Is there any mathematical proof that I can always get a accurate value using composite simpson's rule rather than only using 3 8 simpson's rule? First of all is my claim correct that we yield a better accuracy always with composite simpson's rule?

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Is there any mathematical proof that I can always get a accurate value using composite simpson&#039;s rule rather than only using       3    8   simpson&#039;s rule? First of all is my claim correct that we yield a better accuracy always with composite simpson&#039;s rule?

Paxton James · Accepted Answer

Estimating       ∫    a    b    f  (  x  )  d  x where   a  &amp;lt;  b by the       3    8   Simpson&#039;s rule has error   −            (      b      −      a              )        5              6480        f          (      4      )        (  ξ  ) for some   ξ  ∈  [  a  ,    b  ], but for the composite version the denominator   6480 is changed to   80      n    4   if   3      |    n. Admittedly the exact value of   ξ is bound to change too, but if   [  a  ,    b  ]| is nonzero and not very variable in relative terms on   [  a  ,    b  ], this only causes a small change in the value of   n for which the methods&#039; error terms are closest to matching. In fact,   80      n    4    =  6480 for   n  =  3, so by the time you get to   n  =  5 the composite rule should have a much smaller error term (although, since   5 isn&#039;t quite the same as   6, which is a multiple of   3, we can&#039;t use the above formula for the error term).

Is there any mathematical proof that I can always get a accurate value using composite simpson's rul

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