Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics. It is a subfield of numerical analysis, and a type of linear algebra. Computers use floating-point arithmetic and cannot exactly represent irrational data, so when a computer algorithm is applied to a matrix of data, it can sometimes increase the difference between a number stored in the computer and the true number that it is an approximation of. Numerical linear algebra uses properties of vectors and matrices to develop computer algorithms that minimize the error introduced by the computer, and is also concerned with ensuring that the algorithm is as efficient as possible.
Numerical linear algebra aims to solve problems of continuous mathematics using finite precision computers, so its applications to the natural and social sciences are as vast as the applications of continuous mathematics. It is often a fundamental part of engineering and computational science problems, such as image and signal processing, telecommunication, computational finance, materials science simulations, structural biology, data mining, bioinformatics, and fluid dynamics. Matrix methods are particularly used in finite difference methods, finite element methods, and the modeling of differential equations. Noting the broad applications of numerical linear algebra, Lloyd N. Trefethen and David Bau, III argue that it is "as fundamental to the mathematical sciences as calculus and differential equations",[1]: x even though it is a comparatively small field.[2] Because many properties of matrices and vectors also apply to functions and operators, numerical linear algebra can also be viewed as a type of functional analysis which has a particular emphasis on practical algorithms.[1]: ix
Common problems in numerical linear algebra include obtaining matrix decompositions like the singular value decomposition, the QR factorization, the LU factorization, or the eigendecomposition, which can then be used to answer common linear algebraic problems like solving linear systems of equations, locating eigenvalues, or least squares optimisation. Numerical linear algebra's central concern with developing algorithms that do not introduce errors when applied to real data on a finite precision computer is often achieved by iterative methods rather than direct ones.
^ abTrefethen, Lloyd; Bau III, David (1997). Numerical Linear Algebra (1st ed.). Philadelphia: SIAM. ISBN 978-0-89871-361-9.
^Golub, Gene H. "A History of Modern Numerical Linear Algebra" (PDF). University of Chicago Statistics Department. Retrieved February 17, 2019.
and 25 Related for: Numerical linear algebra information
Numericallinearalgebra, sometimes called applied linearalgebra, is the study of how matrix operations can be used to create computer algorithms which...
Linearalgebra is the branch of mathematics concerning linear equations such as: a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b...
Basic LinearAlgebra Subprograms (BLAS) is a specification that prescribes a set of low-level routines for performing common linearalgebra operations...
mechanics (predicting the motions of planets, stars and galaxies), numericallinearalgebra in data analysis, and stochastic differential equations and Markov...
provide a comparison of linearalgebra software libraries, either specialized or general purpose libraries with significant linearalgebra coverage. Matrix types...
of numericallinearalgebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A system of non-linear equations...
numericallinearalgebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear...
ALGLIB is an open source / commercial numerical analysis library with C++ version Armadillo is a C++ linearalgebra library (matrix and vector maths), aiming...
In linearalgebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express...
LAPACK ("LinearAlgebra Package") is a standard software library for numericallinearalgebra. It provides routines for solving systems of linear equations...
(1997). Applied numericallinearalgebra. SIAM. Ciarlet, P. G., Miara, B., & Thomas, J. M. (1989). Introduction to numericallinearalgebra and optimization...
validated numerics is divided into the following fields: Verification in numericallinearalgebra Validating numerical solutions of a given system of linear equations...
involving π Numericallinearalgebra — study of numerical algorithms for linearalgebra problems Types of matrices appearing in numerical analysis: Sparse...
(2021). "Algebra and Geometry with Python". SpringerLink. doi:10.1007/978-3-030-61541-3. Atkinson, Kendall A. (1989), An Introduction to Numerical Analysis...
Numericalalgebraic geometry is a field of computational mathematics, particularly computational algebraic geometry, which uses methods from numerical...
traditional engineering methods. Numerical methods used in scientific computation, for example numericallinearalgebra and numerical solution of partial differential...
In linearalgebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, A is symmetric ⟺ A = A T . {\displaystyle A{\text{...
approach is taken by the Numerical Recipes library, where emphasis is placed on clear understanding of algorithms. Many computer algebra systems (listed above)...
In numericallinearalgebra, a Givens rotation is a rotation in the plane spanned by two coordinates axes. Givens rotations are named after Wallace Givens...
In linearalgebra, the Cholesky decomposition or Cholesky factorization (pronounced /ʃəˈlɛski/ shə-LES-kee) is a decomposition of a Hermitian, positive-definite...
In linearalgebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row:...
In numericallinearalgebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form...
matrix Tridiagonal matrix Invariant subspace Axler, Sheldon Jay (1997). LinearAlgebra Done Right (2nd ed.). New York: Springer. pp. 86–87, 169. ISBN 0-387-22595-1...
mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure...