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In [[numerical analysis]], a '''sparse matrix''' is a [[Matrix (mathematics)|matrix]] in which most of the elements are zero.

== Description ==

The fraction of non-zero elements over the total number of elements (i.e., that can fit into the matrix, say a matrix of dimension of m x n can accommodate m x n total number of elements) in a matrix is called the sparsity (density).

Conceptually, sparsity corresponds to systems which are loosely coupled. 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 had 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, which have a low density of significant data or connections.

Large sparse matrices often appear in scientific or [[engineering]] applications when solving [[partial differential equations]].

== Computing ==

When storing and manipulating sparse matrices on a [[computer]], it is beneficial and often necessary to use specialized [[Algorithm|algorithms]] and [[Data structure|data structures]] that take advantage of the sparse structure of the matrix.

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 zeroes.

Sparse data is by nature more easily compressed and thus require significantly less storage.

Some very large sparse matrices are infeasible to manipulate using standard dense-matrix algorithms.

== Dense matrix ==

By contrast, if most of the elements are nonzero, then the matrix is considered dense.

== See also ==

* [[Algorithm]]
* [[Computation]]
* [[Computer science]]
* [[Entity–attribute–value model]]
* [[Mathematics]]
* [[Matrix (mathematics)]]
* [[Numerical analysis]]

== External links == 

* [https://en.wikipedia.org/wiki/Sparse_matrix Sparse matrix] @ Wikipedia