![Trapezoidal Rule
Multiple Segment Trapezoidal Rule
Divide [a,b] into n equal segments and apply trapezoidal rule
over each segment. The sum of the results obtained for each
segment is the approx. value for the integral.](https://image.slidesharecdn.com/rombergsmethod-121110095036-phpapp02/85/Romberg-s-method-4-320.jpg)
![Error in Multiple Segment Trapezoidal
Rule
True error is given by
where ∈ [a+(i-1)h , a+ih] for each i
- approx. average of f’’(x) in [a,b].
Because of that, we can say](https://image.slidesharecdn.com/rombergsmethod-121110095036-phpapp02/85/Romberg-s-method-5-320.jpg)
Romberg’s method
- 2. Romberg’s Method An extrapolation formula of the Trapezoidal Rule for integration. It provides a better approximation of the integral by reducing the True Error. We use two estimates of an integral to compute a third integral that is more accurate than the previous integrals. Named after Werner Romberg
- 3. Trapezoidal Rule Trapezoidal Method Given where f(x) - integrand a – lower limit b – upper limit Et = True Value + Approximate Value |∈t | = |True Error / True Value| x 100
- 4. Trapezoidal Rule Multiple Segment Trapezoidal Rule Divide [a,b] into n equal segments and apply trapezoidal rule over each segment. The sum of the results obtained for each segment is the approx. value for the integral.
- 5. Error in Multiple Segment Trapezoidal Rule True error is given by where ∈ [a+(i-1)h , a+ih] for each i - approx. average of f’’(x) in [a,b]. Because of that, we can say
- 6. Richardson’s Extrapolation Formula for Integration by Trapezoidal Rule True error estimated as where C = proportionality constant Recall: Et = TV – In where TV = true value In = approx value. Then = TV – In (*) If we double the no. of segments from n to 2n, (**) Solving (*) and (**), we get
- 7. Romberg Integration From estimate of true error: Recall: estimate of true error exact true error For small h,
- 8. Romberg Integration Result obtained has an error of O(h4) and can be written as (*) double the no. of segments (**)
- 9. Romberg Integration From (*) and (**) now has an error of O(h6)
- 10. Romberg Integration General expression: where k = order of interpolation (i.e. k = 1, values for regular trapezoidal rule k = 2, values using error estimate O(h2) ) j = more/less accurate estimate of integral (i.e. value of integral w/ index j+1 is more accurate than that with index j)
- 11. Examples: The vertical distance in meters covered by a rocket from t=8 to t=30 seconds is given by Use Romberg’s rule to find the distance covered. Use the 1, 2, 4, and 8-segment trapezoidal rule results. Solution: correspond to 1, 2, 4, and 8-segment trapezoidal rule results.
- 12. 1st order extrapolation values: 2nd order extrapolation values 3rd order extrapolation values
- 14. Solution:
- 16. References: http://www.mymathlib.com/quadrature/rombergs_metho d.html http://numericalmethods.eng.usf.edu/mtl/gen/07int/mtl_ gen_int_txt_romberg.pdf