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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 is named after the German mathematicians Carl Friedrich Gauss and Philipp Ludwig von Seidel, and is similar to the Jacobi method. Though it can be applied to any matrix with non-zero elements on the diagonals, convergence is only guaranteed if the matrix is either strictly diagonally dominant, or symmetric and positive definite. It was only mentioned in a private letter from Gauss to his student Gerling in 1823. A publication was not delivered before 1874 by Seidel. Description The Gauss–Seidel method is an iterative technique for solving a square system of linear equations with unknown : A\mathbf x = \mathbf b . It is defined by the iteration L_* \mathbf^ = \mathbf - U \mathbf^, where \mathbf^ is the -th approximation or iteration of \mathbf,\,\mathbf^ is the next or -th i ... [...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] |
Residual (numerical Analysis)
Loosely speaking, a residual is the error in a result. To be precise, suppose we want to find ''x'' such that : f(x)=b. Given an approximation ''x''0 of ''x'', the residual is : b - f(x_0) that is, "what is left of the right hand side" after subtracting ''f''(''x''0)" (thus, the name "residual": what is left, the rest). On the other hand, the error is : x - x_0 If the exact value of ''x'' is not known, the residual can be computed, whereas the error cannot. Residual of the approximation of a function Similar terminology is used dealing with differential, integral and functional equations. For the approximation f_\text of the solution f of the equation : T(f)(x)=g(x) \, , the residual can either be the function : ~g(x)~ - ~T(f_\text)(x) or can be said to be the maximum of the norm of this difference : \max_ , g(x)-T(f_\text)(x), over the domain \mathcal X, where the function f_\text is expected to approximate the solution f , or some integral of a function of the differe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relaxation (iterative Methods)
Relaxation stands quite generally for a release of tension, a return to equilibrium. In the sciences, the term is used in the following ways: * Relaxation (physics), and more in particular: ** Relaxation (NMR), processes by which nuclear magnetization returns to the equilibrium distribution ** Dielectric relaxation, the delay in the dielectric constant of a material ** Vibrational energy relaxation, the process by which molecules in high energy quantum states return to the Maxwell-Boltzmann distribution ** Chemical relaxation methods, related to temperature jump ** Relaxation oscillator, a type of electronic oscillator In mathematics: :* Relaxation (approximation), a technique for transforming hard constraints into easier ones :* Relaxation (iterative method), a technique for the numerical solution of equations :* Relaxation (extension method), a technique for a natural extension in mathematical optimization or variational problems In computer science: :* Relaxation (computin ... [...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), a piece of nonfictional prose that is an independent part of a publication Article may also refer to: Government and law * Article (European Union), articles of treaties of the European Union * Articles of association, the regulations governing a company, used in India, the UK and other countries * 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 * Articles of incorporation, for corporations, U.S. equivalent of articles of association * Articles of organization, for limited liability organizations, a U.S. equivalent of articles of association Other uses * Article, an HTML element, delimited by the tags and * Article of clothing, an i ... [...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] |
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 Jacobi and Gauss–Seidel method. We seek the solution to a set of linear equations, expressed in matrix terms as : A x = b.\, The Richardson iteration is : x^ = x^ + \omega \left( b - A x^ \right), where \omega is a scalar parameter that has to be chosen such that the sequence x^ converges. It is easy to see that the method has the correct fixed points, because if it converges, then x^ \approx x^ and x^ has to approximate a solution of A x = b. Convergence Subtracting the exact solution x, and introducing the notation for the error e^ = x^-x, we get the equality for the errors : e^ = e^ - \omega A e^ = (I-\omega A) e^. Thus, : \, e^\, = \, (I-\omega A) e^\, \leq \, I-\omega A\, \, e^\, , for any vector norm and the corresponding induced matrix norm. Thus, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, for systems of differential equations) depend upon the direct solution of matrix equations involving matrices more general than tridiagonal matrices. These matrix equations can often be solved directly and efficiently when written as a matrix splitting. The technique was devised by Richard S. Varga in 1960. Regular splittings We seek to solve the matrix equation where A is a given ''n'' × ''n'' non-singular matrix, and k is a given column vector with ''n'' components. We split the matrix A into where B and C are ''n'' × ''n'' matrices. If, for an arbitrary ''n'' × ''n'' matrix M, M has nonnegative entries, we write M ≥ 0. If M has only positive entries, we write M > 0. Similarly, if the matrix M1 − M2 has nonnegative entries, we write M1 ≥ M2. Defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaczmarz Method
The Kaczmarz method or Kaczmarz's algorithm is an iterative algorithm for solving linear equation systems A x = b . It was first discovered by the Polish mathematician Stefan Kaczmarz, and was rediscovered in the field of image reconstruction from projections by Richard Gordon, Robert Bender, and Gabor Herman in 1970, where it is called the Algebraic Reconstruction Technique (ART). ART includes the positivity constraint, making it nonlinear. The Kaczmarz method is applicable to any linear system of equations, but its computational advantage relative to other methods depends on the system being sparse. It has been demonstrated to be superior, in some biomedical imaging applications, to other methods such as the filtered backprojection method. It has many applications ranging from computed tomography (CT) to signal processing. It can be obtained also by applying to the hyperplanes, described by the linear system, the method of successive projections onto convex sets (POCS). A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Belief Propagation
A belief is an attitude that something is the case, or that some proposition is true. In epistemology, philosophers use the term "belief" to refer to attitudes about the world which can be either true or false. To believe something is to take it to be true; for instance, to believe that snow is white is comparable to accepting the truth of the proposition "snow is white". However, holding a belief does not require active introspection. For example, few carefully consider whether or not the sun will rise tomorrow, simply assuming that it will. Moreover, beliefs need not be ''occurrent'' (e.g. a person actively thinking "snow is white"), but can instead be ''dispositional'' (e.g. a person who if asked about the color of snow would assert "snow is white"). There are various different ways that contemporary philosophers have tried to describe beliefs, including as representations of ways that the world could be (Jerry Fodor), as dispositions to act as if certain things are true (Rod ... [...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 z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number z^* Mz is positive for every nonzero complex column vector z, where z^* denotes the conjugate transpose of z. Positive semi-definite matrices are defined similarly, except that the scalars z^\textsfMz and z^* Mz 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. A matrix is thus positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it define ... [...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=\begi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sparse Matrix
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 regarding the proportion of zero-value elements for a matrix to qualify as sparse but a common criterion is that the number of non-zero elements is roughly equal to the number of rows or columns. By contrast, if most of the elements are non-zero, the matrix is considered dense. The number of zero-valued elements divided by the total number of elements (e.g., ''m'' × ''n'' for an ''m'' × ''n'' matrix) is sometimes referred to as the sparsity of the matrix. Conceptually, sparsity corresponds to systems with few pairwise interactions. For example, consider a line of balls connected by springs from one to the next: this is a sparse system as only adjacent balls are coupled. By contrast, if the same line of balls were to have springs connecting each ball to all other balls, the system would correspond to a dense matrix. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
