2. Sparse Matrices
Yousef El-Mabruk Saad · Society for Industrial and Applied Mathematics eBooks · 2011
The eigenvalue problems that arise in practice often involve very large matrices. The meaning of ‘large’ is relative and it is changing rapidly with the progress of computer technology. A matrix of size a few tens of thousands can be considered large if one is working on a workstation, while, similarly, a matrix whose size is in the hundreds of millions can be considered large if one is using a high-performance computer. Fortunately, many of these matrices are also sparse, i.e., they have very few nonzeros. Again, it is not clear how ‘few’ nonzeros a matrix must have before it can be called sparse. A commonly used definition due to Wilkinson is to say that a matrix is sparse whenever it is possible to take advantage of the number and location of its nonzero entries. By this definition a tridiagonal matrix is sparse, but so would also be a triangular matrix, which may not be as convincing. It is probably best to leave this notion somewhat vague, since the decision as to whether or not a matrix should be considered sparse is a practical one that is ultimately made by the user.