Turn in an brief proposal of between 1/3 and 1/2 page on
Wednesday, November 25. I will return this
with feedback, especially advice on how to
limit
the project so that you can do it in a reasonable amount of time.
This proposal should include a statement of the subject you'll examine,
a brief statement of what you'll do, and at least two specific
references for it.
The entire project is due at 5:00 pm Friday,
December
18, the day of the Final Exam. The due date and time are
very firm.
It should not take over 15 hours
work total, and should be 5 to 10 pages in length including
programs and
outputs.
Please edit you project to produce a reasonable document with the usual
standards for written English and for readability. Mere length will definitely
not be rewarded.
Suggested subjects:
Investigate one of the subjects not
covered in the course but which
appear in the Kincaid&Cheney text:
IEEE 754 standard for floating point computations (section
2.1 is too old and confusing)
computing roots of polynomials (3.5)
complex Newton's method (3.5)
least squares for solving overdetermined linear systems
QR factorization and the Gram-Schmidt method
Cholesky factorization (4.2)
matrix norms and condition number (4.4)
iterative refinement (4.5)
iterative numerical linear algebra methods (4.6)
numerical eigenvalue methods (5.1)
SVD decomposition of matrices (5.4)
Hermite interpolation (6.3)
B splines (6.5)
multi-variable polynomial interpolation (6.10)
trigonometric interpolation (6.12) and the Fast Fourier
Transform (6.13)
Richardson extrapolation (7.1)
Romberg integration (7.4)
adaptive quadrature (7.5)
multistep ODE solvers (8.4)
Many topics in the Moler text: choose ones we have not covered.
Examine one of Matlab's
numerical abilities. (Note Matlab has much better (on-line,
detailed) documentation than Octave.)
For instance: sparse matrix support, roots, polyfit and
polyval,
fsolve/fzero,
quad,
fminbnd, fminsearch,
ode45,
...
Perhaps you have an existing project or interest, in science,
engineering, economics, etc., for which there is a
numerical
task. I encourage you to do it here, but be aware that such
projects
have frequently become too involved in the application and then have
not had
much mathematical content. (Come talk to me about it soon,
and we will figure out an appropriate form for the project.)
To do:
Step one is to decide on a subject that interests you.
Look at the Kincaid&Cheney textbook only as one possible
source. It is very likely there is a better, easier-to-read
source. See
Moler for a "light" and practical view. Look at wikipedia
pages. Perhaps see Numerical
Recipes, an online text
at http://www.nrbook.com/a/, or
go to the library to find one of many numerical analysis texts.
Step two is to get feedback from me, by turning
in a proposal on Wednesday 11/25. You are also encouraged to email me
at
any point with ideas or questions. I will
return your proposal with comments and suggested
alterations.
Step three is to set-up a standard outline of what to do,
interpreting each of these appropriately for your topic:
State the problem or class of problems.
Describe an algorithm or a couple of algorithms (more than two
is generally unnecessary and/or unwise). This may involve writing
a code, but that is not always required.
Give at least one example showing how it works. This may
involve applying the code, or doing a small version by hand and then a
big version by the code, but for sure you need a worked example, and
two are usually appropriate.
Evaluate the algorithm:
how good is it?
how fast is it?
what features of a problem will make it fail, or help it
succeed?
The parts of item 4 are numerical
analysis, and numerical
analysis is required in your project. Perhaps this means a
theorem to state, but in any case it means careful mathematical
arguments about the properties of the problem and the algorithm(s).
Step four is to do what you have outlined. It may be that
you
discover the need to do things a different way, once you actually start
working
on it. If such an issue comes up and it is a significant
change
to the plan given in your proposal, please let me know, for instance by
email.