56 articles
Matrix decomposition
A matrix decomposition, or matrix factorization, represents a matrix as a product of simpler matrices such as LU, Cholesky, QR, and the singular value decomposition.
Multigrid method
The multigrid method is an iterative solver and preconditioner for linear systems, especially elliptic PDEs, that corrects fine-grid solutions on coarser grids to solve N unknowns in O(N) work.
NAG Library
The NAG Library is a commercial collection of more than 1,600 numerical and statistical algorithms developed by the Numerical Algorithms Group in Oxford, first released in 1971.
Non-linear least squares
Non-linear least squares is a parameter estimation method that fits models nonlinear in their parameters by minimizing squared residuals, typically with the Gauss-Newton or Levenberg-Marquardt algorithm.
NumPy
NumPy, or Numerical Python, is the primary open-source array programming library for Python, first released in 2006 and serving as the foundation of the scientific Python ecosystem.
OpenBLAS
OpenBLAS is an open-source, optimized implementation of the Basic Linear Algebra Subprograms (BLAS), based on Goto's GotoBLAS2 and including LAPACK, with hand-tuned assembly kernels for many CPU architectures.
Orthogonal iteration
Orthogonal iteration, also called subspace iteration or simultaneous iteration, is a numerical linear algebra algorithm that generalizes the power method to compute bases for eigenspaces.
Overdetermined system
An overdetermined system is a system of equations, typically linear, with more equations than unknowns, arising in curve fitting and satellite positioning and usually solved by least squares.
Pivot element
A pivot element, central to complete pivoting and other pivoting strategies, is the entry by which Gaussian elimination divides rows below it to create zero multipliers.
Power iteration
Power iteration, also called the power method or Von Mises iteration, is a numerical linear algebra algorithm that finds a square matrix's dominant eigenvalue and a corresponding eigenvector.
Preconditioner
A preconditioner is a matrix, or any linear operator, applied to a linear system to give it better spectral properties so iterative methods converge faster.
QR decomposition
The QR decomposition, also called QR factorization, expresses a matrix A as A = QR with Q orthogonal and R upper triangular, and underlies least-squares fitting and eigenvalue computation.
Randomized singular value decomposition
The randomized singular value decomposition (randomized SVD, or RSVD) is a numerical linear algebra algorithm that computes an approximate truncated SVD of a large matrix using random sampling.
Recursive least squares filter
The recursive least squares (RLS) filter, or RLS algorithm, is an adaptive filter that recursively minimizes weighted squared error, converging faster than LMS at higher computational cost per step.
Restricted isometry property
The restricted isometry property (RIP) is a condition on a matrix stating that it approximately preserves the length of every sparse vector, proposed by Candès and Tao for compressed sensing.
ScaLAPACK
ScaLAPACK (Scalable Linear Algebra PACKage) is a library of high-performance linear algebra routines for parallel distributed-memory machines, solving linear systems, least squares, eigenvalue, and singular value problems, first released in 1995.
Scilab
Scilab is a free and open-source platform for numerical computation, providing a high-level interpreted programming language, hundreds of mathematical functions, 2-D and 3-D graphics, and the Xcos graphical modeler and simulator.
Singular value thresholding
Singular value thresholding (SVT) is a first-order iterative algorithm for minimizing the nuclear norm of a matrix, introduced by Cai, Candès, and Shen in 2010, whose leading application is matrix completion.
Sparse matrix
A sparse matrix, also known as a sparse array, is a matrix in which most entries are zero; exploiting sparsity makes solving systems with millions of unknowns practical.
Square root of a matrix
A square root of a matrix is a matrix B whose square B² equals the given matrix A. Roots may not exist and are generally not unique.
Strassen algorithm
The Strassen algorithm, or Strassen's method, is a recursive matrix multiplication algorithm Volker Strassen published in 1969 that uses seven multiplications of half-sized submatrices instead of eight, the first with sub-cubic complexity.
Successive over-relaxation
Successive over-relaxation (SOR) is an iterative method for solving linear systems, using a relaxation factor to speed up Gauss–Seidel, especially for sparse systems from discretized PDEs.
SuiteSparse
SuiteSparse is an open-source collection of sparse matrix algorithm packages written or co-authored by Timothy A. Davis of Texas A&M University, providing solvers used in MATLAB, Julia, and Google Earth.
Sum of squares
A sum of squares is an expression in which squared quantities are added, naming ideas in number theory, statistics, and optimization that exploit squares never being negative.
Sylvester equation
The Sylvester equation is the matrix equation AX + XB = C for an unknown matrix X, named after James Joseph Sylvester, who studied it in 1884.
Tridiagonal matrix algorithm
The tridiagonal matrix algorithm, also known as the Thomas algorithm, is a simplified Gaussian elimination for solving tridiagonal linear systems in linear time, widely used in discretized partial differential equations.