MA522 Homework
(due Oct 29, 4:59pm, by email or in my mailbox in SAS 3151).

Problem 1: Richard Brent in 1980 gave a different algorithm than Floyd's for
finding a cycle in a sequence x_0, x_1=f(x), x_2=f(f(x)),... that uses fewer
applications of f and again only 2 funciton values stored for "equality testing."
Describe Brent's approach, possibly giving a Maple procedure, by either
searching the Internet/literature or expanding my hint.  In particular, compare
the number of function applications.

Hint: for all n there exists k such that n <= 2^k < 2n. One compares
for all i the value x_{2^i} with all x_{2^i+1},x_{2^i+2},...,x_{2^{i+1}}.

Problem 2: Expand the fraction-free Gaussian elimination algorithm to
the back-substitution process.

Hint: by Cramer's rule, one knows each solution as a fraction of
determinants of submatrices of the augmented matrix.

Problem 3 is Problem 6.42 [on page 202 in von zur Gathen and Gerhard 2003].

Optional exercise: implement Teske's Pollard rho discrete log method
and compare its performance to the one using 3 different cases of updates.