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Matrix Representation
Matrix representation is a method used by a computer language to store column-vector matrix (mathematics), matrices of more than one dimension in computer storage, memory. Fortran and C (programming language), C use different schemes for their native arrays. Fortran uses "Column Major", in which all the elements for a given column are stored contiguously in memory. C (programming language), C uses "Row Major", which stores all the elements for a given row contiguously in memory. LAPACK defines various matrix representations in memory. There is also Sparse matrix representation and Morton-order matrix representation. According to the documentation, in LAPACK the unitary matrix representation is optimized. Some languages such as Java programming language, Java store matrices using Iliffe vectors. These are particularly useful for storing irregular matrix, irregular matrices. Matrices are of primary importance in linear algebra. Basic mathematical operations An m × n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathematics), matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as line (geometry), lines, plane (geometry), planes and rotation (mathematics), rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to Space of functions, function spaces. Linear algebra is also used in most sciences and fields of engineering because it allows mathematical model, modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Locality Of Reference
In computer science, locality of reference, also known as the principle of locality, is the tendency of a processor to access the same set of memory locations repetitively over a short period of time. There are two basic types of reference locality temporal and spatial locality. Temporal locality refers to the reuse of specific data and/or resources within a relatively small time duration. Spatial locality (also termed ''data locality'')"NIST Big Data Interoperability Framework: Volume 1"urn:doi:10.6028/NIST.SP.1500-1r2 refers to the use of data elements within relatively close storage locations. Sequential locality, a special case of spatial locality, occurs when data elements are arranged and accessed linearly, such as traversing the elements in a one-dimensional Array data structure, array. Locality is a type of predictability, predictable behavior that occurs in computer systems. Systems which exhibit strong ''locality of reference'' are good candidates for performance optimiza ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
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] [Amazon] |
Row- And Column-major Order
In computing, row-major order and column-major order are methods for storing multidimensional arrays in linear storage such as random access memory. The difference between the orders lies in which elements of an array are contiguous in memory. In row-major order, the consecutive elements of a row reside next to each other, whereas the same holds true for consecutive elements of a column in column-major order. While the terms allude to the rows and columns of a two-dimensional array, i.e. a matrix (mathematics), matrix, the orders can be generalized to arrays of any dimension by noting that the terms row-major and column-major are equivalent to lexicographic order, lexicographic and colexicographic orders, respectively. It is also worth noting that matrices, being commonly represented as collections of row or column vectors, using this approach are effectively stored as consecutive vectors or consecutive vector components. Such ways of storing data are referred to as AoS and SoA res ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Transformation Matrix
In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping \mathbb^n to \mathbb^m and \mathbf x is a column vector with n entries, then there exists an m \times n matrix A, called the transformation matrix of T, such that: T( \mathbf x ) = A \mathbf x Note that A has m rows and n columns, whereas the transformation T is from \mathbb^n to \mathbb^m. There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors. Uses Matrices allow arbitrary linear transformations to be displayed in a consistent format, suitable for computation. This also allows transformations to be composed easily (by multiplying their matrices). Linear transformations are not the only ones that can be represented by matrices. Some transformations that are non-linear on an n-dimensional Euclidean space R''n'' can be represented as linear transformations on the ''n''+1-dimensional space R''n''+1. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Basis Vector
In mathematics, a set of elements of a vector space is called a basis (: bases) if every element of can be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to . The elements of a basis are called . Equivalently, a set is a basis if its elements are linearly independent and every element of is a linear combination of elements of . In other words, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Basis vectors find applications in the study of crystal structures and frames of reference. Definition A basis of a vector space over a field (such as th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
SIMD Lane
Single instruction, multiple threads (SIMT) is an execution model used in parallel computing where single instruction, multiple data (SIMD) is combined with zero-overhead multithreading, i.e. multithreading where the hardware is capable of switching between threads on a cycle-by-cycle basis. There are two models of multithreading involved. In addition to the zero-overhead multithreading mentioned, the SIMD execution hardware is virtualized to represent a multiprocessor, but is inferior to a SPMD processor in that instructions in all "threads" are executed in lock-step in the lanes of the SIMD processor which can only execute the same instruction in a given cycle across all lanes. The SIMT execution model has been implemented on several GPUs and is relevant for general-purpose computing on graphics processing units (GPGPU), e.g. some supercomputers combine CPUs with GPUs. The processors, say a number of them, seem to execute many more than tasks. This is achieved by each processor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Dot Product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused with scalar multiplication. is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two Euclidean vector, vectors is widely used. It is often called the inner product (or rarely the projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see ''Inner product space'' for more). It should not be confused with the cross product. Algebraically, the dot product is the sum of the Product (mathematics), products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
SIMD Instruction
In computer science, an instruction set architecture (ISA) is an abstract model that generally defines how software controls the CPU in a computer or a family of computers. A device or program that executes instructions described by that ISA, such as a central processing unit (CPU), is called an ''implementation'' of that ISA. In general, an ISA defines the supported instructions, data types, registers, the hardware support for managing main memory, fundamental features (such as the memory consistency, addressing modes, virtual memory), and the input/output model of implementations of the ISA. An ISA specifies the behavior of machine code running on implementations of that ISA in a fashion that does not depend on the characteristics of that implementation, providing binary compatibility between implementations. This enables multiple implementations of an ISA that differ in characteristics such as performance, physical size, and monetary cost (among other things), but t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
