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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
The total number of multiplications performed in finding the product of two \(64 \times 64\) matrices using the standard matrix multiplication algorithm is 262,144.

Step by step solution

01

Understand Matrix Multiplication Operation

In matrix multiplication, each element in the resulting matrix is computed by a sum of products from input matrices. For instance, assuming A and B are \(2 \times 2\) matrices, then the product C = AB would have its element \(c_{11}\) in the first row and first column computed as follows: \(c_{11} = a_{11}b_{11} + a_{12}b_{21}\). Therefore, you can observe that to calculate each element of the resulting matrix, we perform two multiplication operations.
02

Number of Elements in the Resulting Matrix

The resulting matrix from the multiplication of two \(64 \times 64\) matrices is also a \(64 \times 64\) matrix. That means it has \(64 \times 64 = 4096\) elements in total.
03

Calculate the Total Number of Multiplication Operations

To compute each element in the resulting matrix, we perform 64 multiplication operations, as seen in the first step. Therefore, for all 4096 elements, the total multiplication operations performed would be \(64 \times 4096 = 262,144\) operations.

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Most popular questions from this chapter

Use the divide-and-conquer approach to write an algorithm that finds the largest item in a list of \(n\) items. Analyze your algorithm, and show the results in order notation.

Use Binary Search (Algorithm 2.1 ) to search for the integer 120 in the following list (array) of integers. Show the actions step by step. $$\begin{array}{lllllllll} 12 & 34 & 37 & 45 & 57 & 82 & 99 & 120 & 134 \end{array}$$

When a divide-and-conquer algorithm divides an instance of size \(n\) of a problem into subinstances each of size \(n / c\), the recurrence relation is typically given by \\[ \begin{array}{l} T(n)=a T\left(\frac{n}{c}\right)+g(n) \quad \text { for } n>1 \\ T(1)=d \end{array} \\] where \(g(n)\) is the cost of the dividing and combining processes, and \(d\) is a constant. Let \(n=c^{k}\) (a) Show that \\[ T\left(c^{k}\right)=d \times d^{k}+\sum_{j=1}^{k}\left[a^{k-j} \times g\left(c^{j}\right)\right] \\] (b) Solve the recurrence relation given that \(g(n) \in \Theta(n)\)

Suppose that on a particular computer it takes \(12 n^{2}\), \(\mu\) s to decompose and recombine an instance of size \(n\) in the case of Algorithm 2.8 (Strassen). Note that this time includes the time it takes to do all the additions and subtractions. If it takes \(n^{3} \mu\) s to multiply two \(n \times n\) matrices using the standard algorithm, determine thresholds at which we should call the standard algorithm instead of dividing the instance further. Is there a unique optimal threshold?

Use the divide-and-conquer approach to write a recursive algorithm that finds the maximum sum in any contiguous sublist of a given list of \(n\) real values. Analyze your algorithm, and show the results in order notation.
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