The error bound formulas for trapezoidal rule and simpson's rule say that: Error Bound for the Trap

Alannah Short

Alannah Short

Answered question

2022-06-14

The error bound formulas for trapezoidal rule and simpson's rule say that:
Error Bound for the Trapezoid Rule: Suppose that  Error Bound for the Trapezoid Rule: Suppose that  | f ( x ) | k  for some  k R  where  a x b .  Then  | E T | k ( b a ) 3 12 n 2  Error Bound for Simpson's Rule: Suppose that  | f ( 4 ) ( x ) | k  for some  k R  where  a x b .  Then  | E S | k ( b a ) 5 180 n 4
Using these formulas, is it possible to find functions where Trapezoid Rule is more accurate than Simpson's rule?

Answer & Explanation

pheniankang

pheniankang

Beginner2022-06-15Added 22 answers

Note that "these formulas" give only an error estimate. The actual error may be much smaller.
In order to obtain an example where the formula for E S produces a larger error than the formula for E T consider the function
f ( x ) := sin ( ω x ) ( 0 x 1 )   .
You then have | f ( x ) | ω 2 and | f ( 4 ) ( x ) | ω 4 . This gives
| E T | ω 2 12 n 2 , | E S | ω 4 180 n 4   ,
so that the estimate for E S is worse than the estimate for E T as soon as ω 2 15 n 2 .

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