Iterative methods are indispensable in solving equations where direct algebraic solutions are impractical. They use a function repeatedly to approximate a root, offering a practical approach to problem-solving across various mathematical domains.

**Understanding Iterative Formulas**

Iterative formulas generate a sequence approximating the root of an equation, with the general form $x_{n+1} = F(x_n)$, where $F$ is derived from the original equation.

### Key Concepts:

**Iterative Formula:**Recursive relation where each term is based on the previous one.**Convergence:**Sequence values approach a specific value, the root.**Divergence:**Sequence does not settle, move away, or oscillate.

## Connecting Iterative Formulas to Equations:

Iterative formulas are tailored to the equations they solve, often derived by rearranging the original equation.

### Example:

For the equation $x^2 - 2 = 0$, an iterative formula could be $x_{n+1} = \sqrt{2}$, derived from rearranging.

**Solution:**

**Rearranging the Equation:**Rearranged form: $x = \sqrt{2}$.**Developing the Iterative Formula:**Iterative formula: $x_{n+1} = \sqrt{2}$.**Using the Iterative Formula:**Start with $x_0$, and use the formula for subsequent approximations.

**Iteration Process:**

1. **Initial Guess: **$x_0 = 1$.

2. **First Iteration: **$x_1 = \sqrt{2} \approx 1.414$.

3. **Second Iteration:** $x_2 = \sqrt{2} \approx 1.414$.

4. **Third Iteration: **$x_3 = \sqrt{2} \approx 1.414$.

5. **Subsequent Iterations:** Continue applying the formula, each yielding $\sqrt{2} \approx 1.414$, indicating convergence.

**Conclusion: **The iterative formula consistently approximates the square root of 2, demonstrating convergence to $\sqrt{2}$, validating the equation.

### Techniques for Achieving Accuracy:

- Initial Guess: Choose close to the root.
- Iteration Count: Determine the needed number to reach the desired accuracy.
- Error Estimation: Calculate the difference between successive iterations to gauge convergence.

## Worked Example

Approximate the root of $x^2 - 2 = 0$ using the iterative method.

#### Solution:

1. **Setting Up the Problem:**

- Solve $x^2 - 2 = 0$ using iteration. The root is $\sqrt{2}$.

2. **Choosing an Initial Guess:**

- Initial guess $x_0 = 1$, near the actual root $\sqrt{2}$.

3. **Iterative Method:**

- Use the Babylonian method: $x_{n+1} = \frac{1}{2} \left( x_n + \frac{2}{x_n} \right)$.

4. **Applying the Iterations:**

**Iteration 1:**$x_{1} = 1.5$**Iteration 2:**$x_{2} \approx 1.4167$**Iteration 3:**$x_{3} \approx 1.4143$

5. **Error Estimation:**

- Estimate error as $|x_{n+1} - x_n|$. For the third iteration, error $\approx 0.0024$.

6. **Deciding When to Stop:**

- Continue until error < 10^{-4}. After three iterations, error is within the acceptable range.

**Conclusion:**

After three iterations, the root of $x^2 - 2 = 0$ is approximated as $\approx 1.4143$, illustrating the effectiveness of iterative methods in approximating solutions to equations with desired accuracy.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.