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Orthonormalization
In linear algebra, orthogonalization is the process of finding a Set (mathematics), set of orthogonal vectors that span (linear algebra), span a particular linear subspace, subspace. Formally, starting with a linearly independent set of vectors in an inner product space (most commonly the Euclidean space R''n''), orthogonalization results in a set of Orthogonality, orthogonal vectors that Generator (mathematics), generate the same subspace as the vectors ''v''1, ... , ''v''''k''. Every vector in the new set is orthogonal to every other vector in the new set; and the new set and the old set have the same linear span. In addition, if we want the resulting vectors to all be unit vectors, then we Unit vector, normalize each vector and the procedure is called orthonormalization. Orthogonalization is also possible with respect to any symmetric bilinear form (not necessarily an inner product, not necessarily over real numbers), but standard algorithms may encounter div ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gram–Schmidt Process
In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process or Gram-Schmidt algorithm is a way of finding a set of two or more vectors that are perpendicular to each other. By technical definition, it is a method of constructing an orthonormal basis from a set of vector (geometry), vectors in an inner product space, most commonly the Euclidean space \mathbb^n equipped with the standard inner product. The Gram–Schmidt process takes a finite set, finite, linearly independent set of vectors S = \ for and generates an orthogonal set S' = \ that spans the same k-dimensional subspace of \mathbb^n as S. The method is named after Jørgen Pedersen Gram and Erhard Schmidt, but Pierre-Simon Laplace had been familiar with it before Gram and Schmidt. In the theory of Lie group decompositions, it is generalized by the Iwasawa decomposition. The application of the Gram–Schmidt process to the column vectors of a full column rank (linear algebra), rank mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projection (linear Algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it were applied once (i.e. P is idempotent). It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object. Definitions A projection on a vector space V is a linear operator P\colon V \to V such that P^2 = P. When V has an inner product and is complete, i.e. when V is a Hilbert space, the concept of orthogonality can be used. A projection P on a Hilbert space V is called an orthogonal projection if it satisfies \langle P \mathbf x, \mathbf y \rangle = \langle \mathbf x, P \mathbf y \rangle for all \mathbf x, \mathbf y \in V. A projecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biorthogonal System
In mathematics, a biorthogonal system is a pair of indexed families of vectors \tilde v_i \text E \text \tilde u_i \text F such that \left\langle\tilde v_i , \tilde u_j\right\rangle = \delta_, where E and F form a pair of topological vector spaces that are in duality, \langle \,\cdot, \cdot\, \rangle is a bilinear mapping and \delta_ is the Kronecker delta. An example is the pair of sets of respectively left and right eigenvectors of a matrix, indexed by eigenvalue, if the eigenvalues are distinct. A biorthogonal system in which E = F and \tilde v_i = \tilde u_i is an orthonormal system. Projection Related to a biorthogonal system is the projection P := \sum_ \tilde u_i \otimes \tilde v_i, where (u \otimes v) (x) := u \langle v, x \rangle; its image is the linear span In mathematics, the linear span (also called the linear hull or just span) of a set S of elements of a vector space V is the smallest linear subspace of V that contains S. It is the set of all finite linear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seismic Exploration
Reflection seismology (or seismic reflection) is a method of exploration geophysics that uses the principles of seismology to estimate the properties of the Earth's subsurface from reflection (physics), reflected seismic waves. The method requires a controlled seismic source of energy, such as dynamite or Tovex blast, a specialized Seismic source#Air gun, air gun or a seismic vibrator. Reflection seismology is similar to sonar and acoustic location, echolocation. History Reflections and refractions of seismic waves at geologic Interface (matter), interfaces within the Earth were first observed on recordings of earthquake-generated seismic waves. The basic model of the Earth's deep interior is based on observations of earthquake-generated seismic waves transmitted through the Earth's interior (e.g., Mohorovičić, 1910). The use of human-generated seismic waves to map in detail the geology of the upper few kilometers of the Earth's crust followed shortly thereafter and has deve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signal Processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomography, seismic signals, Altimeter, altimetry processing, and scientific measurements. Signal processing techniques are used to optimize transmissions, Data storage, digital storage efficiency, correcting distorted signals, improve subjective video quality, and to detect or pinpoint components of interest in a measured signal. History According to Alan V. Oppenheim and Ronald W. Schafer, the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century. They further state that the digital refinement of these techniques can be found in the digital control systems of the 1940s and 1950s. In 1948, Claude Shannon wrote the influential paper "A Mathematical Theory of Communication" which was publis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Denoising
