Numerical Integration Chapter 6
Nizar Salim 1 lecture 2
Numerical Integration
6.3.2 Simpson s 1/3 Rule
Simpson s 1/3 rule is obtained by fitting a second-degree polynomial to three
equally spaced discrete points, as illustrated in Figure 6.6. The upper limit of integration
x2 corresponds to s = 2. Thus, Eq. (6.19) gives
Performing the integration, evaluating the result, and introducing the expressions for f0
and 2f0, yields Simpson s 1/3 rule for a single interval of two increments:
Numerical Integration Chapter 6
Nizar Salim 2 lecture 2
The composite Simpson s 1/3 rule for equally spaced points is obtained by applying Eq.
(6.34) over the entire range of integration. Note that the total number of increments must
be even. Thus,
This surprising result does not mean that the error is zero. It simply means that the cubic
term is identically zero, and the error is obtained from the next term in the Newton
forward-difference polynomial. Thus,
Thus, the local error is 0(h5). By an analysis similar to that performed for the trapezoid
rule, it can be shown that the global error is 0(h4).
Example 6.3. Simpson s 1/3 Rule
Let s solve the example problem presented in Section 6.1 using Simpson s 1/3 rule. Recall
that f(x) = l/x. Solving the problem for two increments of h = 0.4, the minimum
permissible number of increments for Simpson s 1 /3 rule, and one interval yields
Numerical Integration Chapter 6
Nizar Salim 3 lecture 2
Breaking the total range of integration into four increments of h = 0.2 and two intervals
and applying the composite rule yields:
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