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Skyline Matrix
In scientific computing, skyline matrix storage, or SKS, or a variable band matrix storage, or envelope storage scheme is a form of a sparse matrix storage format matrix that reduces the storage requirement of a matrix more than band matrix, banded storage. In banded storage, all entries within a fixed distance from the diagonal (called half-bandwidth) are stored. In column-oriented skyline storage, only the entries from the first nonzero entry to the last nonzero entry in each column are stored. There is also row oriented skyline storage, and, for symmetric matrices, only one triangle is usually stored. Skyline storage has become very popular in the finite element codes for structural mechanics, because the skyline is preserved by Cholesky decomposition (a method of solving systems of linear equations with a symmetric, positive-definite matrix; all Sparse_matrix#Reducing_fill-in, fill-in falls within the skyline), and systems of equations from finite elements have a relatively sm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Scientific Computing
Computational science, also known as scientific computing, technical computing or scientific computation (SC), is a division of science, and more specifically the Computer Sciences, which uses advanced computing capabilities to understand and solve complex physical problems. While this typically extends into computational specializations, this field of study includes: * Algorithms ( numerical and non-numerical): mathematical models, computational models, and computer simulations developed to solve sciences (e.g, physical, biological, and social), engineering, and humanities problems * Computer hardware that develops and optimizes the advanced system hardware, firmware, networking, and data management components needed to solve computationally demanding problems * The computing infrastructure that supports both the science and engineering problem solving and the developmental computer and information science In practical use, it is typically the application of computer simu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Banded Matrix
In mathematics, particularly matrix theory, a band matrix or banded matrix is a sparse matrix whose non-zero entries are confined to a diagonal ''band'', comprising the main diagonal and zero or more diagonals on either side. Band matrix Bandwidth Formally, consider an ''n''×''n'' matrix ''A''=(''a''''i,j'' ). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants ''k''1 and ''k''2: :a_=0 \quad\mbox\quad ji+k_2; \quad k_1, k_2 \ge 0.\, then the quantities ''k''1 and ''k''2 are called the and , respectively. The of the matrix is the maximum of ''k''1 and ''k''2; in other words, it is the number ''k'' such that a_=0 if , i-j, > k . Examples *A band matrix with ''k''1 = ''k''2 = 0 is a diagonal matrix, with bandwidth 0. *A band matrix with ''k''1 = ''k''2 = 1 is a tridiagonal matrix, with bandwidth 1. *For ''k''1 = ''k''2 = 2 one has a pentadiagonal matrix and so on. * Triangular matrices **For ''k''1 = 0, ''k''2 = ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Frontal Solver
A frontal solver is an approach to solving sparse linear systems which is used extensively in finite element analysis. Algorithms of this kind are variants of Gauss elimination that automatically avoids a large number of operations involving zero terms due to the fact that the matrix is only sparse. The development of frontal solvers is usually considered as dating back to work by Bruce Irons. A frontal solver builds a LU or Cholesky decomposition of a sparse matrix. Frontal solvers start with one or a few diagonal entries of the matrix, then consider all of those diagonal entries that are coupled to the first set via off-diagonal entries, and so on. In the finite element context, these consecutive sets form "fronts" that march through the domain (and consequently through the matrix, if one were to permute rows and columns of the matrix in such a way that the diagonal entries are ordered by the wave they are part of). Processing the front involves dense matrix operations, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Band Matrix
In mathematics, particularly matrix theory, a band matrix or banded matrix is a sparse matrix whose non-zero entries are confined to a diagonal ''band'', comprising the main diagonal and zero or more diagonals on either side. Band matrix Bandwidth Formally, consider an ''n''×''n'' matrix ''A''=(''a''''i,j'' ). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants ''k''1 and ''k''2: :a_=0 \quad\mbox\quad ji+k_2; \quad k_1, k_2 \ge 0.\, then the quantities ''k''1 and ''k''2 are called the and , respectively. The of the matrix is the maximum of ''k''1 and ''k''2; in other words, it is the number ''k'' such that a_=0 if , i-j, > k . Examples *A band matrix with ''k''1 = ''k''2 = 0 is a diagonal matrix, with bandwidth 0. *A band matrix with ''k''1 = ''k''2 = 1 is a tridiagonal matrix, with bandwidth 1. *For ''k''1 = ''k''2 = 2 one has a pentadiagonal matrix and so on. * Triangular matrices **For ''k''1 = 0, ''k''2 = ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Massively Parallel Computing
Massively parallel is the term for using a large number of computer processors (or separate computers) to simultaneously perform a set of coordinated computations in parallel. GPUs are massively parallel architecture with tens of thousands of threads. One approach is grid computing, where the processing power of many computers in distributed, diverse administrative domains is opportunistically used whenever a computer is available.''Grid computing: experiment management, tool integration, and scientific workflows'' by Radu Prodan, Thomas Fahringer 2007 pages 1–4 An example is BOINC, a volunteer-based, opportunistic grid system, whereby the grid provides power only on a best effort basis.''Parallel and Distributed Computational Intelligence'' by Francisco Fernández de Vega 2010 pages 65–68 Another approach is grouping many processors in close proximity to each other, as in a computer cluster. In such a centralized system the speed and flexibility of the interconnect beco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Reverse Cuthill–McKee Algorithm
Reverse or reversing may refer to: Arts and media * ''Reverse'' (Eldritch album), 2001 * ''Reverse'' (2009 film), a Polish comedy-drama film * ''Reverse'' (2019 film), an Iranian crime-drama film * ''Reverse'' (Morandi album), 2005 * ''Reverse'' (TV series), a 2017–2018 South Korean television series *"Reverse", a 2014 song by SomeKindaWonderful * REVERSE art gallery, in Brooklyn, NY, US *Reverse tape effects including backmasking, the recording of sound in reverse * '' Reversing: Secrets of Reverse Engineering'', a book by Eldad Eilam *''Tegami Bachi: REVERSE'', the second season of the '' Tegami Bachi'' anime series, 2010 Driving * Reverse gear, in a motor or mechanical transmission * Reversing (vehicle maneuver), reversing the direction of a vehicle * Turning a vehicle through 180 degrees Sports and games *Reverse (American football), a trick play in American football *Reverse swing, a cricket delivery * Reverse (bridge), a type of bid in contract bridge Technology *Rever ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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LAPACK
LAPACK ("Linear Algebra Package") is a standard software library for numerical linear algebra. It provides routines for solving systems of linear equations and linear least squares, eigenvalue problems, and singular value decomposition. It also includes routines to implement the associated matrix factorizations such as LU, QR, Cholesky and Schur decomposition. LAPACK was originally written in FORTRAN 77, but moved to Fortran 90 in version 3.2 (2008). The routines handle both real and complex matrices in both single and double precision. LAPACK relies on an underlying BLAS implementation to provide efficient and portable computational building blocks for its routines. LAPACK was designed as the successor to the linear equations and linear least-squares routines of LINPACK and the eigenvalue routines of EISPACK. LINPACK, written in the 1970s and 1980s, was designed to run on the then-modern vector computers with shared memory. LAPACK, in contrast, was designed to eff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Linear Equations
In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficients may be considered as parameters of the equation and may be arbitrary expressions, provided they do not contain any of the variables. To yield a meaningful equation, the coefficients a_1, \ldots, a_n are required to not all be zero. Alternatively, a linear equation can be obtained by equating to zero a linear polynomial over some field, from which the coefficients are taken. The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true. In the case of just one variable, there is exactly one solution (provided that a_1\ne 0). Often, the term ''linear equation'' refers implicitly to this particular case, in which the variable is sensibly called the ''unknown''. In the case of two var ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |