PreconditionerIn 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 linear algebra and numerical analysis, a preconditioner of a matrix is a matrix such that has a smaller condition number than .
Schur complementIn linear algebra and the theory of matrices, the Schur complement of a block matrix is defined as follows. Suppose p, q are nonnegative integers, and suppose A, B, C, D are respectively p × p, p × q, q × p, and q × q matrices of complex numbers. Let so that M is a (p + q) × (p + q) matrix. If D is invertible, then the Schur complement of the block D of the matrix M is the p × p matrix defined by If A is invertible, the Schur complement of the block A of the matrix M is the q × q matrix defined by In the case that A or D is singular, substituting a generalized inverse for the inverses on M/A and M/D yields the generalized Schur complement.
Linear algebraLinear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.
System of linear equationsIn 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 variables x, y, z. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by the ordered triple since it makes all three equations valid. The word "system" indicates that the equations should be considered collectively, rather than individually.
Matrix decompositionIn 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, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations , the matrix A can be decomposed via the LU decomposition.
FactorizationIn mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial factorization of x2 – 4. Factorization is not usually considered meaningful within number systems possessing division, such as the real or complex numbers, since any can be trivially written as whenever is not zero.
Linear functionIn mathematics, the term linear function refers to two distinct but related notions: In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For distinguishing such a linear function from the other concept, the term affine function is often used. In linear algebra, mathematical analysis, and functional analysis, a linear function is a linear map.
Krylov subspaceIn 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 of b under the first r powers of A (starting from ), that is, The concept is named after Russian applied mathematician and naval engineer Alexei Krylov, who published a paper about it in 1931. Vectors are linearly independent until , and . Thus, denotes the maximal dimension of a Krylov subspace. The maximal dimension satisfies and . More exactly, , where is the minimal polynomial of .
Block matrixIn mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices. Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned.
QR decompositionIn 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 orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem and is the basis for a particular eigenvalue algorithm, the QR algorithm. Any real square matrix A may be decomposed as where Q is an orthogonal matrix (its columns are orthogonal unit vectors meaning ) and R is an upper triangular matrix (also called right triangular matrix).
