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Chapter 2: Problem 28

How many multiplications would be performed in finding the product of two \(64 \times 64\) matrices using Strassen's Method (Algorithm 2.8)?

Short Answer

Expert verified
Therefore, the number of multiplications needed to multiply two 64x64 matrices using Strassen's method is 117,649.

Step by step solution

01

Understanding Strassen's Method

Strassen's method is an algorithmic technique used to speed up the multiplication of matrices. Normally, multiplying two matrices of size N x N would require \(N^3\) multiplications. But Strassen's method reduces this to approximately \(7^{log_2N}\) multiplications. The main idea of the method is to minimize the multiplications, which are computationally expensive, at the expense of performing more addition and subtraction operations.
02

Determine the Logarithm

The number of times the matrices are divided is given by \(log_2(64) = 6\) times, given that N=64 per step 1. This is because at each step, the size of the matrices is halved in Strassen's method.
03

Calculate the Number of Multiplications

Since Strassen's method performs 7 multiplications for each division of the matrix, the total number of multiplications for a 64 x 64 matrix will be \(7^{6} = 117,649\).

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Most popular questions from this chapter

Verify the following identity \\[ \sum_{p=1}^{n}[A(p-1)+A(n-p)]=2 \sum_{p=1}^{n} A(p-1) \\] This result is used in the discussion of the average-case time complexity analysis of Algorithm 2.6 (Quicksort).

Show that the recurrence equation for the worst-case time complexity for Mergesort (Algorithms \(2.2 \text { and } 2.4)\) is given by \\[ W(n)=W\left(\left[\frac{n}{2}\right\rfloor\right)+W\left(\left\lceil\frac{n}{2}\right]\right)+n-1 \\] when \(n\) is not restricted to being a power of 2

Given the recurrence relation \\[ \begin{array}{l} T(n)=7 T\left(\frac{n}{5}\right)+10 n \quad \text { for } n>1 \\ T(1)=1 \end{array} \\] find \(T(625)\)

Write algorithms that perform the operations \(u \times 10^{m}\) \(u\) divide \(10^{n}\) \(u\) rem \(10 "\) where \(u\) represents a large integer, \(m\) is a nonnegative integer, divide returns the quotient in integer division, and rem returns the remainder. Analyze your algorithms, and show that these operations can be done in linear time.

Consider procedure solve \((P, I, O)\) given below. This algorithm solves problem \(P\) by finding the output (solution) \(O\) corresponding to any input \(l\) Assume \(g(n)\) basic operations for partitioning and combining and no basic operations for an instance of size 1 (a) Write a recurrence equation \(T(n)\) for the number of basic operations needed to solve \(P\) when the input size is \(n\) (b) What is the solution to this recurrence equation if \(g(n) \in \Theta(n)\) (proof not required \() ?\) (c) Assuming that \(g(n)=n^{2},\) solve the recurrence equation exactly for \(n=\) 27 (d) Find the general solution for \(n\) a power of 3
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