Noise reduction is the process of removing noise from a signal. Noise reduction techniques exist for audio and images. Noise reduction algorithms may distort the signal to some degree. Noise rejection is the ability of a circuit to isolate an undesired signal component from the desired signal component, as with common-mode rejection ratio. All signal processing devices, both analog and digital, have traits that make them susceptible to noise. Noise can be random with an even frequency distribution (white noise), or frequency-dependent noise introduced by a device's mechanism or signal processing algorithms. In electronic systems, a major type of noise is ''hiss'' created by random electron motion due to thermal agitation. These agitated electrons rapidly add and subtract from the output signal and thus create detectable noise. In the case of photographic film and magnetic tape, noise (both visible and audible) is introduced due to the grain structure of the medium. In photogr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Per-Olov Löwdin
Per-Olov Löwdin (October 28, 1916 – October 6, 2000) was a Swedish physicist, professor at the University of Uppsala from 1960 to 1983, and in parallel at the University of Florida until 1993. A former graduate student under Ivar Waller, Löwdin formulated in 1950 the symmetric orthogonalization scheme for atomic and molecular orbital calculations, greatly simplifying the tight-binding method. This scheme is the basis of the zero-differential overlap (ZDO) approximation used in semiempirical theories. In 1956 he introduced the canonical orthogonalization scheme, which is optimal for eliminating approximate linear dependencies of a basis set. These orthogonalization procedures are widely used today in all modern quantum chemistry calculations. The famous 'Löwdin's pairing theorem' used in restricted open-shell Hartree–Fock (ROHF), unrestricted Hartree–Fock (UHF) and generalized valence bond (RES-GVB) theories is not his. According to himself, George G. Hall and Kin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel Computing
Parallel computing is a type of computing, computation in which many calculations or Process (computing), processes are carried out simultaneously. Large problems can often be divided into smaller ones, which can then be solved at the same time. There are several different forms of parallel computing: Bit-level parallelism, bit-level, Instruction-level parallelism, instruction-level, Data parallelism, data, and task parallelism. Parallelism has long been employed in high-performance computing, but has gained broader interest due to the physical constraints preventing frequency scaling.S.V. Adve ''et al.'' (November 2008)"Parallel Computing Research at Illinois: The UPCRC Agenda" (PDF). Parallel@Illinois, University of Illinois at Urbana-Champaign. "The main techniques for these performance benefits—increased clock frequency and smarter but increasingly complex architectures—are now hitting the so-called power wall. The computer industry has accepted that future performance inc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnoldi Iteration
In numerical linear algebra, the Arnoldi iteration is an eigenvalue algorithm and an important example of an iterative method. Arnoldi finds an approximation to the eigenvalues and eigenvectors of general (possibly non- Hermitian) matrices by constructing an orthonormal basis of the Krylov subspace, which makes it particularly useful when dealing with large sparse matrices. The Arnoldi method belongs to a class of linear algebra algorithms that give a partial result after a small number of iterations, in contrast to so-called ''direct methods'' which must complete to give any useful results (see for example, Householder transformation). The partial result in this case being the first few vectors of the basis the algorithm is building. When applied to Hermitian matrices it reduces to the Lanczos algorithm. The Arnoldi iteration was invented by W. E. Arnoldi in 1951. Krylov subspaces and the power iteration An intuitive method for finding the largest (in absolute value) ei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterative Method
In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an "iterate") is derived from the previous ones. A specific implementation with Algorithm#Termination, termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or Quasi-Newton method, quasi-Newton methods like Broyden–Fletcher–Goldfarb–Shanno algorithm, BFGS, is an algorithm of an iterative method or a method of successive approximation. An iterative method is called ''Convergent series, convergent'' if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common. In contrast, direct methods attempt to solve the problem by a finit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Stability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context: one important context is numerical linear algebra, and another is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might cause a large deviation of final answer from the exact solution. Some numerical algorithms may damp out the small fluctuations (errors) in the input data; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called ''numerically stable''. One ... [...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] |
