Let's solve the recurrence relation $T(n) = 9T\left(\frac{n}{6}\right) + n \log^2 n$ using the Master Theorem for divide-and-conquer recurrences.

The recurrence is of the form: $T(n) = aT\left(\frac{n}{b}\right) + f(n)$ Where:

• $a = 9$
• $b = 6$
• $f(n) = n \log^2 n$

To apply the Master Theorem, we compare $f(n)$ with $n^{\log_b a}$.

Step 1: Calculate $\log_b a$

We need to compute $\log_b a = \log_6 9$. Using the change of base formula: $\log_6 9 = \frac{\log 9}{\log 6}$ We approximate: $\log 9 \approx 0.9542, \quad \log 6 \approx 0.7782$ Thus: $\log_6 9 \approx \frac{0.9542}{0.7782} \approx 1.226$ So, $n^{\log_6 9} \approx n^{1.226}$.

Step 2: Compare $f(n) = n \log^2 n$ with $n^{\log_b a}$

We now compare the asymptotic behavior of $f(n) = n \log^2 n$ with $n^{\log_6 9}$:

• $f(n) = n \log^2 n$ grows slightly slower than $n^{1.226}$, because $n \log^2 n$ is $n$ times a polylogarithmic term, while $n^{1.226}$ has a strictly larger exponent of $n$.

Step 3: Apply Master Theorem

The Master Theorem cases are:

1. Case 1: If $f(n) = O(n^{\log_b a - \epsilon})$ for some $\epsilon > 0$, then $T(n) = \Theta(n^{\log_b a})$.
2. Case 2: If $f(n) = \Theta(n^{\log_b a} \log^k n)$ for some $k \geq 0$, then $T(n) = \Theta(n^{\log_b a} \log^{k+1} n)$.
3. Case 3: If $f(n) = \Omega(n^{\log_b a + \epsilon})$ for some $\epsilon > 0$ and if $a f\left(\frac{n}{b}\right) \leq c f(n)$ for some constant $c < 1$ and sufficiently large $n$, then $T(n) = \Theta(f(n))$.

In this case, $f(n) = n \log^2 n$ is asymptotically smaller than $n^{\log_6 9}$ but not polynomially smaller. Therefore, we are in Case 2 of the Master Theorem. Hence, the solution to the recurrence is:

$T(n) = \Theta(n^{\log_6 9} \log^3 n)$

Final Answer:

The solution to the recurrence $T(n) = 9T(n/6) + n \log^2 n$ by the Master Method is: $T(n) = \Theta(n^{1.226} \log^3 n)$

Would you like any additional details or clarifications?

---

Here are 5 related questions:

1. How does the Master Theorem handle logarithmic factors in $f(n)$?
2. What conditions are needed for each case of the Master Theorem to apply?
3. How do we handle non-standard functions like $\log^2 n$ in recurrence relations?
4. Can the Master Theorem be applied to all recurrences?
5. How does the change of base formula help in solving recurrences with non-integer bases?

Tip: When solving recurrence relations, carefully compare the growth rates of the terms to choose the correct case of the Master Theorem.