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Algorithm to multiply matrices
Because matrix multiplication is such a central operation in many numerical algorithms, much work has been invested in making matrix multiplication algorithms
Matrix multiplication algorithm
Matrix_multiplication_algorithm
Recursive algorithm for matrix multiplication
Strassen algorithm, named after Volker Strassen, is an algorithm for matrix multiplication. It is faster than the standard matrix multiplication algorithm for
Strassen_algorithm
Algorithmic runtime requirements for matrix multiplication
complexity of matrix multiplication dictates how quickly the operation of matrix multiplication can be performed. Matrix multiplication algorithms are a central
Computational complexity of matrix multiplication
Computational_complexity_of_matrix_multiplication
Mathematical operation in linear algebra
in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns
Matrix_multiplication
Algorithm to multiply two numbers
A multiplication algorithm is an algorithm (or method) to multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient
Multiplication_algorithm
Randomized algorithm for verifying matrix multiplication
Freivalds' algorithm (named after Rūsiņš Mārtiņš Freivalds) is a probabilistic randomized algorithm used to verify matrix multiplication. Given three
Freivalds'_algorithm
Algorithm for matrix multiplication
In computer science, Cannon's algorithm is a distributed algorithm for matrix multiplication for two-dimensional meshes first described in 1969 by Lynn
Cannon's_algorithm
Mathematics optimization problem
Matrix chain multiplication (or the matrix chain ordering problem) is an optimization problem concerning the most efficient way to multiply a given sequence
Matrix_chain_multiplication
Artificial intelligence system for discovering matrix multiplication algorithms
intelligence system developed by DeepMind for discovering efficient matrix multiplication algorithms using reinforcement learning. Introduced in 2022, the system
AlphaTensor
Array of numbers
and multiplication. For example, [19−13205−6] denotes a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a
Matrix_(mathematics)
In mathematics, invariant of square matrices
determinant by the block matrices in a fast way with the use of fast matrix multiplication algorithms in the time O ( n ω ) {\displaystyle O({n^{\omega }})} for
Determinant
Classification of algorithm
brute-force matrix multiplication (which takes O ( n 3 ) {\displaystyle O(n^{3})} operations) was the Strassen algorithm: a recursive algorithm that takes
Galactic_algorithm
Algorithm for multiplying large numbers
introduced the new algorithm with its low complexity, and Stephen Cook, who cleaned the description of it, is a multiplication algorithm for large integers
Toom–Cook_multiplication
Matrix defined using smaller matrices called blocks
space) Strassen algorithm (algorithm for matrix multiplication that is faster than the conventional matrix multiplication algorithm) Eves, Howard (1980)
Block_matrix
Soviet American mathematician
and computer scientist, known for his research on algorithms for polynomials and matrix multiplication. Pan earned his Ph.D. at Moscow University in 1964
Victor_Pan
Coppersmith–Winograd algorithm: square matrix multiplication Freivalds' algorithm: a randomized algorithm used to verify matrix multiplication Strassen algorithm: faster
List_of_algorithms
Points with no three in a line
bounds on cap sets imply lower bounds on certain types of algorithms for matrix multiplication. The Games graph is a strongly regular graph with 729 vertices
Cap_set
block Lanczos algorithm is an algorithm for finding the nullspace of a matrix over a finite field, using only multiplication of the matrix by long, thin
Block_Lanczos_algorithm
Matrix with shifting rows
{\tilde {O}}({\alpha ^{\omega -1}}n)} ops with the use of fast matrix multiplication algorithms, where α {\displaystyle \alpha } is the rank and ∼ 2.37 ≤ ω
Toeplitz_matrix
Problem in computational complexity theory
computational complexity theory, the online matrix-vector multiplication problem (OMv) asks an online algorithm to return, at each round, the product of
Online matrix-vector multiplication problem
Online_matrix-vector_multiplication_problem
Algorithmic runtime requirements for common math procedures
variety of multiplication algorithms, M ( n ) {\displaystyle M(n)} below stands in for the complexity of the chosen multiplication algorithm. This table
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
{nmk}{CM^{1/2}}}} . Direct computation verifies that the tiling matrix multiplication algorithm reaches the lower bound. Consider the following running-time
Communication-avoiding algorithm
Communication-avoiding_algorithm
Algorithm to calculate eigenvalues
the QR algorithm or QR iteration is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix. The QR algorithm
QR_algorithm
Problem optimization method
dimensions m×q, and will require m*n*q scalar multiplications (using a simplistic matrix multiplication algorithm for purposes of illustration). For example
Dynamic_programming
Algorithm for linear programming
et al. is the representative of a branch of algorithms that apply fast matrix multiplication algorithms to linear programs. Linear–fractional programming
Simplex_algorithm
Matrix class
working with the matrix. For example, there are known algorithms in literature for approximate Cauchy matrix-vector multiplication with O ( n log n
Cauchy_matrix
Mathematical operation on matrices
Min-plus matrix multiplication, also known as distance product, is an operation on matrices. Given two n×n matrices A=(aij) and B=(bij), their distance
Min-plus matrix multiplication
Min-plus_matrix_multiplication
more efficient matrix multiplication algorithms. For 4 × 4 matrices over the finite field with two elements, AlphaTensor found an algorithm requiring 47
List of mathematical discoveries made with artificial intelligence
List_of_mathematical_discoveries_made_with_artificial_intelligence
Extensions to the x86 instruction set architecture
They perform matrix multiplication at the hardware level, making them apt for problems and algorithms that use matrix multiplication as their core.
