6. Refer to the Figure.

(a) Sketch the first two points, x_{1} and x_{2}, generated by applying
Newton's method to the guess x_{0}.

(b) Locate two nearby starting points, A and B, such that Newton's method from A leads to the left root (denoted L), and Newton's method from B leads to the right root (denoted R). Sketch the first few steps of applying Newton's method to both of these guesses.

7.[A] Let f(x) be the function defined by f(x) = x^{2}.

(a) Show that Newton's method for finding the roots of f(x) is equivalent to the iterative
scheme x_{n+1} = _{n}/2

(b) Show x = 0 is the root of f(x).

(c) What happens when Newton's method is iterated, starting with x_{0} = 4?
Answer

(d) Repeat (c) taking as starting points x_{0} = -4, ±2, ±1, and ±1/3. What do you
conclude happens in general?
Answer

(e) Note that iterating Newton's method for x^{2} is different from iterating
x^{2} itself. What is the basin of attraction of x = 0 when iterating x^{2}?
Answer

8.[A] Let f(x) be the function defined by f(x) = ^{2} - 1

(a) Show that Newton's method for finding the roots of f(x) is equivalent to the
iterative scheme x_{n+1} = _{n}^{2} + 1)/(2x_{n})

(b) Show x = 1 and x = -1 are fixed points for this Newton map.

(c) Using a calculator, plot the values of x_{n+1} for x_{n} =
±1/8, ±1/4, ±1/2, ±3/4, ±1, ±5/4, ±3/2, ±2, ±3. Sketch the graph of x_{n+1} as a
function of x_{n}.
Answer

(d) From the graph in (c), deduce x_{n} = 1 and x_{n} = -1 are
*stable* fixed points of the Newton map.
Answer

(e) Using graphical iteration, find the basins of attraction of these roots. Answer

(f) Is your answer to (e) consistent with Cayley's result for ^{2} - 1

9.[A] Let f(x) = ^{2} + 1

(a) Show this f(x) has no roots that are real numbers.

(b) Show that Newton's method for finding the roots of f(x) is equivalent to the
iterative scheme x_{n+1} = _{n}^{2} - 1)/(2x_{n})

(c) Using a calculator, plot the values of x_{n+1} for x_{n} =
±1/8, ±1/4, ±1/2, ±3/4, ±1, ±5/4, ±3/2, ±2, ±3. Sketch the graph of x_{n+1}
as a function of x_{n}.
Answer

(d) From the graph of (c), deduce that the Newton map has no fixed points. Answer

(e) Using a calculator, determine the first ten iterates of Newton's method
starting with x_{0} = 0.333.
Answer

(f) Repeat (e) starting with x_{0} = 0.334.
Answer

(g) What do the results of (e) and (f) suggest to you about the dynamics of this map? Try other starting values, being careful to avoid anything leading to x = 0. Answer

10.[A] This is a generalization of the previous three exercises. Consider the family
of functions f_{c}(x) = ^{2} + c

(a) Show f_{c}(x) has two real roots when c < 0. What are these roots?
Answer

(b) Using a calculator, sketch the graph of the Newton map for c = -1/2. Answer

(c) Use graphical iteration to determine the fate of initial points under this Newton map. Answer

(d) Repeat (b) and (c) for c = -3/2. Answer

(e) What do you think happens for any c < 0? Answer

(f) Repeat (b) and (c) for c = 1/2. Answer

(g) What do you think happens for any c > 0? Answer

11.[A] Suppose f(x^{*}) = 0 and f'(x^{*}) not equal to 0. Show x^{*}
is a fixed point for the associated Newton map.
Answer

12.[A] Show that (z+d)^{3} = ^{3} + 3z^{2}d +
3zd^{2} + d^{3}

13.[A] Show that Equation (4) can be rewritten as

z_{n+1} = _{n})/3 + 1/(3*z_{n}^{2})

14.[A] The complex number

^{2} + B^{2})

(Thus, ^{2} + B^{2})^{2} + B^{2})

15.[A] With the help of exercise 14, show that Newton's method for finding the roots of
^{3} - 1

_{n+1} = (2*x_{n}/3) + (x_{n}^{2} +
y_{n}^{2})/D

and

_{n+1} = (2*y_{n}/3) - 2*x_{n}*y_{n}/D

where D = _{n}^{2} -
y_{n}^{2})^{2} +
4*x_{n}^{2}*y_{n}^{2})_{n}^{2} +
y_{n}^{2})^{2}

Return to Chapter 8 Exercises

Return to Chapter 8 Exercises: Basins of Attraction

Go to Chapter8 Exercises: Another Surprise in Newton's Method

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