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
(**)
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
