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DIIS

DIIS (direct inversion in the iterative subspace or direct inversion of the iterative subspace), also known as Pulay mixing, is a technique for extrapolating the solution to a set of linear equations

Low-rank approximation

In mathematics, low-rank approximation is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable),

Tridiagonal matrix algorithm

In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve

Kaczmarz method

The Kaczmarz method or Kaczmarz's algorithm is an iterative algorithm for solving linear equation systems . It was first discovered by the Polish mathematician Stefan Kaczmarz, and was rediscovered in

Jacobi method

In numerical linear algebra, the Jacobi method is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for

Method of Four Russians

In computer science, the Method of Four Russians is a technique for speeding up algorithms involving Boolean matrices, or more generally algorithms involving matrices in which each cell may take on on

Convergent matrix

In linear algebra, a convergent matrix is a matrix that converges to the zero matrix under matrix exponentiation.

Rybicki Press algorithm

The Rybicki–Press algorithm is a fast algorithm for inverting a matrix whose entries are given by , where and where the are sorted in order. The key observation behind the Rybicki-Press observation is

Stieltjes matrix

In mathematics, particularly matrix theory, a Stieltjes matrix, named after Thomas Joannes Stieltjes, is a real symmetric positive definite matrix with nonpositive off-diagonal entries. A Stieltjes ma

Sparse approximation

Sparse approximation (also known as sparse representation) theory deals with sparse solutions for systems of linear equations. Techniques for finding these solutions and exploiting them in application

QR decomposition

In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthogonal matrix Q and an upper triangular ma

Frobenius inner product

In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a scalar. It is often denoted . The operation is a component-wise inner product of two matrices as

Eigenvalue algorithm

In numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors.

LU reduction

LU reduction is an algorithm related to LU decomposition. This term is usually used in the context of super computing and highly parallel computing. In this context it is used as a benchmarking algori

Block Lanczos algorithm

In computer science, the 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 matrices. Such matrice

Householder transformation

In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane cont

Symmetric successive over-relaxation

In applied mathematics, symmetric successive over-relaxation (SSOR), is a preconditioner. If the original matrix can be split into diagonal, lower and upper triangular as then the SSOR preconditioner

Householder operator

In linear algebra, the Householder operator is defined as follows. Let be a finite dimensional inner product space with inner product and unit vector . Then is defined by This operator reflects the ve

In-place matrix transposition

In-place matrix transposition, also called in-situ matrix transposition, is the problem of transposing an N×M matrix in-place in computer memory, ideally with O(1) (bounded) additional storage, or at

ILNumerics

ILNumerics is a mathematical class library for Common Language Infrastructure (CLI) developers and a domain specific language (DSL) for the implementation of numerical algorithms on the .NET platform.

Preconditioner

In mathematics, preconditioning is the application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for numerical solving methods. Prec

Eigenmode expansion

Eigenmode expansion (EME) is a computational electrodynamics modelling technique. It is also referred to as the mode matching technique or the bidirectional eigenmode propagation method (BEP method).

Mathcad

Mathcad is computer software for the verification, validation, documentation and re-use of mathematical calculations in engineering and science, notably mechanical, chemical, electrical, and civil eng

RRQR factorization

An RRQR factorization or rank-revealing QR factorization is a matrix decomposition algorithm based on the QR factorization which can be used to determine the rank of a matrix. The singular value decom

Interpolative decomposition

In numerical analysis, interpolative decomposition (ID) factors a matrix as the product of two matrices, one of which contains selected columns from the original matrix, and the other of which has a s

Triangular matrix

In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is ca

Matrix multiplication

In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matr

Matrix splitting

In the mathematical discipline of numerical linear algebra, a matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. Many iterative methods (for example,

Frontal solver

A frontal solver, conceived by Bruce Irons, is an approach to solving sparse linear systems which is used extensively in finite element analysis. It is a variant of Gauss elimination that automaticall

Lis (linear algebra library)

Lis (Library of Iterative Solvers for linear systems, pronounced [lis]) is a scalable parallel software library for solving discretized linear equations and eigenvalue problems that mainly arise in th

MATLAB

MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plott

Pseudospectrum

In mathematics, the pseudospectrum of an operator is a set containing the spectrum of the operator and the numbers that are "almost" eigenvalues. Knowledge of the pseudospectrum can be particularly us

Biconjugate gradient stabilized method

In numerical linear algebra, the biconjugate gradient stabilized method, often abbreviated as BiCGSTAB, is an iterative method developed by H. A. van der Vorst for the numerical solution of nonsymmetr

Row echelon form

In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination. A matrix being in row echelon form means that Gaussian elimination has operated on the rows, a

