By I. S. Duff, A. M. Erisman, J. K. Reid
The topic of sparse matrices has its root in such various fields as administration technology, energy structures research, surveying, circuit concept, and structural research. effective use of sparsity is a key to fixing huge difficulties in lots of fields. This publication offers either perception and solutions for these trying to resolve those problems.
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Extra resources for Direct methods for sparse matrices
C Oxford University Press 2017. Published 2017 by Oxford University Press. 1), each of length at least the number of entries. This is called the packed form. We call the operation of transforming from full-length to packed form a gather and the reverse a scatter. 0 do k = 1,tau product = product + value(k)*w(index(k)) end do Fig. 1. Code for the inner product between packed vector (value, index) with tau entries and full-length vector w. To compare the storage requirements for the two methods, we first look at two examples.
15 SPARSE MATRICES: STORAGE SCHEMES AND SIMPLE OPERATIONS Matrix by vector products Any of the storage schemes we have considered is suitable for multiplying a sparse matrix by a full-length vector to give a full-length vector result. 1) where x is sparse, access to A by columns is desirable. In fact, all we need are the columns corresponding to the nonzero components xi , and any data structure that provides access by columns is suitable. 4 is very suitable. If column access is not available, a scan of the whole packed matrix A is inevitable.
Such a block structure can occur though the use of a nested dissection ordering, as discussed in Chapter 9, or when domain decomposition is used in the numerical solution of partial differential equations. Parallelism can also be exploited at the level of the still sparse submatrices, Aij . Finally, we show in Chapters 11–13 how dense kernels can be used at the heart of a sparse code. For example, consider the matrix multiplication problem of the previous paragraph. If the problem is large enough, the data may be redistributed across the processes so that they share the work and each works on submatrices for which streaming occurs.
Direct methods for sparse matrices by I. S. Duff, A. M. Erisman, J. K. Reid