Update 4.2.md

docs/Chap04/4.2.md

@@ -123,17 +123,71 @@ By master theorem, we can find the largest $k$ to satisfy $\log_3 k < \lg 7$ is
 
 > V. Pan has discovered a way of multiplying $68 \times 68$ matrices using $132464$ multiplications, a way of multiplying $70 \times 70$ matrices using $143640$ multiplications, and a way of multiplying $72 \times 72$ matrices using $155424$ multiplications. Which method yields the best asymptotic running time when used in a divide-and-conquer matrix-multiplication algorithm? How does it compare to Strassen's algorithm?
 
-Using what we know from the last exercise, we need to pick the smallest of the following
+### Analyzing Pan’s Methods
+
+Pan has introduced three methods for divide-and-conquer matrix multiplication, each with different parameters. We will analyze the recurrence relations, compute the exponents using the Master Theorem, and compare the resulting asymptotic running times to Strassen’s algorithm.
+
+#### **Method 1:**
+- **Recurrence Relation:**
 
 $$
-\begin{aligned}
-\log_{68} 132464 & \approx 2.795128 \\\\
-\log_{70} 143640 & \approx 2.795122 \\\\
-\log_{72} 155424 & \approx 2.795147.
-\end{aligned}
+T(n) = 132{,}464 \cdot T\left(\frac{n}{68}\right)
+$$
+
+- **Parameters:** $a = 132{,}464$, $b = 68$
+- **Applying Master Theorem:**
+
+$$
+\log_b a = \frac{\log 132{,}464}{\log 68} \approx 2.7951284873613815
+$$
+
+- **Asymptotic Running Time:** $O(n^{2.7951284873613815})$
+
+#### **Method 2:**
+- **Recurrence Relation:**
+
+$$
+T(n) = 143{,}640 \cdot T\left(\frac{n}{70}\right)
+$$
+
+- **Parameters:** $a = 143{,}640$, $b = 70$
+- **Applying Master Theorem:**
+
 $$
+\log_b a = \frac{\log 143{,}640}{\log 70} \approx 2.795122689748337
+$$
+
+- **Asymptotic Running Time:** $O(n^{2.795122689748337})$
+
+#### **Method 3:**
+- **Recurrence Relation:**
+
+$$
+T(n) = 155{,}424 \cdot T\left(\frac{n}{72}\right)
+$$
+
+- **Parameters:** $a = 155{,}424$, $b = 72$
+- **Applying Master Theorem:**
+
+$$
+\log_b a = \frac{\log 155{,}424}{\log 72} \approx 2.795147391093449
+$$
+
+- **Asymptotic Running Time:** $O(n^{2.795147391093449})$
+
+### **Comparison to Strassen’s Algorithm**
+
+Strassen’s algorithm has an asymptotic running time of $O(n^{2.807})$, derived from $\log_2 7 \approx 2.807$.
+
+### **Which Method is Best?**
+
+Now, let's compare the exponents:
+
+- **Method 1:** $\log_b a = 2.7951284873613815$
+- **Method 2:** $\log_b a = 2.795122689748337$
+- **Method 3:** $\log_b a = 2.795147391093449$
 
-The fastest one asymptotically is $70 \times 70$ using $143640$.
+Since **smaller exponents lead to more efficient algorithms**, **Method 2** is the best because it has the smallest exponent, $2.795122689748337$.
 
 ## 4.2-6