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.