# Exercises for Chaos Under Control

## Chapter 8: Newton's Method for Digging Up Roots

6. Refer to the Figure.

(a) Sketch the first two points, x1 and x2, generated by applying Newton's method to the guess x0.

(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) = x2.

(a) Show that Newton's method for finding the roots of f(x) is equivalent to the iterative scheme xn+1 = xn/2. Answer

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

(c) What happens when Newton's method is iterated, starting with x0 = 4? Answer

(d) Repeat (c) taking as starting points x0 = -4, ±2, ±1, and ±1/3. What do you conclude happens in general? Answer

(e) Note that iterating Newton's method for x2 is different from iterating x2 itself. What is the basin of attraction of x = 0 when iterating x2? Answer

8.[A] Let f(x) be the function defined by f(x) = x2 - 1.

(a) Show that Newton's method for finding the roots of f(x) is equivalent to the iterative scheme xn+1 = (xn2 + 1)/(2xn). Answer

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

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

(d) From the graph in (c), deduce xn = 1 and xn = -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 z2 - 1?

9.[A] Let f(x) = x2 + 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 xn+1 = (xn2 - 1)/(2xn).

(c) Using a calculator, plot the values of xn+1 for xn = ±1/8, ±1/4, ±1/2, ±3/4, ±1, ±5/4, ±3/2, ±2, ±3. Sketch the graph of xn+1 as a function of xn. 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 x0 = 0.333. Answer

(f) Repeat (e) starting with x0 = 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 fc(x) = x2 + c, for c and x real numbers.

(a) Show fc(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 = z3 + 3z2d + 3zd2 + d3.

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

zn+1 = (2*zn)/3 + 1/(3*zn2).

14.[A] The complex number 1/(A + B*i) is not in standard form: it has the i downstairs. To put it in standard form multiply both numerator and denominator by A - B*i. Show that

1/(A + B*i) = (A - B*i)/(A2 + B2).

(Thus, 1/(A+B*i) has the real part A/(A2 + B2) and the imaginary part -B/(A2 + B2).)

15.[A] With the help of exercise 14, show that Newton's method for finding the roots of z3 - 1 is equivalent to a set of transformations

xn+1 = (2*xn/3) + (xn2 + yn2)/D

and

yn+1 = (2*yn/3) - 2*xn*yn/D

where D = 3*((xn2 - yn2)2 + 4*xn2*yn2) = 3*(xn2 + yn2)2.