In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero.[1] 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. By contrast, if most of the elements are non-zero, the matrix is considered dense.[1] 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.

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Example of sparse matrix
The above sparse matrix contains only 9 non-zero elements, with 26 zero elements. Its sparsity is 74%, and its density is 26%.
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A sparse matrix obtained when solving a finite element problem in two dimensions. The non-zero elements are shown in black.

Conceptually, sparsity corresponds to systems with few pairwise interactions. 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. 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. 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. Large sparse matrices often appear in scientific or engineering applications when solving partial differential equations.

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. Specialized computers have been made for sparse matrices,[2] as they are common in the machine learning field.[3] Operations using standard dense-matrix structures and algorithms are slow and inefficient when applied to large sparse matrices as processing and memory are wasted on the zeros. Sparse data is by nature more easily compressed and thus requires significantly less storage. Some very large sparse matrices are infeasible to manipulate using standard dense-matrix algorithms.

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