Handout34 - COMPUTER SCIENCE 349A Handout Number 34 ROMBERG...

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1 COMPUTER SCIENCE 349A Handout Number 34 ROMBERG INTEGRATION (Section 22.2) Romberg integration is the application of Richardson's Extrapolation to the composite trapezoidal rule approximations. Notation : let 1 , k I denote the composite trapezoidal rule approximation to f ( x ) dx a b using 2 k 1 subintervals. Then, letting ) ( i i x f f = as usual, [] 4 where , 2 2 2 where , 2 2 where , 2 4 3 2 1 0 1 , 3 2 1 0 1 , 2 1 0 1 , 1 a b h f f f f f h I a b h f f f h I a b h f f h I = + + + + = = + + = = + = and so on. The error term b a f h a b < < μ where ), ( 12 2 , for the composite trapezoidal rule can also be shown to have a series expansion of the form L + + + 6 3 4 2 2 1 h K h K h K , where the K i are constants independent of h . That is, this error term has the form (1) in Handout Number 33, so the Richardson's Extrapolation formulas below are the same as those in Handout Number 33. For example, letting h = b a , (a) L + + + + = 6 3 4 2 2 1 1 , 1 ) ( h K h K h K I dx x f b a (b) L + + + + = 64 16 4 ) ( 6 3 4 2 2 1 1 , 2 h K h K h K I dx x f b a so computing 4 × ( b ) ( a ) gives
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2 ) ( 4 ) ( ) ( 4 4 1 , 1 1 , 2 h O I I dx x f dx x f b a b a + = .
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Handout34 - COMPUTER SCIENCE 349A Handout Number 34 ROMBERG...

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