Lab for Romberg Integration

Module for Romberg Integration

Romberg Integration for Quadrature Quadrature Rule for Numerical Integration.

To approximate the integral

by generating a table of approximations, and using as the final answer.
The approximations are stored in a special lower triangular matrix.
The elements of the first column are computed using the sequential trapezoidal rule
based on subintervals of ; then is computed using Romberg's rule.
Elements of row j are .
The algorithm is terminated early when .

Report to be handed in.

Computer Projects.

Project I. Construct the Romberg table for finding .

Exercise 1. Plot the function over the interval [0, 1.25].

Exercise 2. Construct the Romberg table using tol = 0.001

Exercise 3. The last entry in the table is R_j,j. Let's find it.

Exercise 4. Look at 10 digits in R_j,j.

Exercise 5. Are all 10 digits correct ? Why ?
How many digits can you guarantee based on the computations involving rows 1-4 of the table ?

Exercise 6. Determine how to get Romberg integration to achieve 10 digits of accuracy.
How many rows of the Romberg table do you think will be needed ?
Do it.
Report the answer with 10 digits of accuracy.

Project II. Construct the Romberg table for finding .

Exercise 7. Plot the function over the interval [0, 1].

Exercise 8. Construct the Romberg table using tol = 0.001

Exercise 9. The last entry in the table is R_j,j. Let's find it.

Exercise 10. Look at 10 digits in R_j,j.

Exercise 11. Are all 10 digits correct ? Why ?
How many digits can you guarantee based on the computations involving rows 1-4 of the table ?

Exercise 12. Determine how to get Romberg integration to achieve 10 digits of accuracy.
How many rows of the Romberg table do you think will be needed ?
Do it.
Report the answer with 10 digits of accuracy.

(c) John H. Mathews, 1998