Numerical Methods 2024/5: Individual Project
• This work will count for 50% of your final mark for Numerical Methods.
• You must answer the question assigned to you . No marks will be awarded for answering other questions.
• The mark breakdown is as follows .
Analysis 60
Implementation & testing 20
Good programming practice 15
Overall presentation 5
Total 100
• The work does not require the use of external sources; any sources you do use (aside from the course materials) must be properly cited.
• Store all files on One Drive or the M drive to protect against loss.
• Save your Maple work regularly. Executing incorrect codes may cause Maple to become trapped in an infinite loop. If this happens, you can try pressing the interrupt button (①), but you may be forced to close the application and reload your work.
• There is no requirement to type your analytical work; scans of handwritten work are equally acceptable provided they are properly organised and readable.
• Submit work as a single pdf file. See the project guidance notes for instructions on merging and rearranging pdf files.
• Your final submission must include a pdf export of your Maple worksheet. If you work the numerical calculations into a report (e.g. by copying parts into MS Word and adding appropriate explanations), you still need to include the Maple worksheet; just add it as an appendix at the end.
• Invalid submissions (e.g. files in formats other than pdf) will be deleted. Students who make invalid submissions will be given another chance to submit, but this will be treated as late, and subject to standard university penalties (5% deduction for each day, and a mark of zero after five days) .
This problem is concerned with integrals of the form.
(*)
Here, h(0) exists but may not be equal to zero, so f (x) can be unbounded in the limit x → 0.
(a) Consider the case in which h(x) = 1 for all x, so that the integrand is just f (x) = ln(x) .
(i) Obtain a simplified form of the general formula for quadrature error in this case. Write the error for the whole interval in terms of the function
You may assume that S1 = 0, since this is the case for all nontrivial quadrature rules.
(ii) Include the following table in your submission, and fill in the results with values accurate to at least three significant figures.
(iii) Suppose I is estimated twice using the same quadrature rule, first with N subintervals of equal size and then with 2N . What relationship do you expect the errors in the two estimates to (approximately) satisfy? Justify your answer .
(iv) What do you think is the cause of this?
(b) Using the three-point Gaussian quadrature procedure from the NumericalMethods package (or solutions7. mw), or your own version, estimate I for the case
with N = 20 and then with N = 40 . Obtain numerical estimates for the absolute errors in your approximations. Are the results consistent with your analysis in part (a)? Why (or why not)?
(c) (i) Consider the monic cubic polynomial
where A, B and C are constants. Use integration by parts to show that if r ≥ 0 then
In your answer you may use without proof the fact that limx→0+ xlnx = 0 .
(ii) Find exact values for A, B and C such that the above integral vanishes if r = 0, 1 or 2.
(iii) Let x1 , x2 and x3 represent the roots of u(x) (with A, B and C as in part (ii)), arranged in ascending order . Find weights w1 , w2 and w3 such that
Use Maple to avoid boring algebra; you should find that w3 = -0.0946 . . .
(d) If h(x) is a polynomial of degree d, what is the maximum value for d such that the integral in (*) is always integrated exactly by the quadrature rule from part (c)? Justify your answer.
Hint: the justification is very similar to a result presented in the lecture notes. You don’t need to write out the relevant argument in full.
(e) (i) Write a Maple procedure that approximates integrals of the form (*) using the quadrature rule derived in part (c) . The procedure should takes as its argument a function h, and return the resulting estimate as its result. The procedure should not recalculate the nodes and weights; it should use values stored to at least ten correct significant figures.
(ii) Test your procedure with function h from part (b). Obtain a numerical estimate for the absolute error in this approximation and find the approximate number of subintervals needed to achieve the same level of accuracy using the three point Gaussian rule.