Advanced_Matrix_Extensions
rectangular matrix multiplication algorithm available instead of achieving rectangular multiplication via multiple square matrix multiplications. The best
Seidel's_algorithm
Matrix representing a Euclidean rotation
then the inverse of the example matrix should be used, which coincides with its transpose. Since matrix multiplication has no effect on the zero vector
Rotation_matrix
Matrix in which most of the elements are zero
sparse matrix-vector and matrix-transpose-vector multiplication using compressed sparse blocks (PDF). ACM Symp. on Parallelism in Algorithms and Architectures
Sparse_matrix
Theoretical computer scientist
This improved a previous time bound for matrix multiplication algorithms, the Coppersmith–Winograd algorithm, that had stood as the best known for 24
Virginia_Vassilevska_Williams
Algorithm for computing greatest common divisors
The matrix method is as efficient as the equivalent recursion, with two multiplications and two additions per step of the Euclidean algorithm. Bézout's
Euclidean_algorithm
Longest distance between two vertices
known matrix multiplication algorithms. For sparse graphs, with few edges, repeated breadth-first search is faster than matrix multiplication. Assuming
Diameter_(graph_theory)
Math operation methods
matrix multiplication algorithm that is used internally. Research into matrix multiplication complexity shows that there exist matrix multiplication algorithms
Methods_of_matrix_inversion
Algorithm for determinants of integers
mathematics, the Bareiss algorithm, named after Erwin Bareiss, is an algorithm to calculate the determinant or the echelon form of a matrix with integer entries
Bareiss_algorithm
Improved reduction for specific matrices
In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form
Tridiagonal_matrix_algorithm
Estimate of time taken for running an algorithm
O(nlogn) Monge array calculation, O(nlogn) Schönhage–Strassen algorithm for multiplication, O(nlognloglogn) In many cases, the O(nlogn) running time
Time_complexity
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
Discrete Fourier transform algorithm
include: fast large-integer multiplication algorithms and polynomial multiplication, efficient matrix–vector multiplication for Toeplitz, circulant and
Fast_Fourier_transform
I/O-efficient algorithm regardless of cache size
cache-oblivious algorithms are known for matrix multiplication, matrix transposition, sorting, and several other problems. Some more general algorithms, such as
Cache-oblivious_algorithm
Matrix of geometric progressions
fast matrix multiplication algorithms, where α {\displaystyle \alpha } is just the rank and ω < 2.372 {\displaystyle \omega <2.372} is the matrix multiplication
Vandermonde_matrix
Most widely known generalized inverse of a matrix
the Moore–Penrose inverse in a fast way with the use of fast matrix multiplication algorithms in the time O ( n ω ) {\displaystyle O({n^{\omega }})} for
Moore–Penrose_inverse
Method to solve optimization problems
an algorithm that runs in O(n3) time. In 1989, Vaidya developed an algorithm that runs in O(n2.5) time with the use of fast matrix multiplication algorithms
Linear_programming
Matrix with a multiplicative inverse
n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix B is uniquely determined by A
Invertible_matrix
Overview of and topical guide to algorithms
Singular value decomposition Eigenvalue algorithm Strassen algorithm Matrix chain multiplication Newton's method Gradient descent Conjugate gradient method
Outline_of_algorithms
Optimization algorithm for artificial neural networks
The overall network is a combination of function composition and matrix multiplication: g(x):=fL(WLfL−1(WL−1⋯f1(W1x)⋯)) For a training set there will be
Backpropagation
Real square matrix whose columns and rows are orthogonal unit vectors
and practical. The n × n orthogonal matrices form a group under matrix multiplication, the orthogonal group denoted by O(n), which—with its subgroups—is
Orthogonal_matrix
Square matrix of cyclically shifted rows
{\displaystyle O(n\log n)} arithmetic operations. Algebraically, multiplication by a circulant matrix is circular convolution on the cyclic group Z / n Z {\displaystyle
Circulant_matrix
Matrix of partial derivatives of a vector-valued function
Jacobian determinant, and the multiplicative inverse of the derivative is replaced by the inverse of the Jacobian matrix. The Jacobian determinant is fundamentally
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Directed graph with no directed cycles
be solved in time O(nω) where ω < 2.373 is the exponent for matrix multiplication algorithms; this is a theoretical improvement over the O(mn) bound for
Directed_acyclic_graph
Artificial intelligence (AI) program
system for discovering matrix multiplication algorithms AlphaDev — DeepMind system for discovering faster sorting algorithms AlphaEvolve — DeepMind coding
AlphaGeometry
Matrix of binary truth values
matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1)-matrix is a matrix with entries from the Boolean domain B = {0, 1}. Such a matrix can
Logical_matrix
Quantum algorithm for integer factorization
using fast multiplication, or even O((logn)2(loglogn)) using the asymptotically fastest multiplication algorithm currently known due to
Shor's_algorithm
Algorithm for computing trigonometric, hyperbolic, logarithmic and exponential functions
is a simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots, multiplications, divisions, exponentials
CORDIC
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
Arithmetical operation
peasant multiplication algorithm, does not. The example below illustrates "long multiplication" (the "standard algorithm", "grade-school multiplication"):
Multiplication
Toeplitz Hash Algorithm describes hash functions that compute hash values through matrix multiplication of the key with a suitable Toeplitz matrix. The Toeplitz
Toeplitz_Hash_Algorithm
Approximation method
decompositions and solutions to matrix equations. The central algorithm is the efficient matrix-matrix multiplication, i.e., the computation of Z = Z
Hierarchical_matrix
Special kind of square matrix
decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix L and an upper triangular matrix U if and only
Triangular_matrix
Algorithm for solving systems of linear equations
reduces a single row may be viewed as multiplication by a Frobenius matrix. Then the first part of the algorithm computes an LU decomposition, while the
Gaussian_elimination
Concept from linear programming
the simplex algorithm. Megiddo in co-work with Beling proposed also fast algorithm which uses the fast matrix multiplication algorithms. How to move
Basic_feasible_solution
Routines for performing common linear algebra operations
operations such as vector addition, scalar multiplication, dot products, linear combinations, and matrix multiplication. They are the de facto standard low-level
Basic Linear Algebra Subprograms
Basic_Linear_Algebra_Subprograms
Graph theory problem: find a matching containing the most edges
randomization and is based on the fast matrix multiplication algorithm. This gives a randomized algorithm for general graphs with complexity O ( V 2.372
Maximum-cardinality_matching
Square matrix containing the distances between elements in a set
is the adjacency matrix of G. The distance matrix of G can be computed from W as above; by contrast, if normal matrix multiplication is used, and unlinked
Distance_matrix
Set of edges without common vertices
also possible to find a maximum matching with the use of fast matrix multiplication algorithms in the time O ( n ω ) {\displaystyle O({n^{\omega }})} for
Matching_(graph_theory)
High-performance algorithm
on the columns and rows of the matrix, with multiplication by correction "twiddle factors" in between. The algorithm is named after an article by David
Bailey's_FFT_algorithm
Matrix decomposition
eigenvalue algorithm, the QR algorithm. Any real square matrix A may be decomposed as A = Q R , {\displaystyle A=QR,} where Q is an orthogonal matrix (its columns
QR_decomposition
Technique for speeding up algorithms involving Boolean matrices
Algorithms to which the Method of Four Russians may be applied include: computing the transitive closure of a graph, Boolean matrix multiplication, edit
Method_of_Four_Russians
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
Numerical linear algebra algorithm
Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix (a process known
Jacobi_eigenvalue_algorithm
Four-dimensional number system
such a way that quaternion addition and multiplication correspond to matrix addition and matrix multiplication. One is to use 2 × 2 complex matrices, and
Quaternion
Polynomial whose roots are the eigenvalues of a matrix
characteristic polynomial in a fast way with the use of fast matrix multiplication algorithms in the time O ( n ω ) {\displaystyle O({n^{\omega }})} for
Characteristic_polynomial
Application of computer science in biology
variable.[citation needed] These are attempts to utilize fast matrix multiplication algorithms in computational biology. Examples of this type of work are
Computational_biology
Mapping function that preserves data point locality
"Parallel sparse matrix-vector and matrix-transpose-vector multiplication using compressed sparse blocks", ACM Symp. on Parallelism in Algorithms and Architectures
Z-order_curve
Correspondence between quaternions and 3D rotations
except the commutative law of multiplication (a familiar example of such a noncommutative multiplication is matrix multiplication). From this all of the rules
Quaternions and spatial rotation
Quaternions_and_spatial_rotation
Type of mathematical expression
coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number
Polynomial
Discrete fourier transform expressed as a matrix
which can be applied to a signal through matrix multiplication. An N-point DFT is expressed as the multiplication X = W x {\displaystyle X=Wx} , where x
