 1h a ih and i Mi 11 E t the term can be viewed as an approximate average value of f quotx in n 61 b This leads us to say that the true error E in Equation 2 El E 01L 3 2 7 17 in estimate of the integral I f xdx using the nsegment Trapezoidal Rule Table 1 Values obtained using multiple segment Trapezoidal rule for 30 x j 20001namp 98t dt 8 140000 2100t 71 Value E1 IEI Ea 1 11868 807 7296 2 11266 205 1854 5343 3 11153 914 08265 1019 4 11113 515 04655 03594 5 11094 330 02981 01669 6 11084 229 02070 009082 7 11078 168 01521 005482 8 11074 129 01165 003560 Table 1 shows the results obtained for the integral using multiplesegment Trapezoidal rule 140000 20001n126666i5166 98t The true error for the lsegment Trapezoidal rule is 807 while for the 2segment rule the true error is 205 The true error of 205 is approximately a quarter of 807 The true error gets approximately quartered as the number of segments is doubled from 1 to 2 Same trend is observed when the number of segments is doubled from 2 to 4 true error for 2segments is 205 and for four segments is 515 This follows Equation 3 This information although interesting can also be used to get better approximation of the integral That is the basis of Richardson s extrapolation formula for integration by Trapezoidal Rule The true error E in the nsegment Trapezoidal rule is estimated as l E E a 2 quot2 C E g 2 4 n where C is an approximate constant of proportionality Since E TV In 5 where TV true value In approximate value using nsegments Then from equations 4 and 5 C quot 2 E TV In 6 If the number of segments is doubled from n to 211 in the Trapezoidal rule C 2 E TV I in 7 2quot Equations 1 and 2 can be solved simultaneously to get 12 quot quot 8 3 TVEIM Example 1 The vertical distance covered by a rocket from t 8 to t 30 seconds is given by 30 x j 20001namp 98t dt 8 140000 2100t a Use Richardson s extrapolation to nd the distance covered Use the 2 segment and 4segment Trapezoidal rule results given in Table 1 b Find the true error E for part a c Find the absolute relative true error for part a Solution a 12 11266m 14 11113m Using Richardson s extrapolation formula for Trapezoidal rule TV I I 3Iquot and choosing 712 TV514I42 111131111311266 11062m b The exact value of the above integral is 30 x j 20001namp 98t dt 8 140000 2100t 11061 m so the true error E True Value Approximate Value 11061 11062 1 m c The absolute relative true error 11061 11062 e I would then be e X 100 000904 Table 2 shows the Richardson s extrapolation results using 1 2 4 8 segments Results are compared with those of Trapezoidal rule Table 2 Values obtained using Richardson s extrapolation formula for 30 Trapezoidal rule for x I 20001n 8 140000 140000 2100t 98th n Trapezoidal letl for Trapezoidal Richardson s letl for Richardson s Rule Rule Extrapolation Fa 39 A39 1 11868 7296 2 11266 1854 11065 003616 4 11113 04655 11062 0009041 8 11074 01165 11061 00000
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