If you are using Bisection to determine the root for a given problem, and the difference between
the initial points is Lo = b – a = 2, determine the number of iterations it will take so that the relative
approximate error isεa ≤ 10-6%. Hint: remember the special tolerance relationship for the bisection
method

The **Bisection Method** is an iterative numerical technique used to find roots of a function \( f(x) \). The stopping criterion often involves ensuring that the relative approximate error \( \varepsilon_a \) falls below a certain tolerance level. A key formula associated with the Bisection Method is:

\[
n \geq \frac{\log \left( \frac{b - a}{\varepsilon} \right)}{\log(2)}
\]

Where:
- \( n \) is the number of iterations,
- \( b - a \) is the initial interval length (given as 2 in this case),
- \( \varepsilon \) is the desired tolerance (in this case, \( \varepsilon_a \leq 10^{-6}\% \)).

### Step 1: Convert tolerance to decimal form

The relative approximate error tolerance \( \varepsilon_a \) is \( 10^{-6}\% \), so we need to convert it to a decimal:
\[
\varepsilon = \frac{10^{-6}}{100} = 10^{-8}
\]

### Step 2: Use the Bisection Method formula

Substitute the given values \( b - a = 2 \) and \( \varepsilon = 10^{-8} \) into the formula:
\[
n \geq \frac{\log \left( \frac{2}{10^{-8}} \right)}{\log(2)}
\]

### Step 3: Solve for \( n \)

Now we calculate the value of \( n \).

\[
n \geq \frac{\log(2 \times 10^8)}{\log(2)}
\]
\[
n \geq \frac{\log(2) + 8\log(10)}{\log(2)}
\]
\[
n \geq \frac{0.3010 + 8 \times 1}{0.3010}
\]
\[
n \geq \frac{8.3010}{0.3010} \approx 27.58
\]

Since the number of iterations must be an integer, we round up to the next whole number:
\[
n \geq 28
\]

### Final Answer:
It will take at least **28 iterations** to ensure that the relative approximate error \( \varepsilon_a \leq 10^{-6}\% \).

---

Would you like further details or clarifications on any of the steps? 

Here are 5 related questions that expand on the topic:
1. How does the Bisection Method ensure convergence to a root?
2. What happens if the initial interval does not bracket a root in the Bisection Method?
3. How does the choice of the tolerance \( \varepsilon \) affect the accuracy and speed of convergence?
4. Can the Bisection Method fail? Under what conditions might this happen?
5. How does the Bisection Method compare to other root-finding methods like Newton's method?

**Tip:** The Bisection Method is guaranteed to converge to a root if the function changes sign over the initial interval \( [a, b] \).

If you are using Bisection to determine the root for a given problem, and the difference between the initial points is Lo = b – a = 2, determine the number of iterations it will take so that the relative approximate error is εa ≤ 10^-6%. Hint: remember the special tolerance relationship for the bisection method.

Learn how to determine the number of iterations needed using the Bisection Method for root finding with an initial interval of 2 and a tolerance of εa ≤ 10^-6%. Explore the formula and step-by-step calculations to ensure the relative approximate error is minimized. Suitable for students in undergraduate mathematics or numerical methods.

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