DFT_matrix
Numerical eigenvalue calculation
counting the matrix–vector multiplication, each iteration does O ( n ) {\displaystyle O(n)} arithmetical operations. The matrix–vector multiplication can be
Lanczos_algorithm
Algorithm used to solve non-linear least squares problems
(size of the vector β {\displaystyle {\boldsymbol {\beta }}} ). The matrix multiplication ( J T J ) {\displaystyle \left(\mathbf {J} ^{\mathrm {T} }\mathbf
Levenberg–Marquardt_algorithm
Problem in computer science
Kadane's algorithm as a subroutine, or through a divide-and-conquer approach. Slightly faster algorithms based on distance matrix multiplication have been
Maximum_subarray_problem
Algorithmic technique
The multiplicative weights update method is an algorithmic technique most commonly used for decision making and prediction, and also widely deployed in
Multiplicative weight update method
Multiplicative_weight_update_method
Standard for the encryption of electronic data
GF(28)[x]. The MixColumns step can also be viewed as a multiplication by the shown particular MDS matrix in the finite field GF(28). This process is described
Advanced_Encryption_Standard
Algorithm for shuffling a finite sequence
The Fisher–Yates shuffle is an algorithm for shuffling a finite sequence. The algorithm takes a list of all the elements of the sequence, and continually
Fisher–Yates_shuffle
Elementwise product of two matrices
a matrix of the multiplied corresponding elements. This operation can be thought as a "naive matrix multiplication" and is different from the matrix product
Hadamard_product_(matrices)
Matrix of second derivatives
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function
Hessian_matrix
Mathematics concept
matrix H is skew if H T + H = 2 I . {\displaystyle H^{\textsf {T}}+H=2I.} A skew Hadamard matrix remains a skew Hadamard matrix after multiplication of
Hadamard_matrix
Matrix decomposition
the matrix multiplication exponent and η>0 is any constant, i.e. essentially in matrix multiplication time. The singular values of a 2×2 matrix can
Singular_value_decomposition
Algorithms for polynomial evaluation
parallelizing the computation. A similar idea enables to involve fast matrix multiplication algorithms to evaluate a polynomial in a series of points. Arbitrary polynomials
Polynomial_evaluation
Node ordering for directed acyclic graphs
repeatedly square the adjacency matrix of the given graph, logarithmically many times, using min-plus matrix multiplication with maximization in place of
Topological_sorting
Computational problem of graph theory
linear number of matrix multiplications that takes a total time of O(V4). For undirected and unweighted graphs, Dory et al. gives an algorithm computing a
Shortest_path_problem
For a square matrix, the transpose of the cofactor matrix
,} where I is the identity matrix of the same size as A. Consequently, the multiplicative inverse of an invertible matrix can be found by dividing its
Adjugate_matrix
Matrix decomposition
Rayleigh quotient of the eigenvector). In the QR algorithm for a Hermitian matrix (or any normal matrix), the orthonormal eigenvectors are obtained as a
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
Algorithm in graph theory
Floyd–Warshall algorithm (also known as Floyd's algorithm, the Roy–Warshall algorithm, the Roy–Floyd algorithm, or the WFI algorithm) is an algorithm for finding
Floyd–Warshall_algorithm
Subfield of computational complexity theory
Negative Triangle, and (min,+)-matrix multiplication. A faster algorithm for any one would give a faster algorithm for all of them. This subcubic equivalence
Fine_grained_complexity
Binary arithmetic algorithm
over the field with two elements, the steps in the algorithm can be interpreted as multiplication by 2×2 matrices over the field with two elements. For
XOR_swap_algorithm
Square matrix constructed from a monic polynomial
is possible to calculate the companion matrix in a fast way with the use of fast matrix multiplication algorithms in the time O ( n ω ) {\displaystyle O({n^{\omega
Companion_matrix
Method of calculating ray-triangle intersections in 3D space
u} , and v {\displaystyle v} ), and can be represented as a matrix-vector multiplication. [ | | | − D ( v 2 − v 1 ) ( v 3 − v 1 ) | | | ] [ t u v ] =
Möller–Trumbore intersection algorithm
Möller–Trumbore_intersection_algorithm
Form of a matrix
all matrix entries belong to a field 𝔽 whose characteristic is not equal to 2. That is, we assume that 1 + 1 ≠ 0, where 1 denotes the multiplicative identity
Skew-symmetric_matrix
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MATRIX MULTIPLICATION-ALGORITHM
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MATRIX MULTIPLICATION-ALGORITHM
MATRIX MULTIPLICATION-ALGORITHM
MATRIX MULTIPLICATION-ALGORITHM
MATRIX MULTIPLICATION-ALGORITHM
MATRIX MULTIPLICATION-ALGORITHM
MATRIX MULTIPLICATION-ALGORITHM
MATRIX MULTIPLICATION-ALGORITHM
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