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Sparsity is a concept, useful in combinatorial mathematics and application areas such as network theory, of a low density of significant data or connections. This concept is amenable to quantitative reasoning. It is also noticeable in everyday life.
Huge sparse matrices often appear in science or engineering when solving problems for linear models.
When storing and manipulating sparse matrices on the computer it is often neccessary to modify the standard algorithms and take advantage of the sparse structure of the matrix. Sparse data is by its nature more easily compressed which can yield enormous savings in memory usage. And more importantly manipulating huge sparse matrices with the standard algorithms may be impossible due to their sheer size. The definition of huge depends on the hardware and the computer programs available to manipulate the matrix.
Given a sparse N×M matrix A the row bandwidth for the n-th row is defined as
The bandwidth for the matrix is defined as
A BMP image having only 2 colors, with one of them dominant (say a file that stores signature) can be encoded as a sparse matrix that contains only row no and col no of non-dominant color.
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 n×m matrix we need at least (n*m) / 8 bytes to represent the matrix when assuming 1 bit for each entry.
A sparse matrix contains many zero entries. The basic idea when storing sparse matrices is to only store 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 naive approach.
A very efficient structure for a diagonal matrix is to store just the entries in the main diagonal as a one dimensional array. For n×n matrix we need only n / 8 bytes when assuming 1 bit for each entry.
The Cuthill-McKee_algorithmIn the mathematical subfield of matrix theory the Cuthill-McKee algorithm is an algorithm to reduce the bandwidth of sparse symmetric matrices. Algorithm Given a symmetric n ''n matrix we visualize the matrix as the adjacency matrix of a graph. The Cuthil can be used to reduce the bandwith of a sparse symmetric matrixIn linear algebra, a symmetric matrix is a matrix that is its own transpose. Thus A is symmetric if: : which implies that A is a square matrix. Examples The entries of a symmetric matrix are symmetric with respect to the main diagonal (top left to bottom.
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 decompositionIn the mathematical subfield of numerical analysis the symbolic Cholesky decomposition is an algorithm used estimate the worst possible fill-in for a symmetric sparse matrix when applying the Cholesky decomposition. Algorithm Let : be a symmetric positive can be used to calculate the worst possible fill-in before doing the actual Cholesky decompositionIn mathematics, the Cholesky decomposition named after Andre-Louis Cholesky, is a matrix decomposition of a positive-definite matrix into a lower triangular matrix and the conjugate transpose of the lower triangular matrix. It can be considered as a speci.