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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. Defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Matrix
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are zero. Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the matrix multiplication, product of a lower triangular matrix ''L'' and an upper triangular matrix ''U'' if and only if all its leading principal minor (linear algebra), minors are non-zero. Description A matrix of the form :L = \begin \ell_ & & & & 0 \\ \ell_ & \ell_ & & & \\ \ell_ & \ell_ & \ddots & & \\ \vdots & \vdots & \ddots & \ddots & \\ \ell_ & \ell_ & \ldots & \ell_ & \ell_ \end is called a lower trian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrices (mathematics)
Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryotic organism's cells * Matrix (chemical analysis), the non-analyte components of a sample * Matrix (geology), the fine-grained material in which larger objects are embedded * Matrix (composite), the constituent of a composite material * Hair matrix, produces hair * Nail matrix, part of the nail in anatomy Technology * Matrix (mass spectrometry), a compound that promotes the formation of ions * Matrix (numismatics), a tool used in coin manufacturing * Matrix (printing), a mould for casting letters * Matrix (protocol), an open standard for real-time communication * Matrix (record production), or master, a disc used in the production of phonograph records ** Matrix number, of a gramophone record * D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prentice-Hall
Prentice Hall was a major American educational publisher. It published print and digital content for the 6–12 and higher-education market. It was an independent company throughout the bulk of the twentieth century. In its last few years it was owned by, then absorbed into, Savvas Learning Company. In the Web era, it distributed its technical titles through the Safari Books Online e-reference service for some years. History On October 13, 1913, law professor Charles Gerstenberg and his student Richard Ettinger founded Prentice Hall. Gerstenberg and Ettinger took their mothers' maiden names, Prentice and Hall, to name their new company. At the time the name was usually styled as Prentice-Hall (as seen for example on many title pages), per an orthographic norm for coordinate elements within such compounds (compare also ''McGraw-Hill'' with later styling as ''McGraw Hill''). Prentice-Hall became known as a publisher of trade books by authors such as Norman Vincent Peale; ele ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Wisconsin Press
The University of Wisconsin Press (sometimes abbreviated as UW Press) is a Non-profit organization, non-profit university press publishing Peer review, peer-reviewed books and journals. It publishes work by scholars from the global academic community; works of fiction, memoir and poetry under its imprint, Terrace Books; and serves the citizens of Wisconsin by publishing important books about Wisconsin, the Upper Midwest, and the Great Lakes region (North America), Great Lakes region. UW Press annually awards the Brittingham Prize in Poetry, the Felix Pollak Prize in Poetry, and The Four Lakes Prize in Poetry. The press was founded in 1936 in Madison, Wisconsin, Madison and is one of more than 120 member presses in the Association of University Presses. The Journals Division was established in 1965. The press employs approximately 25 full and part-time staff, produces 40 to 60 new books a year, and publishes 13 journals. It also distributes books and some annual journals for sele ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stieltjes Matrix
In mathematics, particularly matrix theory, a Stieltjes matrix, named after Thomas Joannes Stieltjes, is a real symmetric positive definite matrix with nonpositive off-diagonal entries. A Stieltjes matrix is necessarily an M-matrix. Every ''n×n'' Stieltjes matrix is invertible to a nonsingular symmetric nonnegative matrix, though the converse of this statement is not true in general for ''n'' > 2. From the above definition, a Stieltjes matrix is a symmetric invertible Z-matrix whose eigenvalues have positive real parts. As it is a Z-matrix, its off-diagonal entries are less than or equal to zero. See also * Hurwitz-stable matrix * Metzler matrix In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero): : \forall_\, x_ \geq 0. It is named after the American economist Lloyd Metzler. Metzler matrices appear in st ... References * * Matrices (mathematics) Numerical linear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
M-matrix
In mathematics, especially linear algebra, an ''M''-matrix is a matrix whose off-diagonal entries are less than or equal to zero (i.e., it is a ''Z''-matrix) and whose eigenvalues have nonnegative real parts. The set of non-singular ''M''-matrices are a subset of the class of ''P''-matrices, and also of the class of inverse-positive matrices (i.e. matrices with inverses belonging to the class of positive matrices). The name ''M''-matrix was seemingly originally chosen by Alexander Ostrowski in reference to Hermann Minkowski, who proved that if a Z-matrix has all of its row sums positive, then the determinant of that matrix is positive.. Characterizations An M-matrix is commonly defined as follows: Definition: Let be a real Z-matrix. That is, where for all . Then matrix ''A'' is also an ''M-matrix'' if it can be expressed in the form , where with , for all , where is at least as large as the maximum of the moduli of the eigenvalues of , and is an identity matrix. For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Decomposition
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. Example In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For example, when solving a system of linear equations A \mathbf = \mathbf, the matrix ''A'' can be decomposed via the LU decomposition. The LU decomposition factorizes a matrix into a lower triangular matrix ''L'' and an upper triangular matrix ''U''. The systems L(U \mathbf) = \mathbf and U \mathbf = L^ \mathbf require fewer additions and multiplications to solve, compared with the original system A \mathbf = \mathbf, though one might require significantly more digits in inexact arithmetic such as floating point. Similarly, the QR decomposition expresses ''A'' as ''QR'' with ''Q'' an orthogonal matrix and ''R'' an upp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Operator Splitting Topics
This is a list of operator splitting topics. General *Alternating direction implicit method — finite difference method for parabolic, hyperbolic, and elliptic partial differential equations * GRADELA — simple gradient elasticity model *Matrix splitting — general method of splitting a matrix operator into a sum or difference of matrices * Paul Tseng — resolved question on convergence of matrix splitting algorithms * PISO algorithm — pressure-velocity calculation for Navier-Stokes equations *Projection method (fluid dynamics) — computational fluid dynamics method *Reactive transport modeling in porous media Reactive transport modeling in porous media refers to the creation of computer models integrating chemical reaction with transport of fluids through the Earth's crust. Such models predict the distribution in space and time of the chemical reactio ... — modeling of chemical reactions and fluid flow through the Earth's crust * Richard S. Varga — developed matrix sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strictly Diagonally Dominant
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] |
Eigenvalues
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a constant factor \lambda when the linear transformation is applied to it: T\mathbf v=\lambda \mathbf v. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor \lambda (possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
