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Introduction. The main objective of this chapter is to develop appropriate formulas for approximating the integral of the form. = ∫ f ( x ) dx (1)
Simpson's 3/8 Rule for Numerical Integration. The numerical integration technique known as "Simpson's 3/8 rule" is credited to the mathematician Thomas Simpson (1710-1761) of Leicestershire, England. His also worked in the areas of numerical interpolation and probability theory.
Figure 2: Illustration of (a) Simpson’s 1/3 rule, and (b) Simpson’s 3/8 rule Simpson’s 1 = 3 rule: Given function values at 3 points as ( x 0 ;f ( x 0 )), ( x 1 ;f ( x 1 )), and ( x 2 ;f ( x 2 )), we
Simpson’s Rule is based on the fact that given any three points, you can find the equation of a quadratic through those points. For example, let’s say you had points (3, 12), (1, 5), and (5, 9). Starting with (3, 12) and using y = ax2 + bx + c, you could write: x y. 12 = a(3)2 + b(3) + c.
Computer Algorithm for Mixed Simpson 1/3 and 3/8 Rule for Integration. Based on the earlier discussion on (single and composite) Simpson 1/3 and 3/8 rules, the following “pseudo” step-by-step mixed Simpson rules for estimating \[I = \int_{a}^{b}{f(x){dx}}\] can be given as. Step 1. User inputs information, such as \[f(x) = \text{integrand}\]
Aim: To evaluate a definite integral by Simpson’s 3/8 Rule. Algorithm: Given a function f(x): (Get user inputs) Input a,b=endpoints of interval n=number of intervals(Even) (Do the integration) Set h= (b-a)/n. Set sum=0. *i. If i%3=0 Then Set sum=sum+2*f(x) Else Set sum=sum+3*f( Set sum = sum + f(a)+f(b) Set ans = sum*(3h/8). End.
However, in this problem we can combine the methods by appropriately dividing the interval: 1. We’ll use Simpson’s–1/3 rule on interval [1:0; 1:4] (4 subintervals is divisible by 2), and. 2. we’ll use Simpson’s–3/8 rule on interval [1:4; 1:7] (3 subintervals is divisible by 3).
