November 11, 2004 --- Class 22 --- Project, Numerical Integration in
One Dimension
Project
The project is due on Tuesday, Dec. 14 at 5 p.m., as stated in the
syllabus. You project report will be graded on this basis:
Statement 10
Eq Solved 5
Num. Method 5
Code 5
Results 10
Discussion 10
Critique 5
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Total 50
Note the emphasis on the statement of the problem, analysis of results
and discussion of what has been learned. Also please note that your
report should include a critical discussion of how you might do it
better (or at least differently) and how it could be extended.
Numerical Integration
We talked about some of the standard methods for numerical
integration in one dimension. This is discussed in Section 11.1
of CSM. Additional material is available in Numerical Recipes
or in Abramowitz and Stegun. We compared the accuracy of
various methods like the rectangle rule, trapezoid rule and
Simpson's rule. We indicated how Simpson's rule can simply
be derived from removing the leading 1/N^2 error in the trapeziod
rule with N and 2N points. (This is detailed below.)
This technique can be extended in the method
known as Romberg integration. The previous two points are
expanded upon in Numerical Recipes. Practical codes may be found
there.
Simpson's Rule Rederived
Starting from the trapezoid rule, we used the formulae below
for correcting the leading 1/N^2 errors.
S_N = A + E/N^2
S_2N = A + E/(2N)^2
so A = 4/3 S_2N -1/3 S_N
Applying this to a one-dimensional itegral, we easily found that the
weighting of the sum of two expressions on the right hand side
exactly coincides with the Simpson's rule formula. We previously
found that Simpson's Rule has a leading error of order 1/N^4. Since
the approach above removes the leading 1/N^2 error and since the
error of the trapezoid rule only has even powers of 1/N (asserted, but
not proved), this is not surprising.