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Jacobi Method
In numerical linear algebra, the Jacobi method (a.k.a. the Jacobi iteration method) is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in. The process is then iterated until it converges. This algorithm is a stripped-down version of the Jacobi transformation method of matrix diagonalization. The method is named after Carl Gustav Jacob Jacobi. Description Let A\mathbf x = \mathbf b be a square system of ''n'' linear equations, where:A = \begin a_ & a_ & \cdots & a_ \\ a_ & a_ & \cdots & a_ \\ \vdots & \vdots & \ddots & \vdots \\a_ & a_ & \cdots & a_ \end, \qquad \mathbf = \begin x_ \\ x_2 \\ \vdots \\ x_n \end , \qquad \mathbf = \begin b_ \\ b_2 \\ \vdots \\ b_n \end. When A and \mathbf b are known, and \mathbf x is unknown, we can use the Jacobi method to approximate \mathbf x. The vector \mathbf x^ denotes our initial guess for \mat ... [...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 scienc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Society For Industrial And Applied Mathematics
Society for Industrial and Applied Mathematics (SIAM) is a professional society dedicated to applied mathematics, computational science, and data science through research, publications, and community. SIAM is the world's largest scientific society devoted to applied mathematics, and roughly two-thirds of its membership resides within the United States. Founded in 1951, the organization began holding annual national meetings in 1954, and now hosts conferences, publishes books and scholarly journals, and engages in advocacy in issues of interest to its membership. Members include engineers, scientists, and mathematicians, both those employed in academia and those working in industry. The society supports educational institutions promoting applied mathematics. SIAM is one of the four member organizations of the Joint Policy Board for Mathematics. Membership Membership is open to both individuals and organizations. By the end of its first full year of operation, SIAM had 130 me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Articles With Example Pseudocode
Article often refers to: * Article (grammar), a grammatical element used to indicate definiteness or indefiniteness * Article (publishing) An article or piece is a written work published in a Publishing, print or electronic media, electronic medium, for the propagation of news, research results, academic analysis or debate. News A news article discusses current or recent news of e ..., a piece of nonfictional prose that is an independent part of a publication Article(s) may also refer to: Government and law * Elements of treaties of the European Union * Articles of association, the regulations governing a company, used in India, the UK and other countries; called articles of incorporation in the US * Articles of clerkship, the contract accepted to become an articled clerk * Articles of Confederation, the predecessor to the current United States Constitution * Article of impeachment, a formal document and charge used for impeachment in the United States * Article of ma ... [...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 scienc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Belief Propagation
Belief propagation, also known as sum–product message passing, is a message-passing algorithm for performing inference on graphical models, such as Bayesian networks and Markov random fields. It calculates the marginal distribution for each unobserved node (or variable), conditional on any observed nodes (or variables). Belief propagation is commonly used in artificial intelligence and information theory, and has demonstrated empirical success in numerous applications, including low-density parity-check codes, turbo codes, free energy approximation, and satisfiability. The algorithm was first proposed by Judea Pearl in 1982, who formulated it as an exact inference algorithm on trees, later extended to polytrees. While the algorithm is not exact on general graphs, it has been shown to be a useful approximate algorithm. Motivation Given a finite set of discrete random variables X_1, \ldots, X_n with joint probability mass function p, a common task is to compute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterative Method
In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an "iterate") is derived from the previous ones. A specific implementation with Algorithm#Termination, termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or Quasi-Newton method, quasi-Newton methods like Broyden–Fletcher–Goldfarb–Shanno algorithm, BFGS, is an algorithm of an iterative method or a method of successive approximation. An iterative method is called ''Convergent series, 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 finit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 method can be used for any slowly converging iterative process. It was devised simultaneously by David M. Young Jr. and by Stanley P. Frankel in 1950 for the purpose of automatically solving linear systems on digital computers. Over-relaxation methods had been used before the work of Young and Frankel. An example is the method of Lewis Fry Richardson, and the methods developed by R. V. Southwell. However, these methods were designed for computation by human calculators, requiring some expertise to ensure convergence to the solution which made them inapplicable for programming on digital computers. These aspects are discussed in the thesis of David M. Young Jr. Formulation Given a square system of ''n'' linear equations with unknown x: :A\mathbf x = \mathbf b where: :A=\be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Condition Number
In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input, and how much error in the output results from an error in the input. Very frequently, one is solving the inverse problem: given f(x) = y, one is solving for ''x,'' and thus the condition number of the (local) inverse must be used. The condition number is derived from the theory of propagation of uncertainty, and is formally defined as the value of the asymptotic worst-case relative change in output for a relative change in input. The "function" is the solution of a problem and the "arguments" are the data in the problem. The condition number is frequently applied to questions in linear algebra, in which case the derivative is straightforward but the error could be in many different directions, and is thus computed from the geometry of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive-definite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number \mathbf^\mathsf M \mathbf is positive for every nonzero real column vector \mathbf, where \mathbf^\mathsf is the row vector transpose of \mathbf. More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number \mathbf^* M \mathbf is positive for every nonzero complex column vector \mathbf, where \mathbf^* denotes the conjugate transpose of \mathbf. Positive semi-definite matrices are defined similarly, except that the scalars \mathbf^\mathsf M \mathbf and \mathbf^* M \mathbf are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called ''indefinite''. Some authors use more general definitions of definiteness, permitting the matrices to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 greater than or equal to the sum of the magnitudes of all the other (off-diagonal) entries in that row. More precisely, the matrix A is diagonally dominant if :, a_, \geq \sum_ , a_, \ \ \forall \ i where a_ denotes the entry in the ith row and jth column. This definition uses a weak inequality, and is therefore sometimes called ''weak diagonal dominance''. If a strict inequality (>) is used, this is called ''strict diagonal dominance''. The unqualified term ''diagonal dominance'' can mean both strict and weak diagonal dominance, depending on the context. Variations The definition in the first paragraph sums entries across each row. It is therefore sometimes called ''row diagonal dominance''. If one changes the definition to sum down each column, this is called ''column diagonal dominance''. Any strictly diagonally dominant matr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