GotoBLAS

In scientific computing, GotoBLAS and GotoBLAS2 are open source implementations of the BLAS (Basic Linear Algebra Subprograms) API with many hand-crafted optimizations for specific processor types. Go

Hilbert matrix

In linear algebra, a Hilbert matrix, introduced by Hilbert, is a square matrix with entries being the unit fractions For example, this is the 5 × 5 Hilbert matrix: The Hilbert matrix can be regarded a

Conjugate residual method

The conjugate residual method is an iterative numeric method used for solving systems of linear equations. It's a Krylov subspace method very similar to the much more popular conjugate gradient method

Automatically Tuned Linear Algebra Software

Automatically Tuned Linear Algebra Software (ATLAS) is a software library for linear algebra. It provides a mature open source implementation of BLAS APIs for C and Fortran77. ATLAS is often recommend

Diagonally dominant matrix

In mathematics, a square matrix is said to be diagonally dominant if, for every row of the matrix, the magnitude of the diagonal entry in a row is larger than or equal to the sum of the magnitudes of

Walter Edwin Arnoldi

Walter Edwin Arnoldi (December 14, 1917 – October 5, 1995) was an American engineer mainly known for the Arnoldi iteration, an eigenvalue algorithm used in numerical linear algebra. His main research

Wilkinson matrix

In linear algebra, Wilkinson matrices are symmetric, tridiagonal, order-N matrices with pairs of nearly, but not exactly, equal eigenvalues. It is named after the British mathematician James H. Wilkin

BLIS (software)

In scientific computing, BLIS (BLAS-like Library Instantiation Software)is an open-source framework for implementing a superset of BLAS (Basic Linear Algebra Subprograms) functionality for specific pr

Vandermonde matrix

In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an m × n matrix or for all indices i and j. Som

Gauss–Seidel method

In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a system of linear equations. It

MINRES

The Minimal Residual Method or MINRES is a Krylov subspace method for the iterative solution of symmetric linear equation systems. It was proposed by mathematicians and Michael Alan Saunders in 1975.

EISPACK

EISPACK is a software library for numerical computation of eigenvalues and eigenvectors of matrices, written in FORTRAN. It contains subroutines for calculating the eigenvalues of nine classes of matr

Circulant matrix

In linear algebra, a circulant matrix is a square matrix in which all row vectors are composed of the same elements and each row vector is rotated one element to the right relative to the preceding ro

Librsb

librsb is an open-source parallel library for sparse matrix computations using the Recursive Sparse Blocks (RSB) matrix format. librsb provides cache efficient multi-threaded Sparse BLAS operations vi

Hypre

The Parallel High Performance Preconditioners (hypre) is a library of routines for scalable (parallel) solution of linear systems. The built-in BLOPEX package in addition allows solving eigenvalue pro

Julia (programming language)

Julia is a high-level, dynamic programming language. Its features are well suited for numerical analysis and computational science. Distinctive aspects of Julia's design include a type system with par

Comparison of linear algebra libraries

The following tables provide a comparison of linear algebra software libraries, either specialized or general purpose libraries with significant linear algebra coverage.

Bidiagonalization

Bidiagonalization is one of unitary (orthogonal) matrix decompositions such that U* A V = B, where U and V are unitary (orthogonal) matrices; * denotes Hermitian transpose; and B is upper bidiagonal.

Speakeasy (computational environment)

Speakeasy was a numerical computing interactive environment also featuring an interpreted programming language. It was initially developed for internal use at the Physics Division of Argonne National

System of linear equations

In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, is a system of three equations in the three v

Moore–Penrose inverse

In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in

Matrix-free methods

In computational mathematics, a matrix-free method is an algorithm for solving a linear system of equations or an eigenvalue problem that does not store the coefficient matrix explicitly, but accesses

Bareiss algorithm

In mathematics, the Bareiss algorithm, named after , is an algorithm to calculate the determinant or the echelon form of a matrix with integer entries using only integer arithmetic; any divisions that

Data Analytics Library

oneAPI Data Analytics Library (oneDAL; formerly Intel Data Analytics Acceleration Library or Intel DAAL), is a library of optimized algorithmic building blocks for data analysis stages most commonly a

LAPACK

LAPACK ("Linear Algebra Package") is a standard software library for numerical linear algebra. It provides routines for solving systems of linear equations and linear least squares, eigenvalue problem

Basic Linear Algebra Subprograms

Basic Linear Algebra Subprograms (BLAS) is a specification that prescribes a set of low-level routines for performing common linear algebra operations such as vector addition, scalar multiplication, d

Gradient method

In optimization, a gradient method is an algorithm to solve problems of the form with the search directions defined by the gradient of the function at the current point. Examples of gradient methods a

