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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 ...
The matrix (typically assumed to be full-rank) is referred to as the dictionary, and is a signal of interest. The core sparse representation problem is defined as the quest for the sparsest possible representation α {\displaystyle \alpha } satisfying x = D α {\displaystyle x=D\alpha } .
Consider the following matrix as an example: = [] If we apply the full regular Cholesky decomposition, it yields: = [] And, by definition: = ′ However, by applying Cholesky decomposition, we observe that some zero elements in the original matrix end up being non-zero elements in the decomposed matrix, like elements (4,2), (5,2) and (5,3) in this example.
Pages in category "Sparse matrices" The following 19 pages are in this category, out of 19 total. ... Sparse matrix–vector multiplication; T. Tridiagonal matrix; Z.
LDPC codes functionally are defined by a sparse parity-check matrix. This sparse matrix is often randomly generated, subject to the sparsity constraints—LDPC code construction is discussed later. These codes were first designed by Robert Gallager in 1960. [5] Below is a graph fragment of an example LDPC code using Forney's factor graph notation.
A band matrix with k 1 = k 2 = 0 is a diagonal matrix, with bandwidth 0. A band matrix with k 1 = k 2 = 1 is a tridiagonal matrix, with bandwidth 1. For k 1 = k 2 = 2 one has a pentadiagonal matrix and so on. Triangular matrices. For k 1 = 0, k 2 = n−1, one obtains the definition of an upper triangular matrix
For example, for a 3 × 3 matrix A, ... Sparse-matrix decomposition. Special algorithms have been developed for factorizing large sparse matrices.
Cuthill-McKee ordering of a matrix RCM ordering of the same matrix. In numerical linear algebra, the Cuthill–McKee algorithm (CM), named after Elizabeth Cuthill and James McKee, [1] is an algorithm to permute a sparse matrix that has a symmetric sparsity pattern into a band matrix form with a small bandwidth.