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Bartels–Stewart Algorithm
In numerical linear algebra, the Bartels–Stewart algorithm is used to numerically solve the Sylvester matrix equation AX - XB = C. Developed by R.H. Bartels and G.W. Stewart in 1971, it was the first numerically stable method that could be systematically applied to solve such equations. The algorithm works by using the real Schur decompositions of A and B to transform AX - XB = C into a triangular system that can then be solved using forward or backward substitution. In 1979, G. Golub, C. Van Loan and S. Nash introduced an improved version of the algorithm, known as the Hessenberg–Schur algorithm. It remains a standard approach for solving Sylvester equations when X is of small to moderate size. The algorithm Let X, C \in \mathbb^, and assume that the eigenvalues of A are distinct from the eigenvalues of B. Then, the matrix equation AX - XB = C has a unique solution. The Bartels–Stewart algorithm computes X by applying the following steps: 1.Compute the real Schu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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 ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylvester Equation
In mathematics, in the field of control theory, a Sylvester equation is a Matrix (mathematics), matrix equation of the form: :A X + X B = C. Then given matrices ''A'', ''B'', and ''C'', the problem is to find the possible matrices ''X'' that obey this equation. All matrices are assumed to have coefficients in the complex numbers. For the equation to make sense, the matrices must have appropriate sizes, for example they could all be square matrices of the same size. But more generally, ''A'' and ''B'' must be square matrices of sizes ''n'' and ''m'' respectively, and then ''X'' and ''C'' both have ''n'' rows and ''m'' columns. A Sylvester equation has a unique solution for ''X'' exactly when there are no common eigenvalues of ''A'' and −''B''. More generally, the equation ''AX'' + ''XB'' = ''C'' has been considered as an equation of bounded operators on a (possibly infinite-dimensional) Banach space. In this case, the condition for the uniqueness of a solut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Stability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might cause a large deviation of final answer from the exact solution. Some numerical algorithms may damp out the small fluctuations (errors) in the input data; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called ''numerically stable''. One of the common task ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Decomposition
In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily equivalent to an upper triangular matrix whose diagonal elements are the eigenvalues of the original matrix. Statement The Schur decomposition reads as follows: if ''A'' is an square matrix with complex entries, then ''A'' can be expressed as(Section 2.3 and further at p. 79(Section 7.7 at p. 313 : A = Q U Q^ where ''Q'' is a unitary matrix (so that its inverse ''Q''−1 is also the conjugate transpose ''Q''* of ''Q''), and ''U'' is an upper triangular matrix, which is called a Schur form of ''A''. Since ''U'' is similar to ''A'', it has the same spectrum, and since it is triangular, its eigenvalues are the diagonal entries of ''U''. The Schur decomposition implies that there exists a nested sequence of ''A''-invariant subspaces , and that there exists an or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gene H
In biology, the word gene (from , ; "...Wilhelm Johannsen coined the word gene to describe the Mendelian units of heredity..." meaning ''generation'' or ''birth'' or ''gender'') can have several different meanings. The Mendelian gene is a basic unit of heredity and the molecular gene is a sequence of nucleotides in DNA that is transcribed to produce a functional RNA. There are two types of molecular genes: protein-coding genes and noncoding genes. During gene expression, the DNA is first copied into RNA. The RNA can be directly functional or be the intermediate template for a protein that performs a function. The transmission of genes to an organism's offspring is the basis of the inheritance of phenotypic traits. These genes make up different DNA sequences called genotypes. Genotypes along with environmental and developmental factors determine what the phenotypes will be. Most biological traits are under the influence of polygenes (many different genes) as well as gene– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles F
Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning was "free man". The Old English descendant of this word was '' Ċearl'' or ''Ċeorl'', as the name of King Cearl of Mercia, that disappeared after the Norman conquest of England. The name was notably borne by Charlemagne (Charles the Great), and was at the time Latinized as ''Karolus'' (as in ''Vita Karoli Magni''), later also as '' Carolus''. Some Germanic languages, for example Dutch and German, have retained the word in two separate senses. In the particular case of Dutch, ''Karel'' refers to the given name, whereas the noun ''kerel'' means "a bloke, fellow, man". Etymology The name's etymology is a Common Germanic noun ''*karilaz'' meaning "free man", which survives in English as churl (< Old English ''ċeorl''), which developed its