Gaussian elimination, also known as row reduction, is an algorithm in linear algebra for solving a system of linear equations. It is usually understood as a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix. The method is named after Carl Friedrich Gauss (1777–1855). Some special cases of the method - albeit presented without proof - were known to Chinese mathematicians as early as circa 179 CE.
Linear algebra is the branch of mathematics concerning linear equations such as:
In linear algebra, the Cholesky decomposition or Cholesky factorization is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations. It was discovered by André-Louis Cholesky for real matrices. When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations.
In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.
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 definition how many elements need to be zero for a matrix to be considered sparse but a common criterion is that the number of non-zero elements is roughly the number of rows or columns. By contrast, if most of the elements are nonzero, then the matrix is considered dense. The number of zero-valued elements divided by the total number of elements is sometimes referred to as the sparsity of the matrix.
In numerical linear algebra, the Cuthill–McKee algorithm (CM), named for Elizabeth Cuthill and 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 (RCM) due to Alan George is the same algorithm but with the resulting index numbers reversed. In practice this generally results in less fill-in than the CM ordering when Gaussian elimination is applied.
In the mathematical subfield of graph theory a level structure of an undirected graph is a partition of the vertices into subsets that have the same distance from a given root vertex.
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 of non-zeros in the Cholesky factor. This results in reduced storage requirements and means that the Cholesky factor can be applied with fewer arithmetic operations.
In mathematics, particularly matrix theory, a band matrix is a sparse matrix whose non-zero entries are confined to a diagonal band, comprising the main diagonal and zero or more diagonals on either side.
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. Preconditioning is typically related to reducing a condition number of the problem. The preconditioned problem is then usually solved by an iterative method.
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. The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix. The LU decomposition was introduced by the Polish mathematician Tadeusz Banachiewicz in 1938.
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.
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 the matrix by evaluating matrix-vector products. Such methods can be preferable when the matrix is so big that storing and manipulating it would cost a lot of memory and computing time, even with the use of methods for sparse matrices. Many iterative methods allow for a matrix-free implementation, including:
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 automatically avoids a large number of operations involving zero terms.
In mathematics, a matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns. For example, the dimension of the matrix below is 2 × 3, because there are two rows and three columns:
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 introduced by George (1973); the name was suggested by Garrett Birkhoff.
In numerical mathematics, hierarchical matrices (H-matrices) are used as data-sparse approximations of non-sparse matrices. While a sparse matrix of dimension can be represented efficiently in units of storage by storing only its non-zero entries, a non-sparse matrix would require units of storage, and using this type of matrices for large problems would therefore be prohibitively expensive in terms of storage and computing time. Hierarchical matrices provide an approximation requiring only units of storage, where is a parameter controlling the accuracy of the approximation. In typical applications, e.g., when discretizing integral equations , preconditioning the resulting systems of linear equations , or solving elliptic partial differential equations , a rank proportional to with a small constant is sufficient to ensure an accuracy of . Compared to many other data-sparse representations of non-sparse matrices, hierarchical matrices offer a major advantage: the results of matrix arithmetic operations like matrix multiplication, factorization or inversion can be approximated in operations, where
Matrix Toolkit Java (MTJ) is an open-source Java software library for performing numerical linear algebra. The library contains a full set of standard linear algebra operations for dense matrices based on BLAS and LAPACK code. Partial set of sparse operations is provided through the Templates project. The library can be configured to run as a pure Java library or use BLAS machine-optimized code through the Java Native Interface.
