CSC_349_HANDOUT#39

CSC_349_HANDOUT#39 - COMPUTER SCIENCE 349A Handout Number...

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1 COMPUTER SCIENCE 349A Handout Number 38 ADAPTIVE QUADRATURE Section 22.3 of the 6 th edition only of the textbook Let S 1 denote the (non-composite) Simpson's rule approximation to f ( x ) dx a b . x f(x) x 0 = a x 1 x 2 = b h h o o o Then (1) f ( x ) dx = S 1 a b h 5 90 f (4) ( μ ), where h = b a 2 and a < < b . Let S 2 denote the composite Simpson's rule approximation using two applications of Simpson's rule. x f(x) x 0 = a x x = b x 1 2 x 3 4 h ~ o o o o o Then
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2 (2) . 2 and 2 ~ since ), ~ ( ) 90 )( 16 ( ~ and 4 ~ where ), ~ ( ~ 180 ) ( ) 4 ( 5 2 ) 4 ( 4 2 h a b h h f h S b a a b h f h a b S dx x f b a = = = < < = = μ Question : How can the values of S 1 and S 2 be used to estimate the accuracy of S 2 ? To do this, we need to use S 1 and S 2 to estimate the error term for S 2 , which (from (2) ) is (3) h 5 (16 )(90) f (4) ( ˜ μ ) . In order to do this, we assume that (4) f ( ) = f ( ˜ ). (Note that if h is very small, this is likely a good assumption, and will lead to an accurate estimate of the truncation error of S 2 .) Using (4) in (2), from (1) and (2) we obtain S 1 h 5 90 f ( ) = S 2 h 5 (16)(90) f ( ) , from which it follows that h 5 90 f ( ) = 16 15 ( S 2 S 1 ), and thus from (3), an estimate of the truncation error of S 2 is h 5 f ( ˜ ) 1 15 ( S 2 S 1 ) . Consequently, (2) becomes (5) { 43 42 1 error truncation the of estimate 1 2 integral the to ion approximat 2 ) 4 ( 5 2 ) ( 15 1 ) ~ ( ) 90 )( 16 ( ) ( S S S f h S dx x
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CSC_349_HANDOUT#39 - COMPUTER SCIENCE 349A Handout Number...

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