Computer Algebra, MAS5731, Syllabus, Fall 2000.
Location: MCH 107
The course documents and assignments
Time: Wednesday and Friday 12:30 - 1:45
Instructor: Dr. Mark VanHoeij
You do not need a text book for this course because all documents
will be placed on the web. Symbolic integration will be a large part of this
course. If you want a text book on symbolic integration then take
Integration I. This is an excellent book, it is very clear, and it is written by someone
who played an essential role in the development of symbolic integration.
If you would like to learn other topics in
computer algebra then I highly recommend:
Computer Algebra, which is a pleasure to read, contains all the right
topics, it is the best general book to study computer algebra.
Contents Computer Algebra MAS-5731 in Fall 2000.
- e-mail: firstname.lastname@example.org
- URL: http://www.math.fsu.edu/~hoeij
- Office: 105B LOV. Note: this is very near where the construction is going on.
When the noise gets too much (when they are drilling), talking in my office
could become impossible. In such situations you may find me in the Dirac library
instead of in my office, even during office hours. I apologize for the inconvenience.
- Office hours: Monday, Tuesday, Thursday. 10 - 11. You can also visit my office
during other hours, but send me an e-mail first to make an appointment to make sure I'll
be in my office.
Grading: There will be two tests during the semester and a final test. Each
of these three tests will account for 20% of the final grade. The remaining 40%
of the grade will be determined by the turn in assignments, which
will all (or almost all) be computations or algorithms you have to
write in Maple. So it is essential to learn Maple, and we will spend
time on that during the course.
To have access to Maple you need to have an account on one
of the Unix machines. You can also buy Maple in the book
store for $129. Most of the Unix machines have Maple release 5,
but the new version (in the book store) is Maple 6. So I will
write the Maple documents in such a way that they will work
in Maple 5 as well as Maple 6 (or I will point out what needs
to be changed in the document between the two versions).
- Elementary Integration.
See the help page ?int in Maple for some examples.
How can Maple compute such integrals? We will focus on the problem of
elementary integration. A function will be called an elementary function
if it can be built up with the following:
So for example the following functions are elementary functions (note
that sin(x) and cos(x) can be expressed in terms of the exponential
f := x/(x^2+1);
g := sin(x);
h := exp(cos(x)+x^2)/(ln(x)^2+1)^2;
Now the problem of elementary integration is:
Given an elementary function f, does there exist an elementary function F such
that F'=f? If so, how to find F?
Although this problem looks analytical, the solution to this problem is purely
algebraic in nature.
- Rational functions in x with coefficients in the complex numbers C.
- The functions exp and log.
- Additions, multiplications, divisions, and compositions of functions.
- algebraic extensions
- Linear Differential Equations.
ode := (x^3+x)*diff(diff(y(x),x),x)-diff(y(x),x)+y(x)*x^3 = 0;
dsolve(ode); # Maple's solver for differential equations
How does Maple compute something like this?
It turns out that many ideas used in elementary integration can also be
used in linear differential equations. We will study some of the methods.
Again, the algorithms we will study will be algebraic in nature instead