2 What is Simpson’s 3/8 Rule Simpson’s 3/8 is very similar to the Simpson’s Method that we already learned in class. The different is Simpson’s 3/8 method uses a third degree polynomial (cubic) to estimate the curve you are trying to find the integral of while Simpson’s Method (also called Simpson’s 1/3 Method) uses a second degree polynomial (quadratic)
3 To get the estimate for an integral in this case, we will write out the 3rd degree polynomial using the general equation below and then integrate it.
4 How do we get the formula: We begin by integrating the 3rd degreePolynomial p(t)=c3t3+c2t2+c1t+c0.
6 Substitute 𝑐 0 , 𝑐 2 , 𝑐 3 , and 𝑐 1 values back into the equation. h = (𝑏−𝑎)/N , where N is a positive multiple of 3 so another way to look at this is h = (𝑏−𝑎)/3𝑛 , where n is the number of partitions being used.
7 xyay0=f(a)a+hy1=f(a+h)a+2hy2=f(a+2h)a+3hy3=f(b)Here we are using a single partition.1 Partition = 𝑃 1aa + ha + 2hb=a+3h
9 Conclusion:So we can see that the equation for the Simpson’s 3/8 Rule is:Each of the function values will be multiplied by 3 or 2 except for the first and the last. The number of subinterval points will need to be multiples of 3’s.
11 Simpson’s 1/3 Rule VS Simpson’s 3/8 Rule Advantages:Can use an odd or even number of subintervals rather than just an even number of subintervals as used in the Simpson’s 1/3 Rule.Can converge to estimated integral values much quicker than Simpson’s 1/3 RuleHas the same order of accuracy as the Simpson’s 1/3 Rule when calculating the absolute error, but the Simpson’s 3/8 Rule is actually slightly more accurate. But the fraction used in this error will be smaller and then will provide a more accurate error.Disadvantages:This rule requires multiple of 3 segments only.When calculating the value by hand this rule is more complicated.1/3 Rule:and
12 ReferencesModule for Simpson’s 3/8 Rule for Numerical Integration. (2012). Retrieved fromNguyen, Duc. Chapter 07.08: Simpson 3/8 Rule For Integration. University of South Florida.Wikipedia. Simpson’s Rule. Retrieved from