Math 4401 Numerical Methods Syllabus (2008)
Course Prerequisites
To succeed in this course you will need to have mastered basic calculus (Calculus I and Calculus II), differential equations, and certain topics in linear algebra (notably Gaussian elimination and eigensystems). You should have a good background in Mathematica.
Goals
Upon completion of this course a student should be able to:
- Demonstrate understanding of the numerical techniques we have studied in class.
- Communicate the results of a numerical method effectively.
- Be able to investigate numerical techniques which we have not studied in class, and then communicate this understanding.
- Know the strengths and limitations of the numerical techniques we have studied.
- Exhibit competence with the use of technology in the study of numerical analysis.
Textbook
Numerical Analysis 8th Ed. by Richard L. Burden & J. Douglas Faires.
There is a webpage for the text: http://www.as.ysu.edu/~faires/Numerical-Analysis/. There is a list of typos that made their way into the first printing of the eighth edition, so you should visit the errata part of the site and make the corrections to your text if necessary.
We will be covering Chapters 1, 2, 3, 4, 8, 5, 6, 7,and 9 (in that order) from the text, with some sections omitted. I will also be provided some notes for Padé approximants and Monte Carlo Integration, which are not covered in the text.
Programming Expectations
The website for the textbook contains programs for all the algorithms in the text in C, fortran, maple, Mathematica, MATLAB, and Pascal. This makes the textbook very useful to you once the course is completed. If you do any programming in the future this text will provide you with an excellent place to start.
You will have to do a certain amount of simple programming in the course, mainly using looping structures like Do or While. These capabilities exist in Mathematica, and I will help you with coding in Mathematica. Your peers in the course are also an excellent resource if you have a question about the course.
If you prefer to use other coding languages such a fortran or java, go right ahead-however, I will not be able to help you with these languages as much.
Mathematica
Mathematica has many useful features to someone interested in numerical analysis. Many of Mathematica's internal routines use numerical methods, but of course in this class we are more interested in how the method works than simply implementing it. Mathematica can be used to help you with the arithmetic manipulations you need to do, but your solutions should also contain legible, hand written discussions about what it is you are doing.
I have quite a few Mathematica notebooks on my web page from the various courses I have taught over the years. These may serve as a useful guide. On my web page I will be placing Mathematica notebooks for you to download and experiment with, especially any notebooks I use during lectures. As well, there is an abundance of Mathematica assistance available on the web, and you should use those resources as you see fit. My web page has links to some relevant sites, and the UMM math department web page has links as well.
As the focus of this course is understanding numerical methods rather than simply implementing them, you should use Mathematica essentially as a simple CAS to help you do arithmetic, take derivatives or integrals where necessary, and plot graphs. You will also need to do simple programming with Mathematica.
For example, if I ask you to find a root of x5 - cos x = 0 using Newton's method, I would not accept as an answer
FindRoot[x^5 - Cos[x] == 0, {x, 1}]
but I would accept
f[x_] = x^5 - Cos[x]
xnew[0] = 1.0
xnew[n_] := xnew[n] = xnew[n - 1] - f[xnew[n - 1]]/f'[xnew[n - 1]]
TableForm[Table[{n, xnew[n]}, {n, 0, 10}]]
If you have any questions about what my expectations are at any time during the course, make sure you ask me.
Course Components
Assignments. There will be a selection of homework questions assigned throughout the semester. Since an important aspect of the course is the communication of ideas, you should concentrate on first solving the problem, and then communicating the solution in a manner understandable to others. You may work together on assignments, but each student turns in their own assignment, and any group work should involve proper collaboration and not simply copying of another student`s work.
Midterm Exam and Final Exam. These exams will have questions similar to the assignment questions and be a take home exam. You will not be allowed to work in groups or discuss the test with your peers.
The exams will be handed out during class, and handed in to me in my office (Sci 1380). While you are working on a take home exam there will be no lectures scheduled. The final exam will be due at the end of the scheduled exam time for the course, which is 3:30pm May 15.
Analysis Project. The analysis project is an important part of the course, since it will require you to investigate a numerical method on your own (no group work on this project). Acquiring this skill is one of the main goals of the course. The project should be based on one of the methods from the text which we did not study in class, or some other numerical technique that is not in the textbook but that you find interesting.
The project will consist of two components, a paper (60 marks) and oral presentation (40 marks). The Applied Project will be graded out of 100 marks.
