Use of High Order Newton Cotes

• One could also generate a formula in terms of the same function values by using the Newton-Cotes formula appropriate to 2n+1 points.

• This gains an extra derivative in error just like Simpson, and error is where c is some known number.

• In practice, this is not often used because

  (a) the implied interpolation of over a wide range by a high order polynomial can produce disastrous results when not polynomial (e.g., has similarities in its value derivatives).

  On the other hand, local interpolation by by a low order polynomial (used in (NI.9)) is much safer;

  (b) it is much easier to estimate error in iterated Simpson of equation (NI.9) as we will describe in Section NI.2.

Geoffrey Fox, Northeast Parallel Architectures Center at Syracuse University, gcf@npac.syr.edu