 
Summary: Math 581, Fall 2006
Project 1. Calculating the number .
Due November 16, 2005
Topics covered. Cancellation. Running error analysis.
Problem 1. Consider the sequence of numbers defined by the fol
lowing recurrence relation:
x0 = 2,
xi = 2  4  (xi1)2, i = 1, 2, 3, . . .(1)
It can be verified by a geometrical argument that
= lim
i
xi2i
· Write a code implementing algorithm (1).
· Use this code to calculate approximately
xn2n
for values of n = 5, 10, 15, 20, 25, 30. Explain what you observe.
· Modify (1) to avoid cancellation. Implement the new algorithm
in the computer code. Check that the new code computes value
of better.
Problem 2 A rather fast approximation of can be obtained by
