Find the positive square root of 5 using the fixed point iteration method.

To find the positive square root of 5 using the fixed-point iteration method, we need to express the equation \( x = \sqrt{5} \) in the form \( x = g(x) \). One possible iteration function is:

\[ g(x) = \frac{5}{x} \]

We can use this function to iterate and approximate the square root of 5. The fixed-point iteration formula is:

\[ x_{n+1} = g(x_n) \]

Let's choose an initial guess \( x_0 \). A reasonable starting point is \( x_0 = 2 \).

Now we iterate using the formula \( x_{n+1} = \frac{5}{x_n} \) until we converge to a sufficient accuracy.

1. \( x_0 = 2 \)
2. \( x_1 = \frac{5}{x_0} = \frac{5}{2} = 2.5 \)
3. \( x_2 = \frac{5}{x_1} = \frac{5}{2.5} = 2 \)
4. \( x_3 = \frac{5}{x_2} = \frac{5}{2} = 2.5 \)

Notice that in this case, the sequence oscillates between 2 and 2.5. This does not seem to converge directly. Let's consider a modified approach by using the average of our current guess and the result of \( g(x) \):

\[ g(x) = \frac{x + \frac{5}{x}}{2} \]

This new function should help us achieve convergence:

1. \( x_0 = 2 \)
2. \( x_1 = \frac{x_0 + \frac{5}{x_0}}{2} = \frac{2 + \frac{5}{2}}{2} = \frac{2 + 2.5}{2} = 2.25 \)
3. \( x_2 = \frac{x_1 + \frac{5}{x_1}}{2} = \frac{2.25 + \frac{5}{2.25}}{2} = \frac{2.25 + 2.2222}{2} \approx 2.2361 \)
4. \( x_3 = \frac{x_2 + \frac{5}{x_2}}{2} = \frac{2.2361 + \frac{5}{2.2361}}{2} = \frac{2.2361 + 2.2361}{2} = 2.2361 \)

After a few iterations, the value converges to approximately 2.2361, which is the positive square root of 5.

Would you like more details on this method or have any questions?

Here are some related questions you might find interesting:

1. What are other methods to approximate square roots?
2. How does the convergence rate of the fixed-point iteration method compare with the Newton-Raphson method?
3. Can fixed-point iteration be used for solving other types of equations?
4. What are the potential issues with choosing the initial guess in the fixed-point iteration method?
5. How does one determine if a fixed-point iteration will converge?
6. What are the applications of fixed-point iteration in numerical analysis?
7. Can fixed-point iteration be applied to systems of equations?
8. How is the error in each iteration of the fixed-point iteration method estimated?

**Tip:** When using fixed-point iteration, it's essential to ensure that the function \( g(x) \) is chosen such that it converges. A common practice is to look for a function \( g(x) \) that satisfies the condition \( |g'(x)| < 1 \) in the interval of interest.

Discover how to find the positive square root of 5 using the fixed-point iteration method. Step-by-step solution with iteration function and formulas included. Suitable for advanced high school students and anyone interested in numerical methods.

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