Discussion:
Q: The equation x - 3 ln x = 2 has exactly two solutions A and B, with 0 < A < B.
(a) Show that A is in [0.5,0.7], and B is in [8.3,8.5].
(b) Consider the following fixed-point iteration for finding a solution of the given equation: xn+1 = e(1/3(Xn-2)): Show that if X0 = B+ E, where E is a positive real number, then X1 > X0. Deduce that this Fixed point iteration does not converge to ¯Before all X0 > B. (Hint: You may need to use the facts that B is a solution of x-3 ln x = 2,
and that B> 3.)
(c) Consider the following fixed point iteration for finding a solution of the given equation:
Xn+1 = 2 + 3 ln Xn:
(i) Show that this fixed point iteration converges to B for all X0 is in [6.5,9.5].
(ii) Starting with X0 = 8.4, use this fixed point iteration to find B correct to four significant figures.