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Matrix in which most of the elements are zero
In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. There is no strict
Sparse_matrix
Matrix with non-zero elements only in a diagonal band
In mathematics, particularly matrix theory, a band matrix or banded matrix is a sparse matrix whose non-zero entries are confined to a diagonal band, comprising
Band_matrix
Computation routine
Sparse matrix–vector multiplication (SpMV) of the form y = Ax is a widely used computational kernel existing in many scientific applications. The input
Sparse matrix–vector multiplication
Sparse_matrix–vector_multiplication
The Sparse Matrix Converter is an AC/AC converter which offers a reduced number of components, a low-complexity modulation scheme, and low realization
Sparse_matrix_converter
Approximation of a matrix's Cholesky factorization
incomplete Cholesky factorization of a symmetric positive definite matrix is a sparse approximation of the Cholesky factorization. An incomplete Cholesky
Incomplete Cholesky factorization
Incomplete_Cholesky_factorization
Matrix whose only nonzero elements are on its main diagonal
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices
Diagonal_matrix
Square matrix used to represent a graph or network
adjacency matrix and the time needed to perform operations on them is dependent on the matrix representation chosen for the underlying matrix. Sparse matrix representations
Adjacency_matrix
Connectivity measure in graph theory
to the star height of a regular language. It has also found use in sparse matrix computations (see Bodlaender et al. 1995) and logic (Rossman 2008).
Cycle_rank
Representation learning method
directional gradient of a rasterized matrix. Once a matrix or a high-dimensional vector is transferred to a sparse space, different recovery algorithms
Sparse_dictionary_learning
Approximation method
hierarchical matrices (H-matrices) are used as data-sparse approximations of non-sparse matrices. While a sparse matrix of dimension n {\displaystyle n} can be represented
Hierarchical_matrix
Storage method in computer memory
LAPACK defines various matrix representations in memory. There is also Sparse matrix representation and Morton-order matrix representation. According
Matrix_representation
Several types of mathematical matrix containing zeroes
a hollow matrix may refer to one of several related classes of matrix: a sparse matrix; a matrix with a large block of zeroes; or a matrix with diagonal
Hollow_matrix
Concept in statistics
statistics, the projection matrix ( P ) {\displaystyle (\mathbf {P} )} , sometimes also called the influence matrix or hat matrix ( H ) {\displaystyle (\mathbf
Projection_matrix
Algorithmic runtime requirements for matrix multiplication
Matrix multiplication algorithm, for practical implementation details Sparse matrix–vector multiplication Not to be confused with Jean-François Le Gall
Computational complexity of matrix multiplication
Computational_complexity_of_matrix_multiplication
sign matrix, a matrix of 0, 1, and −1 coefficients with the nonzeros in each row or column alternating between 1 and −1 and summing to 1 Sparse matrix, is
Combinatorial_matrix_theory
library for sparse matrix computations using the Recursive Sparse Blocks (RSB) matrix format. librsb provides cache efficient multi-threaded Sparse BLAS operations
Librsb
Mapping function that preserves data point locality
Charles E. (2009), "Parallel sparse matrix-vector and matrix-transpose-vector multiplication using compressed sparse blocks", ACM Symp. on Parallelism
Z-order_curve
Mathematical optimization algorithm
those whose matrix is positive-semidefinite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that
Conjugate_gradient_method
be shown through adjacency matrix. If the most elements in the matrix are zero, such matrix is referred as sparse matrix. In contrast, if most of the
Sparse_network
Array of numbers
be sparse, that is, contain few nonzero entries. Therefore, specifically tailored matrix algorithms can be used in network theory. The Hessian matrix of
Matrix_(mathematics)
Statistical analysis technique
following equivalent definition is in matrix form. Let V {\displaystyle V} be a p×p symmetric matrix, one can rewrite the sparse PCA problem as max T r ( Σ V )
Sparse_PCA
Routines for performing common linear algebra operations
to BLAS for handling sparse matrices have been suggested over the course of the library's history; a small set of sparse matrix kernel routines was finally
Basic Linear Algebra Subprograms
Basic_Linear_Algebra_Subprograms
Technique in photogrammetry and computer vision
Adjustment of observations Stereoscopy Levenberg–Marquardt algorithm Sparse matrix Collinearity equation Structure from motion Simultaneous localization
Bundle_adjustment
Type of data model
Therefore, this type of data model relates to the mathematical notion of a sparse matrix. EAV is also known as object–attribute–value model, vertical database
Entity–attribute–value_model
Numerical linear algebra algorithm
James McKee, is an algorithm to permute a sparse matrix that has a symmetric sparsity pattern into a band matrix form with a small bandwidth. The reverse
