1. Numerical integration continued --- Simpson’s rules - We can add more segments OR - We can use a higher order polynomial

2. Simpson’s 1/3 rule use a second order interpolating polynomial If we use Lagrange form

3. Integrate and do some algebra

4. If we use a=x0 and b=x2, and x1=(b+a)/2 widthAverage height Error for Simpson’s 1/3 rule

5. As with Trapezoidal rule, can use multiple applications of Simpson’s 1/3 rule need even number of segments, odd number of points 9 points, 4 segments

6. As in multiple trapzoid, break integral up Substitute Simpson’s 1/3 rule for each integral and collect terms

7. Example: Numerically integrate from 0 to 1 using 1) single trapezoid, 2) multiple trapezoid, 3) single Simpson’s 1/3 and 4) multiple Simpson’s 1/3

8. True, analytic value of I is 0.4749

9. Really quite bad 1) Single trapezoidal rule

10. 2) Multiple trapezoidal rule

11. 3) Single Simpson’s 1/3 rule

12. 4) Multiple Simpson’s 1/3 rule

14. Simpson’s 1/3 rule is limited to applications with equally-spaced data even number of segments odd number of points Simpson’s 3/8 rule used when there are odd number of segments even number of points

15. Simpson’s 3/8 rule uses a third order Lagrange polynomial Four equally spaced points, separated by or

16. Can do multiple segment application of Simpson’s 3/8 rule. Can also mix and match Simpson’s 1/3 and 3/8 to fill up segments

17. Example: 12 points, 11 segments Each 3/8 rule application takes 3 segments Each 1/3 rule application takes 2 segments

18. Neither 2 nor 3 go into 11 But 3 3’s and a 2 do. 1/3 rule 3/8 rule

20. Higher order Newton-Cotes closed formulas Simpson’s 1/3 - 2nd order Lagrange Simpson’s 3/8 - 3rd order Lagrange we can keep going but don’t usually - Simpson is accurate enough when applied in multiple segments

21. Integration with unequal segments If all unequal, stuck with multiple trapezoid rule application If you can find some sets of equal segments, use Simpson’s rules