Portable, Extensible Toolkit for Scientific Computation

The Portable, Extensible Toolkit for Scientific Computation (PETSc, pronounced PET-see; the S is silent), is a suite of data structures and routines developed by Argonne National Laboratory for the sc

Pivot element

The pivot or pivot element is the element of a matrix, or an array, which is selected first by an algorithm (e.g. Gaussian elimination, simplex algorithm, etc.), to do certain calculations. In the cas

Jacobi method for complex Hermitian matrices

In mathematics, the Jacobi method for complex Hermitian matrices is a generalization of the Jacobi iteration method. The Jacobi iteration method is also explained in "Introduction to Linear Algebra" b

Derivation of the conjugate gradient method

In numerical linear algebra, the conjugate gradient method is an iterative method for numerically solving the linear system where is symmetric positive-definite. The conjugate gradient method can be d

Iterative refinement

Iterative refinement is an iterative method proposed by James H. Wilkinson to improve the accuracy of numerical solutions to systems of linear equations. When solving a linear system due to the compou

Numerical methods for linear least squares

Numerical methods for linear least squares entails the numerical analysis of linear least squares problems.

Krylov subspace

In linear algebra, the order-r Krylov subspace generated by an n-by-n matrix A and a vector b of dimension n is the linear subspace spanned by the images of b under the first r powers of A (starting f

Samuelson–Berkowitz algorithm

In mathematics, the Samuelson–Berkowitz algorithm efficiently computes the characteristic polynomial of an matrix whose entries may be elements of any unital commutative ring. Unlike the Faddeev–LeVer

Matrix multiplication algorithm

Because matrix multiplication is such a central operation in many numerical algorithms, much work has been invested in making matrix multiplication algorithms efficient. Applications of matrix multipl

Kernel (linear algebra)

In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map L

Biconjugate gradient method

In mathematics, more specifically in numerical linear algebra, the biconjugate gradient method is an algorithm to solve systems of linear equations Unlike the conjugate gradient method, this algorithm

Armadillo (C++ library)

Armadillo is a linear algebra software library for the C++ programming language. It aims to provide efficient and streamlined base calculations, while at the same time having a straightforward and eas

LU decomposition

In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomp

Divide-and-conquer eigenvalue algorithm

Divide-and-conquer eigenvalue algorithms are a class of eigenvalue algorithms for Hermitian or real symmetric matrices that have recently (circa 1990s) become competitive in terms of stability and eff

Singular value decomposition

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis

Conjugate gradient method

In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradie

Modal analysis using FEM

The goal of modal analysis in structural mechanics is to determine the natural mode shapes and frequencies of an object or structure during free vibration. It is common to use the finite element metho

Arnoldi iteration

In numerical linear algebra, the Arnoldi iteration is an eigenvalue algorithm and an important example of an iterative method. Arnoldi finds an approximation to the eigenvalues and eigenvectors of gen

Power iteration

In mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix , the algorithm will produce a number , which is the greatest (in absolute va

Least-squares spectral analysis

Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the mos

Block Wiedemann algorithm

The block Wiedemann algorithm for computing kernel vectors of a matrix over a finite field is a generalization by Don Coppersmith of an algorithm due to Doug Wiedemann.

Block matrix pseudoinverse

In mathematics, a block matrix pseudoinverse is a formula for the pseudoinverse of a partitioned matrix. This is useful for decomposing or approximating many algorithms updating parameters in signal p

LOBPCG

Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is a matrix-free method for finding the largest (or smallest) eigenvalues and the corresponding eigenvectors of a symmetric generalized

LINPACK

LINPACK is a software library for performing numerical linear algebra on digital computers. It was written in Fortran by Jack Dongarra, Jim Bunch, Cleve Moler, and Gilbert Stewart, and was intended fo

Chebyshev iteration

In numerical linear algebra, the Chebyshev iteration is aniterative method for determining the solutions of a system of linear equations. The method is named after Russian mathematician Pafnuty Chebys

Numerical linear algebra

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

GraphBLAS

GraphBLAS (/ˈɡræfˌblɑːz/) is an API specification that defines standard building blocks for graph algorithms in the language of linear algebra. GraphBLAS is built upon the notion that a sparse matrix

OpenBLAS

In scientific computing, OpenBLAS is an open-source implementation of the BLAS (Basic Linear Algebra Subprograms) and LAPACK APIs with many hand-crafted optimizations for specific processor types. It

Jacobi eigenvalue algorithm

In numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix (a process known as diagonalizati

SequenceL

SequenceL is a general purpose functional programming language and auto-parallelizing (Parallel computing) compiler and tool set, whose primary design objectives are performance on multi-core processo