Your Paper
Your paper should be about 10 pages long, and written with proper sectioning, numbered equations, and include an abstract and bibliography. It should be typed, using Word, LaTeX, or Mathematica (or other typesetting software like OpenOffice). If you have lots of complicated equations or figures to typeset, you can write those in neatly by hand. Your paper will be graded based on neatness, organization, grammar, and mathematical content. Your paper should include the following:
- The Method
- describe in words and general pictures what the method does
- describe (derived if possible) in math
- An Application of the Method
- worked through by hand (entirely, or one step of the solution)
- a more complex problem solved using a computer (include computer code)
- Conclusion
- strengths, weaknesses of method
- when it should be used
- possible improvements
- References
I keep the reports you submit as part of the Math discipline's assessment efforts, so if you want a copy for yourself make one before you hand your paper in. I will provide a feedback sheet explaining your grade for the paper.
Your Presentation
The presentations will be 30 minutes long, so you will probably have to edit what is in your paper and determine what is the best way to get your main points across. You may use overheads, the whiteboard, powerpoint, LaTeX, Mathematica, or webpages in your presentation. Remember, the presentation is about communicating information, not about flashy effects. Some of the best presentations I have seen used simple, neatly written overheads. Keep the following in mind as you prepare your presentation:
- Introduction.
- Description of problem (with sufficient mathematical detail that can be conveyed to audience in short time).
- Clarity and organization of material (avoid showing anything the audience doesn't have time to follow or understand).
- Enthusiasm doesn't just mean being loud. Are you keeping your audience intellectually engaged?
- Most importantly, think about what you want your fellow classmates to take away from your presentation (it probably won't be everything that is in your paper). The audience should understand the broad picture of what you have done, and if they want more detail they can read your paper or you can go out for coffee with them after to talk about it. Really, that's the goal--get them interested enough to want to learn more!
Grading
Here is the University-wide uniform grading policy.
- A: Represents achievement that is outstanding relative to the level necessary to meet course requirements.
- B: Represents achievement that is significantly above the level necessary to meet course requirements.
- C: Represents achievement that meets the course requirements in every respect.
- D: Represents achievement that is worthy of credit even though it fails to fully meet the course requirements.
- F: Represents failure and indicates that the coursework was completed but at a level unworthy of credit, or was not completed and there was no agreement between the instructor and student that the student would be temporarily given an incomplete.
- I: See the catalog.
You will be graded on assignments, an analysis project, a midterm exam and a final exam. The course marks will be split in the following fashion:
| Assignments (4 at 15% each) | 60% |
| Final Exam | 15% |
| Midterm Exam | 15% |
| Analysis Project | 10% |
Your numerical grades will be converted to letter grades and finally Grade Points via the following cutoffs (see the UMM Catalog for more on Grades and Grading Policy). If you are taking the course S-N, you will need to earn a C- or better to obtain an S grade.
| Numerical | 95% | 90% | 87% | 83% | 80% | 77% | 73% | 70% | 65% | 60% | Below 60% |
| Letter | A | A- | B+ | B | B- | C+ | C | C- | D+ | D | F |
| Grade Point | 4.00 | 3.67 | 3.33 | 3.00 | 2.67 | 2.33 | 2.00 | 1.67 | 1.33 | 1.00 | 0.00 |
Please note that you are not competing against your fellow students. I will adjust the difficulty of the questions and the severity of the grading so that, for example, a B+ score corresponds to what I consider B+ achievement.
Expectations
- Cooperation is vital to your future success, which ever path you take. I encourage cooperation amongst students where ever possible, but the act of copying or other forms of cheating will not be tolerated. Academic dishonesty in any portion of the academic work for a course is grounds for awarding a grade of F or N for the entire course. Any act of plagiarism that is detected will result in a mark of zero on the entire assignment or test for both parties. Please come and talk to me if you are in any way unclear about what constitutes academic dishonesty. UMM's Academic Integrity policy and procedures can be found at www.morris.umn.edu/Scholastic/AcademicIntegrity/.
- Since the assignments are handed out days in advance, only under exceptional circumstances (which are officially documented) will I accept late work. You will receive a mark of zero if an assignment or test is submitted late.
- Solutions to problems should be presented at a level that is appropriate to an upper level university course. This means solutions should be written legibly, contain diagrams where appropriate, and should state the problem and explain the solution. Good solutions should include hand written explanations with attached output from Mathematica sessions showing any computations you did.
- If you have any special needs or requirements to
help you succeed in the class, come and talk to me as soon as
possible, or visit the appropriate University service yourself.
Some resources include:
- The Academic Assistance Center www.morris.umn.edu/services/dsoaac/aac/
- Student Counseling www.morris.umn.edu/services/counseling/
- Disability Services www.morris.umn.edu/services/dsoaac/dso
- Multi-Ethnic Student Program www.morris.umn.edu/services/msp/