Answered: (Sparse matrix–vector product) Recall…

Computer Networking: A Top-Down Approach (7th Edition)

7th Edition

ISBN:9780133594140

Author:James Kurose, Keith Ross

Publisher:James Kurose, Keith Ross

Chapter1: Computer Networks And The Internet

Section: Chapter Questions

Problem R1RQ: What is the difference between a host and an end system? List several different types of end...

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(Sparse matrix–vector product) Recall from Section 3.4.2 that a matrix is said to be sparse if most of its entries are zero. More formally, assume a m × n matrix A has sparsity coefficient γ(A) ≪ 1, where γ(A) ≐ d(A)/s(A), d(A) is the number of nonzero elements in A, and s(A) is the size of A (in this case, s(A) = mⁿ).

1. Evaluate the number of operations (multiplications and additions) that are required to form the matrix– vector product Ax, for any given vector x ∈ Rⁿ and generic, non-sparse A. Show that this number is reduced by a factor γ(A), if A is sparse.

2. Now assume that A is not sparse, but is a rank-one modification of a sparse matrix. That is, A is of the form Ã + uv^⊤, where Ã ∈ R^m,n is sparse, and u ∈ R^m, v ∈ R^m are given. Devise a method to compute the matrix–vector product Ax that exploits sparsity.

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