Method of Iteration to Compute a Real Root

The method of iteration, also known as the fixed-point iteration, is a numerical technique to find a root of the equation $f(x) = 0$. The procedure involves rewriting the equation in the form $x = g(x)$ and then using the iterative formula:

$x_{n+1} = g(x_n)$

Here are the key steps and conditions for convergence:

1. Rewrite the Equation: Express the equation $f(x) = 0$ in the form $x = g(x)$.

2. Initial Guess: Choose an initial guess $x_0$.

3. Iteration Process: Compute successive approximations using $x_{n+1} = g(x_n)$.

4. Convergence Check: The sequence $\{x_n\}$ converges to a fixed point $\alpha$ if $g(\alpha) = \alpha$.

5. Condition for Convergence: The method converges if $|g'(x)| < 1$ in the neighborhood of the fixed point.

Now, let's apply the method to the given equations:

(a) $e^x = 3x$

Rewrite the equation as:

$x = \frac{e^x}{3}$

Choose $g(x) = \frac{e^x}{3}$.

Let's iterate using an initial guess, $x_0 = 1$:

1. $x_1 = g(x_0) = \frac{e^1}{3} \approx 0.9061$
2. $x_2 = g(x_1) = \frac{e^{0.9061}}{3} \approx 0.9622$
3. Continue this process until $x_n$ converges to four significant figures.

(b) $x = \frac{1}{(x+1)^2}$

This equation is already in the form $x = g(x)$ where:

$g(x) = \frac{1}{(x+1)^2}$

Choose an initial guess, $x_0 = 0.5$:

1. $x_1 = g(x_0) = \frac{1}{(0.5+1)^2} = \frac{1}{1.5^2} \approx 0.4444$
2. $x_2 = g(x_1) = \frac{1}{(0.4444+1)^2} \approx 0.4762$
3. Continue this process until $x_n$ converges to four significant figures.

(c) $1 + x^2 = x^3$

Rewrite the equation as:

$x = \sqrt[3]{1 + x^2}$

Choose $g(x) = \sqrt[3]{1 + x^2}$.

Choose an initial guess, $x_0 = 1$:

1. $x_1 = g(x_0) = \sqrt[3]{1 + 1^2} \approx 1.2599$
2. $x_2 = g(x_1) = \sqrt[3]{1 + (1.2599)^2} \approx 1.2931$
3. Continue this process until $x_n$ converges to four significant figures.

(d) $x - \sin x = \frac{1}{2}$

Rewrite the equation as:

$x = \sin x + \frac{1}{2}$

Choose $g(x) = \sin x + \frac{1}{2}$.

Choose an initial guess, $x_0 = 0.5$:

1. $x_1 = g(x_0) = \sin(0.5) + \frac{1}{2} \approx 0.9794$
2. $x_2 = g(x_1) = \sin(0.9794) + \frac{1}{2} \approx 1.3195$
3. Continue this process until $x_n$ converges to four significant figures.

Summary

• Iteration Formula: $x_{n+1} = g(x_n)$
• Convergence Condition: $|g'(x)| < 1$ near the fixed point

Would you like to see the detailed calculations or have any questions?

Suggested Questions

1. Can you explain the convergence condition in more detail?
2. How do you determine a good initial guess for the iteration?
3. What happens if the sequence does not converge?
4. Can you show the iteration steps for a specific example?
5. How is the derivative $g'(x)$ used in practice to check convergence?

Tip

When using the method of iteration, always check the derivative $g'(x)$ to ensure the convergence condition $|g'(x)| < 1$ is satisfied near the expected root.