Cuthill–McKee_algorithm
API for graph data and graph operations
built upon the notion that a sparse matrix can be used to represent graphs as either an adjacency matrix or an incidence matrix. The GraphBLAS specification
GraphBLAS
Device converting an AC waveform to another AC waveform
indirect energy conversion by employing the Indirect Matrix Converter (Fig. 5) or the Sparse matrix converter which was invented by Prof. Johann W. Kolar
AC-to-AC_converter
Matrix in linear algebra
especially in linear algebra and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix into the vectorized form of
Commutation_matrix
number of operations involving zero terms due to the fact that the matrix is only sparse. The development of frontal solvers is usually considered as dating
Frontal_solver
designed to store sparse matrices, first described in 1982 as the format for the Harwell-Boeing collection of sparse matrix test problems. Matrix Market exchange
Harwell-Boeing_file_format
American economist and Nobel Laureate (1927–2023)
three fields: portfolio theory; sparse matrix methods; and simulation language programming (SIMSCRIPT). Sparse matrix methods are now widely used to solve
Harry_Markowitz
Eigenvalue algorithm
algorithm is the multiplication of matrix A {\displaystyle A} by a vector, so it is effective for a very large sparse matrix with appropriate implementation
Power_iteration
bandwidth of a symmetric sparse matrix Minimum degree algorithm: permute the rows and columns of a symmetric sparse matrix before applying the Cholesky
List_of_algorithms
Topics referred to by the same term
complexity theory Sparse matrix, in numerical analysis, a matrix populated primarily with zeros Sparse file, a computer file mostly empty Sparse network, a network
Sparse_(disambiguation)
Type of matrix factorization
factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix multiplication and matrix decomposition)
LU_decomposition
Topics referred to by the same term
Hierarchical matrix, a data-sparse approximation of a non-sparse matrix Hilbert matrix, a square matrix with entries being the unit fractions Hollow matrix, a square
H-matrix
Triangular matrix Tridiagonal matrix Block matrix Sparse matrix Hessenberg matrix Hessian matrix Vandermonde matrix Stochastic matrix Toeplitz matrix Circulant
Outline_of_linear_algebra
Norm on a vector space of matrices
Consider the matrix as a rectangular array of numbers; then a matrix norm may be defined as a function of the entries of the matrix. Such matrix norms are
Matrix_norm
Algorithms for matrix decomposition
Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra
Non-negative matrix factorization
Non-negative_matrix_factorization
Algorithm
non-zero pattern for the L {\displaystyle L} factors of a symmetric sparse matrix when applying the Cholesky decomposition or variants. Let A = ( a i
Symbolic Cholesky decomposition
Symbolic_Cholesky_decomposition
Generalization of graph theory
multiple edges between two vertices P system – Computational model Sparse matrix–vector multiplication – Computation routine Petri Net – Model to describe
Hypergraph
Concept in mathematics
Sparse approximation (also known as sparse representation) theory deals with sparse solutions for systems of linear equations. Techniques for finding
Sparse_approximation
Matrix equal to its conjugate-transpose
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose
Hermitian_matrix
Table of terms in a collection of documents
document. For this reason, document-term matrices are usually stored in a sparse matrix format. As a result of the power-law distribution of tokens in nearly
Document-term_matrix
Method for data management
Document-term matrix Used in latent semantic analysis, stores the occurrences of words in documents in a two-dimensional sparse matrix. A major challenge
Search_engine_indexing
Form of a matrix storage
skyline matrix storage, or SKS, or a variable band matrix storage, or envelope storage scheme is a form of a sparse matrix storage format matrix that reduces
Skyline_matrix
take advantage of coefficient matrix to be a sparse matrix. The LU decomposition of a sparse matrix is usually not sparse, thus, for a large system of
Stone's_method
Matrix with a different number of elements in each row
Regular matrix (disambiguation) Empty matrix Sparse matrix Paul E. Black, Ragged matrix, from Dictionary of Algorithms and Data Structures, Paul
Irregular_matrix
Numerical linear algebra algorithm
inherently a dense matrix algorithm: it draws little or no advantage from being applied to a sparse matrix, and it will destroy sparseness by creating fill-in
Jacobi_eigenvalue_algorithm
Statistics models class
reduction) or by finding sparse representations of the smooths using Markov random fields, which are amenable to the use of sparse matrix methods for computation
Generalized_additive_model
Concept in numerical linear algebra
(abbreviated as ILU) of a matrix is a sparse approximation of the LU factorization often used as a preconditioner. Consider a sparse linear system A x = b
Incomplete_LU_factorization
matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries
List_of_named_matrices
Numerical eigenvalue calculation
elements of the nullspace of a large sparse matrix over GF(2); since the set of people interested in large sparse matrices over finite fields and the set
Lanczos_algorithm
In mathematics, invariant of square matrices
the determinant is a scalar-valued function of the entries of a square matrix that has many properties which make it fundamental for the study of square
Determinant
of linear equations, and an edge represents a nonzero entry in the sparse matrix representing the system. Recursively partition the graph into subgraphs
Nested_dissection
Analog electronic circuit simulator
instead used sparse tableau approach (STA) to construct and solve a sparse matrix. The sparse tableau formulation produced very large, very sparse matrices
ASTAP
Subset of artificial intelligence
assumed to be a sparse matrix. The method is strongly NP-hard and difficult to solve approximately. A popular heuristic method for sparse dictionary learning
Machine_learning
Algorithm to multiply matrices
matrix multiplication CYK algorithm § Valiant's algorithm Matrix chain multiplication Method of Four Russians Multiplication algorithm Sparse matrix–vector
Matrix multiplication algorithm
Matrix_multiplication_algorithm
Data organization and storage formats
Dynamic array Gap buffer Hashed array tree Lookup table Matrix Parallel array Sorted array Sparse matrix Iliffe vector Variable-length array Doubly linked list
List_of_data_structures
Method by which information is represented in the brain
dictionary learning, a representation learning method which aims to find a sparse matrix representation of the input data in the form of a linear combination
Neural_coding
Python library for numerical programming
MATLAB can perform sparse matrix operations, NumPy alone cannot perform such operations and requires the use of the scipy.sparse library. Internally
NumPy
client LibreSSL — fork of OpenSSL for TLS Librsb — parallel library for sparse matrix computations Librsvg — SVG rendering library libsndfile — reading and
List_of_C_software_and_tools
Linear programming algorithm
of the matrix representing the constraints. The matrix-oriented approach allows for greater computational efficiency by enabling sparse matrix operations
Revised_simplex_method
Class of algorithms for solving constrained optimization problems
have been given more attention, in part because they more easily use sparse matrix subroutines from numerical software libraries, and in part because IPMs
Augmented_Lagrangian_method
Statistical software package
efficiently, due to its use of the average information algorithm and sparse matrix methods. It was originally developed by Arthur Gilmour. ASREML can be
ASReml
Projection of data onto lower-dimensional manifolds
including faster optimization when implemented to take advantage of sparse matrix algorithms, and better results with many problems. LLE also begins by
Nonlinear dimensionality reduction
Nonlinear_dimensionality_reduction
Techniques for lossy compression of neural networks
number of parameters. This allows the use of sparse matrix operations, which are faster than dense matrix operations. Pruning criteria can be based on
Model_compression
In mathematics, a bidiagonal matrix is a banded matrix with non-zero entries along the main diagonal and either the diagonal above or the diagonal below
Bidiagonal_matrix
Stochastic matrix representing links between entities
each matrix column is equal to unity. The numerical coefficient α {\displaystyle \alpha } is known as a damping factor. Usually S is a sparse matrix and
Google_matrix
Turkish-born Norwegian computer scientist
Norwegian computer scientist known for her research on graph algorithms, sparse matrix computations, and parameterized complexity. Until July 2025, she was
Pinar_Heggernes
Statistical model containing both fixed effects and random effects
formulation for numerical computation in order to take advantage of sparse matrix methods (e.g. lme4 and MixedModels.jl). In the context of Bayesian methods
Mixed_model
Electronic systems with a continuously variable signal
in the 1970s which used an unusual (compared to other simulators) sparse matrix method of circuit analysis. Analogue circuits can be entirely passive
Analogue_electronics
Visual performance model
Kornilios; Goumas, Georgios; Koziris, Nectarios (2008-01-01). "Optimizing sparse matrix-vector multiplication using index and value compression". Proceedings
Roofline_model
N x N matrix describing a linear power system with N buses
realistic systems which contain thousands of buses, the admittance matrix is quite sparse. Each bus in a real power system is usually connected to only a
Nodal_admittance_matrix
Matrix decomposition
dominated by noise. The first mitigation method is similar to a sparse sample of the original matrix, removing components that are not considered valuable. However
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
parallel numerical software library for partial differential equations and sparse matrix computations. PETSc received an R&D 100 Award in 2009. The PETSc Core
