Simpson’s Rule

Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals.Simpson’s rule is a method for approximating the area under a curve over a given interval that involves partitioning the interval by an odd number n+1 of equally spaced ordinates and adding the areas of the n/2  figures formed by pairs of successive odd-numbered ordinates and the parabolas which they determine with their included even-numbered ordinates. In Simpson's Rule, we will use parabolas to approximate each part of the curve. This proves to be very efficient since it's generally more accurate than the other numerical methods we've seen.

The method is credited to the mathematician Thomas Simpson (1710–1761) of Leicestershire, England. Kepler used similar formulas over 100 years prior. For this reason the method is sometimes called Kepler's rule, or Keplersche Fassregel in German.

Derivation of Simpson’s Rule

·       Quadratic Interpolation

·        Averaging The Midpoint and The Trapezoidal Rules

The Simpson’s Rule is generally more accurate than the Trapezoidal and Midpoint Rules. If f is not linear on a subinterval, then it can be seen that the errors for the midpoint and trapezoid rules behave in a very predictable way, they have opposite sign. For example, if the function is concave up the Tn will be too high, while Mn will be too low. Thus it makes sense that a better estimate would be to average Tn and Mn. However, in this case we can do better than a simple average. The error will be minimized if we use a weighted average. To find the proper weight we take advantage of the fact that for a quadratic function the errors ETn and EMn are exactly related by

Thus we take the following weighted average of the two, which is called Simpson’s Rule:

If we use this weighting on a quadratic function the two errors will exactly cancel.

Notice that we write the subscript as 2n. That is because we usually think of 2n subintervals in the approximation; the n subintervals of the trapezoid are further subdivided by the midpoints. We can the number all the points using integers. If we number them this way we notice that the number of subintervals must be an even number.