|
Hessian Automatic Differentiation
In applied mathematics, Hessian automatic differentiation are techniques based on automatic differentiation (AD) that calculate the second derivative of an n-dimensional function, known as the Hessian matrix. When examining a function in a neighborhood of a point, one can discard many complicated global aspects of the function and accurately approximate it with simpler functions. The quadratic approximation is the best-fitting quadratic in the neighborhood of a point, and is frequently used in engineering and science. To calculate the quadratic approximation, one must first calculate its gradient and Hessian matrix. Let f: \mathbb^n \rightarrow \mathbb , for each x \in \mathbb^n the Hessian matrix H(x) \in \mathbb^ is the second order derivative and is a symmetric matrix. Reverse Hessian-vector products For a given u \in \mathbb^n, this method efficiently calculates the Hessian-vector product H(x)u . Thus can be used to calculate the entire Hessian by calculating H(x)e_i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Applied Mathematics
Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the profession, professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics. History Historically, applied mathematics consisted principally of Mathematical analysis, applied analysis, most notably differential equations; approximation theory (broadly construed, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automatic Differentiation
In mathematics and computer algebra, automatic differentiation (auto-differentiation, autodiff, or AD), also called algorithmic differentiation, computational differentiation, and differentiation arithmetic Hend Dawood and Nefertiti Megahed (2023). Automatic differentiation of uncertainties: an interval computational differentiation for first and higher derivatives with implementation. PeerJ Computer Science 9:e1301 https://doi.org/10.7717/peerj-cs.1301. Hend Dawood and Nefertiti Megahed (2019). A Consistent and Categorical Axiomatization of Differentiation Arithmetic Applicable to First and Higher Order Derivatives. Punjab University Journal of Mathematics. 51(11). pp. 77-100. doi: 10.5281/zenodo.3479546. http://doi.org/10.5281/zenodo.3479546. is a set of techniques to evaluate the partial derivative of a function specified by a computer program. Automatic differentiation is a subtle and central tool to automatize the simultaneous computation of the numerical values of arbitrarily ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hessian Matrix
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued Function (mathematics), function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Otto Hesse, Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants". The Hessian is sometimes denoted by H or \nabla\nabla or \nabla^2 or \nabla\otimes\nabla or D^2. Definitions and properties Suppose f : \R^n \to \R is a function taking as input a vector \mathbf \in \R^n and outputting a scalar f(\mathbf) \in \R. If all second-order partial derivatives of f exist, then the Hessian matrix \mathbf of f is a square n \times n matrix, usually defined and arranged as \mathbf H_f= \begin \dfrac & \dfrac & \cdots & \dfrac \\[2.2ex] \dfrac & \dfrac & \cdots & \dfrac \\[2.2ex] \vdots & \vdot ... [...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] |
Partial Derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function f(x, y, \dots) with respect to the variable x is variously denoted by It can be thought of as the rate of change of the function in the x-direction. Sometimes, for the partial derivative of z with respect to x is denoted as \tfrac. Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in: f'_x(x, y, \ldots), \frac (x, y, \ldots). The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hessian Matrix With Columns Coloured
A Hessian is an inhabitant of the German state of Hesse. Hessian may also refer to: Named from the toponym *Hessian (soldier), eighteenth-century German regiments in service with the British Empire **Hessian (boot), a style of boot **Hessian fabric, coarse woven material **Hessian fly or barley midge, a species of fly (thought to be introduced by Hessian soldiers) *Hessian dialects, West Central German group of dialects *Hessian crucible, a type of ceramic crucible * Hessian Cup, a regional cup competition in German football Named for Otto Hesse *Hessian matrix, in mathematics, is a matrix of second partial derivatives ** Hessian affine region detector, a feature detector used in the fields of computer vision and image analysis **Hessian automatic differentiation ** Hessian equations, partial differential equations (PDEs) based on the Hessian matrix *Hessian pair or Hessian duad in mathematics *Hessian form of an elliptic curve *Hessian group * Hessian polyhedron * Glossary o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Matrix Hessian Matrix
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British North America * Compact of Free Association whereby the sovereign states of the Federated States of Micronesia, the Republic of the Marshall Islands and the Republic of Palau have entered into as associated states with the United States. * Mayflower Compact, the first governing document of Plymouth Colony * United Nations Global Compact * Global Compact for Migration, a UN non-binding intergovernmental agreement Mathematics * Compact element, those elements of a partially ordered set that cannot be subsumed by a supremum of any directed set that does not already contain them * Compact operator, a linear operator that takes bounded subsets to relatively compact subsets, in functional analysis * C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. The primary objects of study in differential calculus are the derivative of a Function (mathematics), function, related notions such as the Differential of a function, differential, and their applications. The derivative of a function at a chosen input value describes the Rate (mathematics)#Of_change, rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent, tangent line to the graph of a function, graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
