Suppose we want to estimate an integral of a function between two limits with a weighted sum of values of the function at different points:
One way of doing this is to ensure that for
coefficients, the approximation is exact for a polynomial of degree
For example, Simpson's rule in it's simplest form uses three coefficients. The x values below are easily space.
Suppose
Take
in succession.
We solve the simultaneous equations
(2)
(3)
(4)
(3)+(4) give
then (3) gives
and finally (1) gives
Substitution of these values into (1) gives the general form with arbitraru limits for the simplest form of Simpson's rule with only three coefficients.
