Specialized computers have been made for sparse matrices, as they are common in the machine learning field. When storing and manipulating sparse matrices on a computer, it is beneficial and often necessary to use specialized algorithms and data structures that take advantage of the sparse structure of the matrix. Large sparse matrices often appear in scientific or engineering applications when solving partial differential equations. The concept of sparsity is useful in combinatorics and application areas such as network theory and numerical analysis, which typically have a low density of significant data or connections. By contrast, if the same line of balls were to have springs connecting each ball to all other balls, the system would correspond to a dense matrix. For example, consider a line of balls connected by springs from one to the next: this is a sparse system as only adjacent balls are coupled. The number of zero-valued elements divided by the total number of elements (e.g., m × n for an m × n matrix) is sometimes referred to as the sparsity of the matrix.Ĭonceptually, sparsity corresponds to systems with few pairwise interactions. By contrast, if most of the elements are non-zero, the matrix is considered dense. There is no strict definition regarding the proportion of zero-value elements for a matrix to qualify as sparse but a common criterion is that the number of non-zero elements is roughly equal to the number of rows or columns. In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. The non-zero elements are shown in black. Its sparsity is 74%, and its density is 26%.Ī sparse matrix obtained when solving a finite element problem in two dimensions. The above sparse matrix contains only 9 non-zero elements, with 26 zero elements.
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