Stone's method

In numerical analysis, Stone's method, also known as the strongly implicit procedure or SIP, is an algorithm for solving a sparse linear system of equations. The method uses an incomplete LU decomposi

Backfitting algorithm

In statistics, the backfitting algorithm is a simple iterative procedure used to fit a generalized additive model. It was introduced in 1985 by Leo Breiman and Jerome Friedman along with generalized a

Fangcheng (mathematics)

Fangcheng (sometimes written as fang-cheng or fang cheng) (Chinese: 方程; pinyin: fāng chéng) is the title of the eighth chapter of the Chinese mathematical classic Jiuzhang suanshu (The Nine Chapters o

Bartels–Stewart algorithm

In numerical linear algebra, the Bartels–Stewart algorithm is used to numerically solve the Sylvester matrix equation . Developed by R.H. Bartels and G.W. Stewart in 1971, it was the first numerically

SLEPc

SLEPc is a software library for the parallel computation of eigenvalues and eigenvectors of large, sparse matrices. It can be seen as a module of PETSc that provides solvers for different types of eig

DADiSP

DADiSP (Data Analysis and Display, pronounced day-disp) is a numerical computing environment developed by DSP Development Corporation which allows one to display and manipulate data series, matrices a

Modified Richardson iteration

Modified Richardson iteration is an iterative method for solving a system of linear equations. Richardson iteration was proposed by Lewis Fry Richardson in his work dated 1910. It is similar to the Ja

Successive over-relaxation

In numerical linear algebra, the method of successive over-relaxation (SOR) is a variant of the Gauss–Seidel method for solving a linear system of equations, resulting in faster convergence. A similar

Nested dissection

In numerical analysis, nested dissection is a divide and conquer heuristic for the solution of sparse symmetric systems of linear equations based on graph partitioning. Nested dissection was introduce

Gaussian elimination

In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matri

Jacobi rotation

In numerical linear algebra, a Jacobi rotation is a rotation, Qkℓ, of a 2-dimensional linear subspace of an n-dimensional inner product space, chosen to zero a symmetric pair of off-diagonal entries o

Rayleigh quotient iteration

Rayleigh quotient iteration is an eigenvalue algorithm which extends the idea of the inverse iteration by using the Rayleigh quotient to obtain increasingly accurate eigenvalue estimates. Rayleigh quo

ABS methods

ABS methods, where the acronym contains the initials of Jozsef Abaffy, Charles G. Broyden and Emilio Spedicato, have been developed since 1981 to generate a large class of algorithms for the following

Folded spectrum method

In mathematics, the folded spectrum method (FSM) is an iterative method for solving large eigenvalue problems.Here you always find a vector with an eigenvalue close to a search-value . This means you

Relaxation (iterative method)

In numerical mathematics, relaxation methods are iterative methods for solving systems of equations, including nonlinear systems. Relaxation methods were developed for solving large sparse linear syst

Generalized minimal residual method

In mathematics, the generalized minimal residual method (GMRES) is an iterative method for the numerical solution of an indefinite nonsymmetric system of linear equations. The method approximates the

Cholesky decomposition

In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced /ʃəˈlɛski/ shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower tria

Incomplete LU factorization

In numerical linear algebra, an incomplete LU factorization (abbreviated as ILU) of a matrix is a sparse approximation of the LU factorization often used as a preconditioner.

Lanczos algorithm

The Lanczos algorithm is an iterative method devised by Cornelius Lanczos that is an adaptation of power methods to find the "most useful" (tending towards extreme highest/lowest) eigenvalues and eige

Givens rotation

In numerical linear algebra, a Givens rotation is a rotation in the plane spanned by two coordinates axes. Givens rotations are named after Wallace Givens, who introduced them to numerical analysts in

Incomplete Cholesky factorization

In numerical analysis, an incomplete Cholesky factorization of a symmetric positive definite matrix is a sparse approximation of the Cholesky factorization. An incomplete Cholesky factorization is oft

Basis function

In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as e

QR algorithm

In numerical linear algebra, 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 was developed

Minimum degree algorithm

In numerical analysis, the minimum degree algorithm is an algorithm used to permute the rows and columns of a symmetric sparse matrix before applying the Cholesky decomposition, to reduce the number o

Inverse iteration

In numerical analysis, inverse iteration (also known as the inverse power method) is an iterative eigenvalue algorithm. It allows one to find an approximateeigenvector when an approximation to a corre

SPIKE algorithm

The SPIKE algorithm is a hybrid parallel solver for banded linear systems developed by Eric Polizzi and Ahmed Sameh

Dune (software)

DUNE (Distributed and Unified Numerics Environment) is a modular C++ library for the solution of partial differential equations using grid-based methods. The DUNE library is divided into modules. In v

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