depr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 in the late 1950s by John G. F. Francis and by Vera N. Kublanovskaya, working independently. The basic idea is to perform a QR decomposition, writing the matrix as a product of an orthogonal matrix and an upper triangular matrix, multiply the factors in the reverse order, and iterate. The practical QR algorithm Formally, let ''A'' be a real matrix of which we want to compute the eigenvalues, and let ''A''0:=''A''. At the ''k''-th step (starting with ''k'' = 0), we compute the QR decomposition ''A''''k''=''Q''''k''''R''''k'' where ''Q''''k'' is an orthogonal matrix (i.e., ''Q''''T'' = ''Q''−1) and ''R''''k'' is an upper triangular matrix. We then form ''A''''k''+1 = ''R''''k''''Q''''k''. Note that : A_ = R_k Q_k = Q_k^ Q_k R_k Q_k = Q_k^ A_k Q_k = Q_k^ A_k Q_k, so all the ''A''''k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hessenberg Matrix
In linear algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular. To be exact, an upper Hessenberg matrix has zero entries below the first subdiagonal, and a lower Hessenberg matrix has zero entries above the first superdiagonal. They are named after Karl Hessenberg. Definitions Upper Hessenberg matrix A square n \times n matrix A is said to be in upper Hessenberg form or to be an upper Hessenberg matrix if a_=0 for all i,j with i > j+1. An upper Hessenberg matrix is called unreduced if all subdiagonal entries are nonzero, i.e. if a_ \neq 0 for all i \in \. Lower Hessenberg matrix A square n \times n matrix A is said to be in lower Hessenberg form or to be a lower Hessenberg matrix if its transpose is an upper Hessenberg matrix or equivalently if a_=0 for all i,j with j > i+1. A lower Hessenberg matrix is called unreduced if all superdiagonal entries are nonzero, i.e. if a_ \neq 0 for all i \in \. Examples Consider the following matri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 containing the origin. The Householder transformation was used in a 1958 paper by Alston Scott Householder. Its analogue over general inner product spaces is the Householder operator. Definition Transformation The reflection hyperplane can be defined by its ''normal vector'', a unit vector v (a vector with length 1) that is orthogonal to the hyperplane. The reflection of a point x about this hyperplane is the linear transformation: : x - 2\langle x, v\rangle v = x - 2v\left(v^\textsf x\right), where v is given as a column unit vector with Hermitian transpose v^\textsf. Householder matrix The matrix constructed from this transformation can be expressed in terms of an outer product as: : P = I - 2vv^\textsf is known as the Householder matrix, where I is the identity matrix. Prope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krylov Subspace Method
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the previous ones. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method. An iterative method is called convergent if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common. In contrast, direct methods attempt to solve the problem by a finite sequence of operations. In the absence of rounding errors, direct methods would deliver an exact solution (for example, solving a linear system of equations A\mathbf=\mathbf by Gaussian elimination). Iterative methods are often the only choice for nonlinear equations. Howe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Direction Implicit Method
In numerical linear algebra, the alternating-direction implicit (ADI) method is an iterative method used to solve Sylvester matrix equations. It is a popular method for solving the large matrix equations that arise in systems theory and control, and can be formulated to construct solutions in a memory-efficient, factored form. It is also used to numerically solve parabolic and elliptic partial differential equations, and is a classic method used for modeling heat conduction and solving the diffusion equation in two or more dimensions.. It is an example of an operator splitting method. ADI for matrix equations The method The ADI method is a two step iteration process that alternately updates the column and row spaces of an approximate solution to AX - XB = C. One ADI iteration consists of the following steps:1. Solve for X^, where \left( A - \beta_ I\right) X^ = X^\left( B - \beta_ I \right) + C. 2. Solve for X^, where X^\left( B - \alpha_ I \right) = \left( A - \alpha_ I\r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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), subject to a constraint that the approximating matrix has reduced rank. The problem is used for mathematical modeling and data compression. The rank constraint is related to a constraint on the complexity of a model that fits the data. In applications, often there are other constraints on the approximating matrix apart from the rank constraint, e.g., non-negativity and Hankel structure. Low-rank approximation is closely related to: * principal component analysis, * factor analysis, * total least squares, * latent semantic analysis * orthogonal regression, and * dynamic mode decomposition. Definition Given * structure specification \mathcal : \mathbb^ \to \mathbb^, * vector of structure parameters p\in\mathbb^, * norm \, \cdot \, , and * desired rank r, : \text \quad \text ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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