Question

Problem 4. Let A be an n x n matrix, and m be an integer parameter. Develop a divide conquer algorithm to compute Am, which is the product of m As (for example, A³ = A x A x A).

You may use Strassen's algorithm to compute the product of two matrices with time O(nlog, 7). Show how your algorithm time depends on the two parameters n and m

Answer 1

To compute Am for an n x n matrix A using a divide and conquer algorithm, follow these steps: check if m is 0, negative, even, or odd and perform the corresponding computations. The time complexity of the algorithm is O(n^(log2(7))log(m)).

Step-by-step explanation:

To develop a divide and conquer algorithm to compute Am for an n x n matrix A and an integer parameter m, you can use the following approach:

1. If m is 0, return the identity matrix of size n.
2. If m is negative, compute A^-m by finding the inverse of A and taking its -m power.
3. If m is even, compute Am/2 by recursively calling the divide and conquer algorithm for A^(m/2), and then compute the result as the product of the two halves using Strassen's algorithm.
4. If m is odd, compute Am-1 by recursively calling the divide and conquer algorithm for A^((m-1)/2), and then compute the result as the product of the two halves using Strassen's algorithm. Finally, multiply the result by A to obtain Am.

The time complexity of this algorithm depends on the dimensions of the matrix n and the exponent m. Since Strassen's algorithm has a time complexity of O(nlog7), the overall time complexity of the divide and conquer algorithm for matrix exponentiation is O(n^(log2(7))log(m)).