Sparse arrays are those in which most of the elements (sparse matrices, or vectors, or tensors) are zero or--more generally--a common value.
Mathematica's SparseArray technology offers a uniquely efficient way to
store and manipulate large sparse objects of any dimension. Sparse
algorithms are automatically used where they would improve performance,
without user intervention, and can be manually evoked. In fact,
Mathematica can perform very-large-scale linear algebra operations at
higher speeds--and with smaller memory footprints--than specialized
numerical systems can.
Dramatic time savings can result from using sparse data
representation. The graph compares the time taken to solve systems of
linear equations utilizing sparse (red) and dense (gray)
representations. The effect becomes particularly pronounced as the
size of the calculation grows.
Note: Times are for a 3.6 GHz Pentium 4 processor running Windows XP.