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Packed Storage Matrix
A packed storage matrix, also known as packed matrix, is a term used in programming for representing an m\times n matrix. It is a more compact way than an m-by-n rectangular array by exploiting a special structure of the matrix. Typical examples of matrices that can take advantage of packed storage include: * symmetric or hermitian matrix * Triangular matrix * Banded matrix. Triangular packed matrices The packed storage matrix allows a matrix to be converted to an array, shrinking the matrix significantly. In doing so, a square n \times n matrix is converted to an array of length . Consider the following upper matrix: :\mathbf = \begin a_ & a_ & a_ & a_ \\ & a_ & a_ & a_ \\ & & a_ & a_ \\ & & & a_ \\ \end which can be packed into the one array: : \mathbf = (\underbrace\ \underbrace\ \underbrace\ \underbrace) Similarly the lower matrix: :\mathbf = \begin a_ & & & \\ a_ & a_ & & \\ a_ & a_ & a_ & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Programming
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. Optimization problems Optimization problems can be divided into two categories, depending on whether the variables ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (: matrices) is a rectangle, rectangular array or table of numbers, symbol (formal), symbols, or expression (mathematics), expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a " matrix", or a matrix of dimension . Matrices are commonly used in linear algebra, where they represent linear maps. In geometry, matrices are widely used for specifying and representing geometric transformations (for example rotation (mathematics), rotations) and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if a_ denotes the entry in the ith row and jth column then for all indices i and j. Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator represented in an orthonormal basis over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and : A \text \quad \iff \quad a_ = \overline or in matrix form: A \text \quad \iff \quad A = \overline . Hermitian matrices can be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A is denoted by A^\mathsf, then the Hermitian property can be written concisely as A \text \quad \iff \quad A = A^\mathsf Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are A^\mathsf = A^\dagger = A^\ast, although in quantum mechanics, A^\ast typically means the complex conjugate onl ... [...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] |
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] |
Sparsity
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. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
Arrays
An array is a systematic arrangement of similar objects, usually in rows and columns. Things called an array include: {{TOC right Music * In twelve-tone and serial composition, the presentation of simultaneous twelve-tone sets such that the sums of their horizontal segments form a succession of twelve-tone aggregates * Array mbira, a musical instrument * Spiral array model, a music pitch space Science Astronomy A telescope array, also called astronomical interferometer. Biology * Various kinds of multiple biological arrays called microarrays * Visual feature array, a model for the visual cortex Computer science Generally, a collection of same type data items that can be selected by indices computed at run-time, including: * Array (data structure), an arrangement of items at equally spaced addresses in computer memory * Array (data type), used in a programming language to specify a variable that can be indexed * Associative array, an abstract data structure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
