# Dictionary Definition

sparsity n : the property of being scanty or
scattered; lacking denseness [syn: sparseness, spareness]

# User Contributed Dictionary

#### Translations

the property of being sparse

- Spanish: escasez

# Extensive Definition

In the mathematical subfield of
numerical
analysis a sparse matrix is a matrix
populated primarily with zeros.

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. By contrast,
if the same line of balls had springs connecting every ball to
every other ball, the system would be represented by a dense
matrix. The concept of sparsity is useful in combinatorics and
application areas such as network
theory, of a low density of significant data or
connections.

Huge sparse matrices often appear in science or engineering 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. Operations using standard matrix structures and algorithms
are slow and consume large amounts of memory when applied to large
sparse matrices. Sparse data is by nature easily compressed,
and this compression almost always results in significantly less
memory
usage. Indeed, some very large sparse matrices are impossible to
manipulate with the standard algorithms.

## Storing a sparse matrix

The naive data
structure for a matrix is a two-dimensional array. Each entry
in the array represents an element ai,j of the matrix and can be
accessed by the two indices i and j. For a m×n matrix we need at
least enough memory to store (m×n) entries to represent the
matrix.

Many if not most entries of a sparse matrix are
zeros. The basic idea when storing sparse matrices is to store only
the non-zero entries as opposed to storing all entries. Depending
on the number and distribution of the non-zero entries, different
data
structures can be used and yield huge savings in memory
when compared to a naïve approach.

One example of such a sparse matrix format is the
(old) Yale Sparse Matrix Format[1]. It stores an initial sparse m×n
matrix, M, in row form using three one-dimensional arrays. Let NNZ
denote the number of nonzero entries of M. The first array is A,
which is of length NNZ, and holds all nonzero entries of M in
left-to-right top-to-bottom order. The second array is IA, which is
of length m + 1 (i.e., one entry per row, plus one). IA(i) contains
the index in A of the first nonzero element of row i. Row i of the
original matrix extends from A(IA(i)) to A(IA(i+1)-1). The third
array, JA, contains the column index of each element of A, so it
also is of length NNZ.

For example, the matrix

[ 1 2 0 0 ] [ 0 3 9 0 ] [ 0 1 4 0 ]

is a three-by-four matrix with six nonzero
elements, so A = [ 1 2 3 9 1 4 ] IA = [ 1 3 5 7 ] JA = [ 1 2 2 3 2
3 ]

Another possibility is to use quadtrees.

### Example

A bitmap image having only 2
colors, with one of them dominant (say a file that stores a
handwritten signature)
can be encoded as a sparse matrix that contains only row and column
numbers for pixels with
the non-dominant color.

### Diagonal matrices

A very efficient structure for a diagonal matrix is to store just the entries in the main diagonal as a one-dimensional array, so a diagonal n×n matrix requires only n entries.## Bandwidth

The lower bandwidth of a matrix A is the smallest
number p such that the entry aij vanishes whenever i > j + p.
Similarly, the upper bandwidth is the smallest p such that aij = 0
whenever i < j − p . For example, a tridiagonal
matrix has lower bandwidth 1 and upper bandwidth 1.

Matrices with small upper and lower bandwidth are
known as band matrices
and often lend themselves to simpler algorithms than general sparse
matrices; one can sometimes apply dense matrix algorithms and
simply loop over a reduced number of indices.

### Reducing bandwidth

The Cuthill-McKee
algorithm can be used to reduce the bandwidth of a sparse
symmetric
matrix. There are, however, matrices for which the
Reverse Cuthill-McKee algorithm performs better.

The
U.S. National Geodetic Survey (NGS) uses Dr. Richard Snay's
"Banker's" algorithm because on realistic sparse matrices used in
Geodesy work it has better performance.

There are many other methods in use.

## Reducing fill-in

- "Fill-in" redirects here. For the puzzle, see Fill-In (puzzle).

The fill-in of a matrix are those entries which
change from an initial zero to a non-zero value during the
execution of an algorithm. To reduce the memory requirements and
the number of arithmetic operations used during an algorithm it is
useful to minimize the fill-in by switching rows and columns in the
matrix. The
symbolic Cholesky decomposition can be used to calculate the
worst possible fill-in before doing the actual Cholesky
decomposition.

There are other methods than the Cholesky
decomposition in use. Orthogonalization methods (such as QR
factorization) are common, for example, when solving problems by
least squares methods. While the theoretical fill-in is still the
same, in practical terms the "false non-zeros" can be different for
different methods. And symbolic versions of those algorithms can be
used in the same manner as the symbolic Cholesky to compute worst
case fill-in.

## Solving sparse matrix equations

Both iterative
and direct methods exist for sparse matrix solving. One popular
iterative method is the conjugate
gradient method.

## See also

## References

- Tewarson, Reginald P, Sparse Matrices (Part of the Mathematics in Science & Engineering series), Academic Press Inc., May 1973. (This book, by a professor at the State University of New York at Stony Book, was the first book exclusively dedicated to Sparse Matrices. Graduate courses using this as a textbook were offered at that University in the early 1980s).
- Sparse Matrix Multiplication Package, Randolph E. Bank, Craig C. Douglas http://www.mgnet.org/~douglas/Preprints/pub0034.pdf
- Pissanetzky, Sergio 1984, "Sparse Matrix Technology", Academic Press
- R. A. Snay. Reducing the profile of sparse symmetric matrices. Bulletin Géodésique, 50:341–352, 1976. Also NOAA Technical Memorandum NOS NGS-4, National Geodetic Survey, Rockville, MD.

## Further reading

- Matrix Computations .
- Sparse Matrix Algorithms Research at the University of Florida, containing the UF sparse matrix collection.

sparsity in Czech: Řídká matice

sparsity in German: Dünnbesetzte Matrix

sparsity in French: Matrice creuse

sparsity in Italian: Matrice sparsa

sparsity in Japanese: 疎行列

sparsity in Chinese: 稀疏矩阵

# Synonyms, Antonyms and Related Words

chinchiness, chintziness, dearth, exiguity, fewness, infrequence, infrequency, meagerness, miserliness, niggardliness, occasionalness, paucity, poverty, rareness, rarity, restrictedness, scant
sufficiency, scantiness, scarceness, scarcity, scrimpiness, seldomness, skimpiness, slowness, smallness, sparseness, stinginess, stringency, thinness, tightness, uncommonness, unfrequentness, uniqueness, unusualness