The Golden Section Search is an iterative method used to find the minimum (or maximum) of a unimodal function within a specified interval. In this case, we want to find the minimum of $f(x) = x(x - 1.5)$ in the interval $[0, 1]$, with an interval of uncertainty of 0.3. Here's a step-by-step outline of how to apply the method:

Step 1: Define the Golden Ratio

The Golden Ratio, denoted by $\phi$, is approximately 1.618. The inverse of the Golden Ratio, $1 - \frac{1}{\phi}$, is approximately 0.618.

Step 2: Initialization

• Define the initial interval as $[a, b] = [0, 1]$.
• The interval of uncertainty is 0.3, meaning we stop the process when the length of the interval is less than or equal to 0.3.

Step 3: Compute the Two Interior Points

Let the two interior points be $x_1$ and $x_2$ within the interval $[a, b]$:

• $x_1 = b - 0.618(b - a)$
• $x_2 = a + 0.618(b - a)$

For the first iteration:

• $a = 0$
• $b = 1$

Thus,

• $x_1 = 1 - 0.618(1 - 0) = 1 - 0.618 = 0.382$
• $x_2 = 0 + 0.618(1 - 0) = 0.618$

Step 4: Evaluate the Function at $x_1$ and $x_2$

We now evaluate the function $f(x) = x(x - 1.5)$ at $x_1 = 0.382$ and $x_2 = 0.618$:

• $f(0.382) = 0.382(0.382 - 1.5) = 0.382(-1.118) = -0.426$
• $f(0.618) = 0.618(0.618 - 1.5) = 0.618(-0.882) = -0.545$

Step 5: Determine the New Interval

Since $f(x_2) < f(x_1)$, the minimum must lie in the interval $[x_1, b]$. Therefore, update the interval to:

• $a = 0.382$
• $b = 1$

Step 6: Repeat the Process

Now we repeat the process with the updated interval. Compute the new interior points:

• $x_1 = b - 0.618(b - a) = 1 - 0.618(1 - 0.382) = 0.764$
• $x_2 = a + 0.618(b - a) = 0.382 + 0.618(1 - 0.382) = 0.618$

Evaluate the function at $x_1 = 0.764$ and $x_2 = 0.618$:

• $f(0.764) = 0.764(0.764 - 1.5) = 0.764(-0.736) = -0.562$
• $f(0.618) = -0.545$ (already computed)

Since $f(x_1) < f(x_2)$, the minimum lies in the interval $[x_2, b] = [0.618, 1]$.

Step 7: Stopping Criterion

At this point, the interval length $b - a = 1 - 0.618 = 0.382$, which is larger than the interval of uncertainty (0.3), so we need one more iteration.

Repeat the process with the updated interval $[0.618, 1]$, and continue until the interval length is less than or equal to 0.3.

Conclusion:

After enough iterations, we will reach a point where the interval of uncertainty is less than 0.3. The value of $x$ at this point will be close to the minimum of the function within the specified interval.

Do you need more details or have any questions?

Here are 5 related questions:

1. What is the convergence rate of the Golden Section Search method?
2. How does the Golden Section Search compare to other optimization methods like Newton's method?
3. Can the Golden Section Search be used for multi-dimensional optimization?
4. What happens if the function has multiple minima in the given interval?
5. How do you choose the initial uncertainty interval for the Golden Section Search?

Tip: The Golden Section Search is particularly useful when the derivative of the function is difficult to calculate or unavailable.