Simpson’s 3/8 Rule By: Mufan Yang. 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

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Simpsons 3/8 Rule By: Mufan Yang Slide 2 What is Simpsons 3/8 Rule Simpsons 3/8 is very similar to the Simpsons Method that we already learned in class. The different is Simpsons 3/8 method uses a third degree polynomial (cubic) to estimate the curve you are trying to find the integral of while Simpsons Method (also called Simpsons 1/3 Method) uses a second degree polynomial (quadratic) Slide 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. Slide 4 How do we get the formula: We begin by integrating the 3rd degree Polynomial p(t)=c 3 t 3 +c 2 t 2 +c 1 t+c 0. Slide 5 Slide 6 h = ()/N, where N is a positive multiple of 3 so another way to look at this is h = ()/3, where n is the number of partitions being used. Slide 7 Here we are using a single partition. xy ay 0 =f(a) a+hy 1 =f(a+h) a+2hy 2 =f(a+2h) a+3hy 3 =f(b) ab=a+3ha + h a + 2h Slide 8 a+3h a+4h a+5hb = a+6h Slide 9 Conclusion: So we can see that the equation for the Simpsons 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 3s. Slide 10 Example: Slide 11 Simpsons 1/3 Rule VS Simpsons 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 Simpsons 1/3 Rule. Can converge to estimated integral values much quicker than Simpsons 1/3 Rule Has the same order of accuracy as the Simpsons 1/3 Rule when calculating the absolute error, but the Simpsons 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 Slide 12 References Module for Simpsons 3/8 Rule for Numerical Integration. (2012). Retrieved from http://math.fullerton.edu/mathews/n2003/Simpson38RuleMod.html. http://math.fullerton.edu/mathews/n2003/Simpson38RuleMod.html Nguyen, Duc. Chapter 07.08: Simpson 3/8 Rule For Integration. University of South Florida. http://numbericalmethods.eng.usf.edu.http://numbericalmethods.eng.usf.edu Wikipedia. Simpsons Rule. Retrieved from http://en.wikipedia.org/wiki/Simpson%27s_rule. http://en.wikipedia.org/wiki/Simpson%27s_rule