An Introduction to Sparse Matrices

Abstract

Consider the simple matrix A on the left in Figure 1.1. Many of its entries are zero (and so are omitted). This is an example of a sparse matrix. The problem we are interested in is that of solving linear systems of equations Ax = b, where the square sparse matrix A and the vector b are given and the solution vector x is required. Such systems arise in a huge range of practical applications, including in areas as diverse as quantum chemistry, computer graphics, computational fluid dynamics, power networks, machine learning, and optimization. The list is endless and constantly growing, together with the sizes of the systems. For efficiency and to enable large systems to be solved, the sparsity of A must be exploited and operations with the zero entries avoided. To achieve this, sophisticated algorithms are required.