Let n be even. Show how the composite Simpson rule with 2n equally spaced nodes can be computed from the case of n equally spaced nodes with a minimum amount of additional work.

Question

Let   n be even. Show how the composite Simpson rule with 2n equally spaced nodes can be computed from the case of n equally spaced nodes with a minimum amount of additional work.

glapaso7ng5 · Accepted Answer

The starting Simpson&#039;s rule on   [  0  ,  1  ] is       I    1    ≈      1    6    f  (  0  )  +      4    6    f  (      1    2    )  +      1    6    f  (  1  ) When you subdivide the interval it is       I    2    ≈      1    12    f  (  0  )  +      4    12    f  (      1    4    )  +      2    12    f  (      1    2    )  +      4    12    f  (      3    4    )  +      1    12    f  (  1  )  =      1    2        I    1    +      4    12    (  f  (      1    4    )  +  f  (      3    4    )  ) Then when you double again, you divide       I    2   in half, then add the contributions for the new points. You are being asked to show that this continues, so all you need to do is evaluate   f at the new points, sum the values, multiply by an appropriate constant, and add half the previous approximation to get the new approximation.

Let n be even. Show how the composite

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