What is the relation between Lagrange interpolation and Simpson's rule to integrate some function with some points x 0 , f ( x 0 ); ... x n , f ( x n ) ?

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What is the relation between Lagrange interpolation and Simpson&#039;s rule to integrate some function with some points       x    0    ,  f  (      x    0    ); ...       x    n    ,  f  (      x    n    ) ?

Ashly Kaufman · Accepted Answer

There are various approaches to deriving Simpson&#039;s Rule. A common one uses a special case of Lagrange interpolation. Recall that we used *evenly spaced points   a  =      x    0    ,      x    1    ,  …      x          2      m        =  b.Then for   i  =  0 to   m  −  1, we find a polynomial       P    (    i  )  (  x  ) of degree   ≤  2 that passes through the three points       A          2      i        =  (      x          2      i        ,  f  (      x          2      i        )  ),       A          2      i      +      1        =  (      x          2      i      +      1        ,  f  (      x          2      i      +      1        )  ) and       A          2      i      +      2        =  (      x          2      i      +      2        ,  f  (      x          2      i      +      2        )  ).This is ordinary three point Lagrange interpolation, though the term is ordinarily not used in this very special context.We then integrate       P    i    (  x  ) from       x          2      i       to       x          2      i      +      2      , and add up over all   i from   0 to   m  −  1. When the calculation is done, we end up with the ordinary Simpson&#039;s Rule formula.

What is the relation between Lagrange interpolation and Simpson's rule to integrate some function wi

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