Computer Algebra, Fall 2000.

Week 1 and 2.

• We will look at this Maple worksheet on sequences in class.
• We will look at this worksheet on polynomials and the use of the quotes: ' ' in class. If there's enough time we'll look at one of the following worksheets: worksheet on rational functions, worksheet on matrices, worksheet on modular arithmetic, Study the remaining worksheets for next time.
• Home work for week 1. Study the worksheets we did in class very carefully. If anything is not clear, ask about it in the next class.
  Click on the Help menu (on the right at the top of the Maple window), select Introduction, and click on the "New User's Tour". This also contains useful information for new users. Spend a few hours this week to find out what kind of commands are available in Maple. Every hour you spend now will save you a lot of time in the programming exercises later in the semester.
  Make yourself familiar with the help system of Maple. Suppose you are interested in differentiation, then type the following in Maple: ?differentiate and you will quickly find out what the syntax for differentiation is in Maple, and that the name of the command is diff. Particularly useful in these help pages are the examples, and the "See Also" section, which gives you the names of related commands. This way you also find out quickly what the syntax for integration is. Look at the help pages of the following commands, and try examples for each of them: expand, factor, factors, sqrfree, gcd, collect, ifactor, igcd, solve, fsolve, isolve, dsolve, plot, plot3d, tubeplot, add, mul, seq, print, convert, simplify, series.
  Turn in home work: Compute the set of all cubes of all integers from 1 up to 25. Then compute the set of all numbers that are the sum of three of such cubes. Then determine the intersection so that you find: The set of all cubes of integers 1 .. 25 that are itself the sum of three (not necessarily distinct) cubes. Then e-mail me (hoeij@math.fsu.edu) the Maple commands that you used to find this set. Hint: something very similar (but then the sum of 2 squares instead of 3 cubes) was done in the worksheet on sequences.
• Homework for next Friday (Sep 08): Factor the polynomial F:=x^15+4*x^14+2*x^13-19*x^12-57*x^11-54*x^10+66*x^9+267*x^8+369*x^7+170*x^6-244* x^5-611*x^4-656*x^3-456*x^2-192*x-48;
  as far as you can without using any Maple command for factoring, use just the commands gcd, normal and diff.

• Integration of rational functions. Homework: do the exercise in this worksheet.
• Integration of rational functions, part 2.
• Short introduction to resultants. For more on resultants see the course from fall 1999.
• The residue.

• The residues, contains assignment for Friday.
• Differential fields.

• Logarithmic extensions, polynomial case, contains assignment.
• Logarithmic extensions, general case, contains assignment.

• Valuations.
• One more example for integration with a logarithmic extension. Study this worksheet!
• Liouvilles principle.

• The exponential case..
• Programming: Study the following worksheet intro and rational functions.

• Wednesday: We will practise writing programs in class. The following worksheet contains an assignment: Examples of if-then-elif-fi and do-od.
• Friday: I will be absent on Friday November 3, so there will be no class on that day. Study the following worksheet: Linear Differential Operators..

• Multiplication, division, and GCRD
• Exponents of an operator.
• Rational Solutions.

• Integration of solutions differentiation, integration, more examples.
  Assignment: Write a procedure sigma like in the worksheet. Then write a procedure IntSol:=proc(f,M,Dx,x) ..... end;

  The input of this procedure is the following: f is a function in x, and M in C(x)[Dx] is an operator for which M(f)=0. The output of the procedure IntSol should be either FAIL (if there exists no L in C(x)[Dx] for which D:V(L)-->V(M) is 1-1 and onto) or an antiderivative r(f) of f otherwise. Give an example where the procedure returns FAIL and another example where it returns an antiderivative of f. Use your procedure to integrate f:=erf(x) (you first have to find an operator M in C(x)[Dx] for which M(f)=0. This operator turns out to have order 2. Find M by computing f, diff(f,x), diff(f,x,x) and then searching for rational functions a0,a1,a2 for which a0*f+a1*diff(f,x)+a2*diff(f,x,x)=0).

• Sample test 2.
• Answers sample test 2. Do not look at the answers until you have solved as much of the sample test as you can.