Does there exist a monic x n , n > 4, for which Simpson’s rule...

Eden Solomon

Eden Solomon

Answered

2022-06-27

Does there exist a monic x n , n > 4, for which Simpson’s rule is exact? If not, why?
S ( f ) = b a 6 f ( a ) + 2 ( b a ) 3 f ( a + b 2 ) + b a 6 f ( b )

Answer & Explanation

Aaron Everett

Aaron Everett

Expert

2022-06-28Added 18 answers

If you take, for sake of simplicity, a = 0 , b = 1 (and f ( x ) = x n ) you would require that
1 6 f ( 0 ) + 2 3 f ( 1 / 2 ) + 1 6 f ( 1 ) = 1 n + 1
i.e. 2 3 ( 1 / 2 ) n + 1 6 = 1 n + 1 1 2 n 2 = 5 n n + 1
However, if n > 4 this last equality is impossible.

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