4. [1+1+1+1pt] The equation

f(x):= 2² - 5 = 0,

has a single root = √5 ≈ 2.2361... in the interval [1,3]. Consider the fixed point iteration

xk+1 = g(x), where g is defined as one of the following options:

• 9₁(x) = 5 + x - x²,

¹Note that we typically thing of local convergence, i.e., close to the limit point.

92(x) = 1 + x - x²,

93(x) = x + 2

2

(a) Identify the fixed point functions for which the fixed point is also a root of f.

(b) For the cases where computing the fixed point is equivalent with solving f(x) = 0, discuss

whether the fixed point iteration is guaranteed to converge in some neighborhood of §.

(c) If the iteration in b) is guaranteed to converge, compute the value of

|æk+1 −

lim

k→∞ |æk - €

Hint: Everything is easier with the mean value theorem!

(d) Give Newton's method for solving f(x) = 0 and, for given starting value = 2 compute

the first Newton iterate 2₁.