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Handout32 - COMPUTER SCIENCE 349A Handout Number 32...

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1 COMPUTER SCIENCE 349A Handout Number 32 ROUNDOFF ERROR ANALYSIS FOR COMPOSITE NEWTON-COTES FORMULAS For simplicity consider the composite Simpson’s rule ( ) + + + = = m j m j m j j b a f f f f h x d x f 1 2 1 2 1 1 2 0 4 2 3 , where f i = f ( x i ) and h = b a 2 m . When this formula is evaluated using floating- point arithmetic, there are two main sources of roundoff error : in the evaluation of the values i f , and in calculating the summations. For now, we'll consider only the effect on the computed approximation to f ( x ) dx a b of roundoff errors that occur in evaluating the values i f . Suppose ˜ f i denotes the computed approximation to i f (that is, ˜ f i is inexact due to roundoff error). Denote the roundoff error by e i = ˜ f i f i and suppose that ε i e for all i (that is, we are assuming that ε is the largest possible roundoff error when evaluating ) ( i x f regardless of the value of x i ).

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Handout32 - COMPUTER SCIENCE 349A Handout Number 32...

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