Simpson.ppt

# Since for simpsons 13rd rule the interval a b is

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Since for Simpson’s 1/3rd Rule, the interval [a, b] is broken into 2 segments, the segment width 2 a b h

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9 Basis of Simpson’s 1/3 rd Rule ) b ( f b a f ) a ( f h dx ) x ( f b a 2 4 3 2 Hence Because the above form has 1/3 in its formula, it is called Simpson’s 1/3rd Ru
10 Example 1 a) Use Simpson’s 1/3rd Rule to find the approximate value of x The distance covered by a rocket from t=8 to t=30 is given by 30 8 8 9 2100 140000 140000 2000 dt t . t ln x b) Find the true error, t E c) Find the absolute relative true error, t

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11 Solution a) 30 8 ) ( dt t f x ) b ( f b a f ) a ( f a b x 2 4 6 ) ( f ) ( f ) ( f 30 19 4 8 6 8 30 6740 901 7455 484 4 2667 177 6 22 . ) . ( . m . 72 11065
12 Solution (cont) b) The exact value of the above integral is 30 8 8 9 2100 140000 140000 2000 dt t . t ln x m . 34 11061 True Error 72 11065 34 11061 . . E t m . 38 4

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13 Solution (cont) a) c) Absolute relative true error, % . . . t 100 34 11061 72 11065 34 11061 % . 0396 0
14 Multiple Segment Simpson’s 1/3rd Rule

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15 Multiple Segment Simpson’s 1/3 rd Rule ust 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 n a b h n x x b a dx ) x ( f dx ) x ( f 0 where a x 0 b x n
16 Multiple Segment Simpson’s 1/3 rd Rule Apply Simpson’s 1/3rd Rule over each interval, ... ) x ( f ) x ( f ) x ( f ) x x ( dx ) x ( f b a 6 4 2 1 0 0 2 ... ) x ( f ) x ( f ) x ( f ) x x ( 6 4 4 3 2 2 4 f(x) . . .

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• 0.0000%, 0.30 M, 0.0001%, 0.0005%, 4.38 m

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