Question

In this project you are going to experience a \"chaos\" using Newton method applied to the function $f(x) = x^3 - x = x(x - 1)(x + 1)$ You will use the Newton method (also known as Newton-Raphson method) to investigate what happens as you change the starting point $x_0$ to solve $f(x) = 0$. Recall that the sequence of points $x_1, x_2, x_3,...$ can be generated using the following formula $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ What would we choose as a starting value? a) Explain why one would not want to select the starting point to be $x_0 = \pm \frac{1}{\sqrt{3}}$ (3 pts.) b) What happens if one selects the starting point $x_0 = \frac{1}{\sqrt{5}}$? (You must carry out the computations in the exact form.) (3 pts.) c) Use an Excel sheet to generate a good number of iterations. Use the following $x_0$ values until the convergence occurs and report what happens: 0.447503, 0.447262, 0.447222, and 0.4472215. Include how many iterations are required for the convergence to occur. (4 pts.) d) Select several values greater than $\frac{1}{\sqrt{3}}$ and numerically verify that $x_n \to 1$ (2 pts.) e) Select several values less than $\frac{1}{\sqrt{3}}$ and numerically verify that $x_n \to 0$. (2 pts.) f) You have shown in (b) that $\frac{1}{\sqrt{5}}$ should not be selected as a starting value. Try several numbers close to $\frac{1}{\sqrt{5}}$ (as close as 7, 8, and 9 decimal places). Record your observation and write a comment. (4 pts.) g) Use Excel to verify your answer in (b) by choosing $x_0$ to be exactly $-\frac{1}{\sqrt{5}}$ (2 pts.)

          In this project you are going to experience a \"chaos\" using Newton method applied to the function
$f(x) = x^3 - x = x(x - 1)(x + 1)$
You will use the Newton method (also known as Newton-Raphson method) to investigate what happens as you change the starting point $x_0$ to solve $f(x) = 0$. Recall that the sequence of points $x_1, x_2, x_3,...$ can be generated using the following formula
$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$
What would we choose as a starting value?
a) Explain why one would not want to select the starting point to be $x_0 = \pm \frac{1}{\sqrt{3}}$ (3 pts.)
b) What happens if one selects the starting point $x_0 = \frac{1}{\sqrt{5}}$?
(You must carry out the computations in the exact form.) (3 pts.)
c) Use an Excel sheet to generate a good number of iterations. Use the following $x_0$ values until the convergence occurs and report what happens: 0.447503, 0.447262, 0.447222, and 0.4472215. Include how many iterations are required for the convergence to occur. (4 pts.)
d) Select several values greater than $\frac{1}{\sqrt{3}}$ and numerically verify that $x_n \to 1$ (2 pts.)
e) Select several values less than $\frac{1}{\sqrt{3}}$ and numerically verify that $x_n \to 0$. (2 pts.)
f) You have shown in (b) that $\frac{1}{\sqrt{5}}$ should not be selected as a starting value. Try several numbers close to $\frac{1}{\sqrt{5}}$ (as close as 7, 8, and 9 decimal places). Record your observation and write a comment. (4 pts.)
g) Use Excel to verify your answer in (b) by choosing $x_0$ to be exactly $-\frac{1}{\sqrt{5}}$ (2 pts.)

In this project you are going to experience a c̈haosüsing Newton method applied to the function
f(x) = x^3 - x = x(x - 1)(x + 1)
You will use the Newton method (also known as Newton-Raphson method) to investigate what happens as you change the starting point x0 to solve f(x) = 0. Recall that the sequence of points x1, x2, x3,... can be generated using the following formula
xn+1 = xn - (f(xn))/(f'(xn))
What would we choose as a starting value?
a) Explain why one would not want to select the starting point to be x0 = ±(1)/(√(3)) (3 pts.)
b) What happens if one selects the starting point x0 = (1)/(√(5))?
(You must carry out the computations in the exact form.) (3 pts.)
c) Use an Excel sheet to generate a good number of iterations. Use the following x0 values until the convergence occurs and report what happens: 0.447503, 0.447262, 0.447222, and 0.4472215. Include how many iterations are required for the convergence to occur. (4 pts.)
d) Select several values greater than (1)/(√(3)) and numerically verify that xn → 1 (2 pts.)
e) Select several values less than (1)/(√(3)) and numerically verify that xn → 0. (2 pts.)
f) You have shown in (b) that (1)/(√(5)) should not be selected as a starting value. Try several numbers close to (1)/(√(5)) (as close as 7, 8, and 9 decimal places). Record your observation and write a comment. (4 pts.)
g) Use Excel to verify your answer in (b) by choosing x0 to be exactly -(1)/(√(5)) (2 pts.)

Added by Todd B.

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