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Bidiagonal Matrix
In mathematics, a bidiagonal matrix is a banded matrix with non-zero entries along the main diagonal and ''either'' the diagonal above or the diagonal below. This means there are exactly two non-zero diagonals in the matrix. When the diagonal above the main diagonal has the non-zero entries the matrix is upper bidiagonal. When the diagonal below the main diagonal has the non-zero entries the matrix is lower bidiagonal. For example, the following matrix is upper bidiagonal: :\begin 1 & 4 & 0 & 0 \\ 0 & 4 & 1 & 0 \\ 0 & 0 & 3 & 4 \\ 0 & 0 & 0 & 3 \\ \end and the following matrix is lower bidiagonal: :\begin 1 & 0 & 0 & 0 \\ 2 & 4 & 0 & 0 \\ 0 & 3 & 3 & 0 \\ 0 & 0 & 4 & 3 \\ \end. Usage One variant of the QR algorithm starts with reducing a general matrix into a bidiagonal one, and the singular value decomposition (SVD) uses this method as well. Bidiagonalization Bidiagonalization allows guaranteed accuracy when using floating-point arithmetic to compute singular values. See ... [...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] |
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] |
QR Algorithm
In numerical linear algebra, the QR algorithm or QR iteration is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a Matrix (mathematics), matrix. The QR algorithm was developed in the late 1950s by John G. F. Francis and by Vera N. Kublanovskaya, working independently. The basic idea is to perform a QR decomposition, writing the matrix as a product of an orthogonal matrix and an upper triangular matrix, multiply the factors in the reverse order, and iterate. The practical QR algorithm Formally, let be a real matrix of which we want to compute the eigenvalues, and let . At the -th step (starting with ), we compute the QR decomposition where is an orthogonal matrix (i.e., ) and is an upper triangular matrix. We then form . Note that A_ = R_k Q_k = Q_k^ Q_k R_k Q_k = Q_k^ A_k Q_k = Q_k^ A_k Q_k, so all the are Similar matrix, similar and hence they have the same eigenvalues. The algorithm is numerical stability, numerically stable b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Value Decomposition
In linear algebra, the singular value decomposition (SVD) is a Matrix decomposition, factorization of a real number, real or complex number, complex matrix (mathematics), matrix into a rotation, followed by a rescaling followed by another rotation. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any matrix. It is related to the polar decomposition#Matrix polar decomposition, polar decomposition. Specifically, the singular value decomposition of an m \times n complex matrix is a factorization of the form \mathbf = \mathbf, where is an complex unitary matrix, \mathbf \Sigma is an m \times n rectangular diagonal matrix with non-negative real numbers on the diagonal, is an n \times n complex unitary matrix, and \mathbf V^* is the conjugate transpose of . Such decomposition always exists for any complex matrix. If is real, then and can be guaranteed to be real orthogonal matrix, orthogonal matrices; in such contexts, the SVD ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floating-point Arithmetic
In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a Sign (mathematics), signed sequence of a fixed number of digits in some Radix, base) multiplied by an integer power of that base. Numbers of this form are called floating-point numbers. For example, the number 2469/200 is a floating-point number in base ten with five digits: 2469/200 = 12.345 = \! \underbrace_\text \! \times \! \underbrace_\text\!\!\!\!\!\!\!\overbrace^ However, 7716/625 = 12.3456 is not a floating-point number in base ten with five digits—it needs six digits. The nearest floating-point number with only five digits is 12.346. And 1/3 = 0.3333… is not a floating-point number in base ten with any finite number of digits. In practice, most floating-point systems use Binary number, base two, though base ten (decimal floating point) is also common. Floating-point arithmetic operations, such as addition and division, approximate the correspond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Matrices
A list is a set of discrete items of information collected and set forth in some format for utility, entertainment, or other purposes. A list may be memorialized in any number of ways, including existing only in the mind of the list-maker, but lists are frequently written down on paper, or maintained electronically. Lists are "most frequently a tool", and "one does not ''read'' but only ''uses'' a list: one looks up the relevant information in it, but usually does not need to deal with it as a whole".Lucie Doležalová,The Potential and Limitations of Studying Lists, in Lucie Doležalová, ed., ''The Charm of a List: From the Sumerians to Computerised Data Processing'' (2009). Purpose It has been observed that, with a few exceptions, "the scholarship on lists remains fragmented". David Wallechinsky, a co-author of ''The Book of Lists'', described the attraction of lists as being "because we live in an era of overstimulation, especially in terms of information, and lists help us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Hessenberg Form
In linear algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular. To be exact, an upper Hessenberg matrix has zero entries below the first subdiagonal, and a lower Hessenberg matrix has zero entries above the first superdiagonal. They are named after Karl Hessenberg. A Hessenberg decomposition is a matrix decomposition of a matrix A into a unitary matrix P and a Hessenberg matrix H such that PHP^*=A where P^* denotes the conjugate transpose. Definitions Upper Hessenberg matrix A square n \times n matrix A is said to be in upper Hessenberg form or to be an upper Hessenberg matrix if a_=0 for all i,j with i > j+1. An upper Hessenberg matrix is called unreduced if all subdiagonal entries are nonzero, i.e. if a_ \neq 0 for all i \in \. Lower Hessenberg matrix A square n \times n matrix A is said to be in lower Hessenberg form or to be a lower Hessenberg matrix if its transpose is an upper Hessenberg matrix or equivalently if a_=0 for all i, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
