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Chapter 2: Problem 27

How many multiplications would be performed in finding the product of two \(64 \times 64\) matrices using the standard algorithm?

Short Answer

Expert verified
Therefore, the total number of multiplications performed when multiplying two 64×64 matrices using the standard matrix multiplication algorithm is \( 64*4096 = 262,144 \)

Step by step solution

01

Understanding Matrix Multiplication

The process of multiplying two matrices, in this case, two 64x64 matrices, involves multiplying each element in a row of the first matrix with the corresponding element in a column of the second matrix and then adding the results. This is done for each pair of rows and columns.
02

Calculation of Single Element

To calculate a single element in the product matrix, we perform 64 multiplications. This is because each element in a row of the first matrix is multiplied by the corresponding element in a column of the second matrix, and the results are added together. As the matrices are 64x64, there are 64 elements in each row and each column.
03

Calculation of All Elements

In our case, there are 64x64 = 4096 elements in the product matrix, as it inherits dimensions from the input matrices. Since each element requires 64 multiplications to calculate, the total number of multiplications performed is 64 (multiplications per element) * 4096 (total number of elements).

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Most popular questions from this chapter

Write an algorithm that searches a sorted list of \(n\) items by dividing it into three sublists of almost \(n / 3\) items. This algorithm finds the sublist that might contain the given item and divides it into three smaller sublists of almost equal size. The algorithm repeats this process until it finds the item or concludes that the item is not in the list. Analyze your algorithm and give the results using order notation.

Show that if \\[W(n) \leq \frac{(p-1)(p-2)}{2}+\frac{(n-p)(n-p-1)}{2}+n-1\\] then \\[W(n) \leq \frac{n(n-1)}{2} \quad \text { for } 1 \leq p \leq n\\] This result is used in the discussion of the worst-case time complexity analysis of Algorithm 2.6 (Quicksort).

Suppose that, even unrealistically, we are to search a list of 700 million items using Binary Search, Recursion (Algorithm 2.1). What is the maximum number of comparisons that this algorithm must perform before finding a given item or concluding that it is not in the list?

Use Mergesort (Algorithms 2.2 and 2.4 ) to sort the following list. Show the actions step by step. \(\begin{array}{llllllll}123 & 34 & 189 & 56 & 150 & 12 & 9 & 240\end{array}\)

Write a nonrecursive algorithm for Quicksort (Algorithm 2.6). Analyze your algorithm, and give the results using order notation.
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