Portable, Extensible Toolkit for Scientific Computation
Portable,_Extensible_Toolkit_for_Scientific_Computation
Israeli mathematician
probabilistic method to subgraph isomorphism.[A] His work with Zwick on sparse matrix multiplication received the 2023 European Symposium on Algorithms Test-of-Time
Raphael_Yuster
Method of data analysis
aims to recover a low-rank matrix L0 from highly corrupted measurements M = L0 +S0. This decomposition in low-rank and sparse matrices can be achieved by
Robust principal component analysis
Robust_principal_component_analysis
Factorization algorithm
elimination does not give the optimal run time of the algorithm. Instead, sparse matrix solving algorithms such as Block Lanczos or Block Wiedemann are used
General_number_field_sieve
Topics referred to by the same term
analysis, the entries of a matrix which change from zero to a non-zero value in the execution of an algorithm; see Sparse matrix § Reducing fill-in An issue
Fill-in
Length of shortest path between two nodes of a graph
See the shortest path problem for more details and algorithms. Many sparse matrix and graph algorithms require a starting vertex of high eccentricity
Distance_(graph_theory)
therefore does not need to be independently compiled. Some support for sparse matrix storage is provided. The source code is in the public domain. TNT is
Template_Numerical_Toolkit
Neural network that learns efficient data encoding in an unsupervised manner
representations assume useful properties. Examples are regularized autoencoders (sparse, denoising and contractive autoencoders), which are effective in learning
Autoencoder
Clustering methods
distance-based similarity. Algorithms to construct the graph adjacency matrix as a sparse matrix are typically based on a nearest neighbor search, which estimate
Spectral_clustering
Area of discrete mathematics
and matrix structures but in concrete applications the best structure is often a combination of both. List structures are often preferred for sparse graphs
Graph_theory
Vector satisfying some of the criteria of an eigenvector
algebra, a generalized eigenvector of an n × n {\displaystyle n\times n} matrix A {\displaystyle A} is a vector which satisfies certain criteria which are
Generalized_eigenvector
Java library for linear algebra
structured sparse matrices are built on top of BLAS and LAPACK, and include the following intrinsic operations: Matrix/vector multiplication. Matrix/matrix multiplication
Matrix_Toolkit_Java
Java software and development tools
linear algebra and random numbers. Efficient Java Matrix Library (EJML) – dense and sparse matrix computations and linear algebra Easy Java Simulations
List of Java software and tools
List_of_Java_software_and_tools
matrix regularization penalty. The function R ( W ) {\displaystyle R(W)} is typically chosen to be convex and is often selected to enforce sparsity (using
Matrix_regularization
Topics referred to by the same term
with each field stored as a separate array Sparse array, with most elements omitted, to store a sparse matrix Variable-length array Associative array, an
Array
Programming technique on linked lists
complexity O(n) to O(1), Knuth implemented a sparse matrix where only 1's are stored. At all times, each node in the matrix will point to the adjacent nodes to
Dancing_links
Machine learning system
algorithms (model-types / representations) OLS regression Matrix factorization (sparse matrix SVD) Single layer neural net (with user specified hidden
Vowpal_Wabbit
computation time, even with the use of methods for sparse matrices. Many iterative methods allow for a matrix-free implementation, including: the power method
Matrix-free_methods
Signal processing technique
Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and
Compressed_sensing
Numerical method for solving physical or engineering problems
most of the entries of the matrix L {\displaystyle L} , which we need to invert, are zero. Such matrices are known as sparse matrices, and there are efficient
Finite_element_method
Statistical Markov model
density or sparseness of the resulting transition matrix. A choice of 1 yields a uniform distribution. Values greater than 1 produce a dense matrix, in which
Hidden_Markov_model
high performance sparse matrix computations providing multi-threaded primitives to build iterative solvers (implements also the Sparse BLAS standard).
List_of_numerical_libraries
Algorithm for linear programming
large linear-programming problems A is typically a sparse matrix and, when the resulting sparsity of B is exploited when maintaining its invertible representation
Simplex_algorithm
Degree of connectedness within a graph
in a dense adjacency matrix representation of the graph, and for edges takes Θ ( E ) {\displaystyle \Theta (E)} in a sparse matrix representation. The
Centrality
Technique in natural language processing
(LSI). LSA can use a document-term matrix which describes the occurrences of terms in documents; it is a sparse matrix whose rows correspond to terms and
Latent_semantic_analysis
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