Simpson's Three-Eighth Method
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1. The document describes Simpson's 3/8 rule for numerical integration and presents an M-file script to implement the rule. 2. The rule uses third-order polynomials to fit four points for integration, simplifying to a formula involving the function values at those points. 3. The script prompts the user for limits and intervals, calculates the function values, applies the Simpson's 3/8 formula, and compares the result to the true integral to determine the error.
Simpson's Three-Eighth Method
- 1. PRACTICAL Name- Saloni Singhal M.Sc. (Statistics) II-Sem. Roll No: 2046398 Course- MATH-409 L Numerical Analysis Lab Submitted To: Dr. S.C. Pandey
- 2. OBJECTIVE 1. Create an M-file to implement Simpson’s Three-Eighth Rule
- 3. Theory Simpson’s 3/8 (Simpson’s second) rule corresponds to using third-order polynomials to fit four points. Integration over the four points simplifies to
- 4. Error in Trapezoidal Rule Local Error Global Error Global Error O(h4)is sum total of all n/3 local error and is of an order less than Local error O(h5)
- 5. Script File a=input('Enter lower limit: '); b=input('Enter upper limit: '); n=input('Enter total no of subintervals: '); if rem(n,3) error('subintervals should be of third multiple') end h=(b-a)/n for i=0:n X(i+1)=a+1*h; end X Y=exp(X) ans=Y(1)+Y(n+1); for j=2:3:n-1 ans=ans+(3*Y(j)); end for k=3:3:n ans=ans+(3+Y(k)); end for l=4:3:n ans=ans+(2*Y(l)); end I=(3*h/8)*ans syms x f=exp(x) true=int(f,a,b) true e=abs(I-true); fprintf('Error is %f',e)
- 6. Output
- 7. Conclusion • As in Simpson’s 1/3 and 3/8 rule, the even segment-odd-point formulas have truncation errors that are the same order as formulas adding one more point. For this reason, the even-segment odd-point formulas are usually the methods of preference • the error term is of same order but smaller than Simpson’s 1/3 rule and thus provides more accurate result. It converges to estimated integral value more quickly. • 3/8ths rule is used for uniformly sampled function integration. Suppose we have a function at equally spaced points. If the number of points is odd, then the composite Simpson's rule works fine. If the number of points is even, then one solution is to use the 3/8ths rule. For example, if the user passed 6 samples, then we use Simpson's for the first three points, and 3/8ths for the last 4 (the middle point is common to both). This preserves the order of accuracy without putting an arbitrary constraint on the number
- 8. Comparison of three methods F(x) Y=Exp(x) h, a, b 30, -1, 5 Trapezoidal Rule I= 148.47 err=0.1173 Simpson 1/3 I=140.7802 err= 7.2650 Simpson 3/8 I=175.6348 err=28.5894 True value=148.045279661 I=Integrated Value by the Rule
- 9. Caveats • The number of subintervals should be a multiple of 3 • When calculated manually, this formula is more complicated