Prove: Let S ( n ) and T ( n ) be the approximations of a function using n intervals by using Simpson's rule and the Trapezoid rule respectfully. S ( 2 n ) = 4 T ( 2 n ) − T ( n ) 3

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Prove: Let  S  (  n  ) and   T  (  n  ) be the approximations of a function using n intervals by using Simpson&#039;s rule and the Trapezoid rule respectfully.  S  (  2  n  )  =            4      T      (      2      n      )      −      T      (      n      )        3

trajeronls · Accepted Answer

Consider       x    0    ≤  x  ≤      x    0    +  2  h, where   h  =  (  b  −  a  )      /    (  2  n  ). This will be one interval in the trapezoidal rule for   n intervals, and so we have      T    1    (  n  )  =  2  h            f      (              x        0            )      +      f      (              x        0            +      2      h      )        2  for the trapezoidal approximation over this single interval. The same interval counts as two intervals if we are looking at   2  n intervals overall, so we then get the trapezoidal approximation      T    1    (  2  n  )  =  h            f      (              x        0            )      +      f      (              x        0            +      h      )        2    +  h            f      (              x        0            +      h      )      +      f      (              x        0            +      2      h      )        2  and the Simpson&#039;s approximation      S    1    (  2  n  )  =  h            f      (              x        0            )      +      4      f      (              x        0            +      h      )      +      f      (              x        0            +      2      h      )        3       .It is easy to check that      S    1    (  2  n  )  =            4              T        1            (      2      n      )      −              T        1            (      n      )        3       ,and adding these for all n intervals gives what you want.

Prove: Let S ( n ) and T ( n ) be the approximations of a function using n interv

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